I am done

rl/Lib/site-packages/sympy/polys/tests/test_appellseqs.py

@@ -0,0 +1,91 @@
+"""Tests for efficient functions for generating Appell sequences."""
+from sympy.core.numbers import Rational as Q
+from sympy.polys.polytools import Poly
+from sympy.testing.pytest import raises
+from sympy.polys.appellseqs import (bernoulli_poly, bernoulli_c_poly,
+    euler_poly, genocchi_poly, andre_poly)
+from sympy.abc import x
+
+def test_bernoulli_poly():
+    raises(ValueError, lambda: bernoulli_poly(-1, x))
+    assert bernoulli_poly(1, x, polys=True) == Poly(x - Q(1,2))
+
+    assert bernoulli_poly(0, x) == 1
+    assert bernoulli_poly(1, x) == x - Q(1,2)
+    assert bernoulli_poly(2, x) == x**2 - x + Q(1,6)
+    assert bernoulli_poly(3, x) == x**3 - Q(3,2)*x**2 + Q(1,2)*x
+    assert bernoulli_poly(4, x) == x**4 - 2*x**3 + x**2 - Q(1,30)
+    assert bernoulli_poly(5, x) == x**5 - Q(5,2)*x**4 + Q(5,3)*x**3 - Q(1,6)*x
+    assert bernoulli_poly(6, x) == x**6 - 3*x**5 + Q(5,2)*x**4 - Q(1,2)*x**2 + Q(1,42)
+
+    assert bernoulli_poly(1).dummy_eq(x - Q(1,2))
+    assert bernoulli_poly(1, polys=True) == Poly(x - Q(1,2))
+
+def test_bernoulli_c_poly():
+    raises(ValueError, lambda: bernoulli_c_poly(-1, x))
+    assert bernoulli_c_poly(1, x, polys=True) == Poly(x, domain='QQ')
+
+    assert bernoulli_c_poly(0, x) == 1
+    assert bernoulli_c_poly(1, x) == x
+    assert bernoulli_c_poly(2, x) == x**2 - Q(1,3)
+    assert bernoulli_c_poly(3, x) == x**3 - x
+    assert bernoulli_c_poly(4, x) == x**4 - 2*x**2 + Q(7,15)
+    assert bernoulli_c_poly(5, x) == x**5 - Q(10,3)*x**3 + Q(7,3)*x
+    assert bernoulli_c_poly(6, x) == x**6 - 5*x**4 + 7*x**2 - Q(31,21)
+
+    assert bernoulli_c_poly(1).dummy_eq(x)
+    assert bernoulli_c_poly(1, polys=True) == Poly(x, domain='QQ')
+
+    assert 2**8 * bernoulli_poly(8, (x+1)/2).expand() == bernoulli_c_poly(8, x)
+    assert 2**9 * bernoulli_poly(9, (x+1)/2).expand() == bernoulli_c_poly(9, x)
+
+def test_genocchi_poly():
+    raises(ValueError, lambda: genocchi_poly(-1, x))
+    assert genocchi_poly(2, x, polys=True) == Poly(-2*x + 1)
+
+    assert genocchi_poly(0, x) == 0
+    assert genocchi_poly(1, x) == -1
+    assert genocchi_poly(2, x) == 1 - 2*x
+    assert genocchi_poly(3, x) == 3*x - 3*x**2
+    assert genocchi_poly(4, x) == -1 + 6*x**2 - 4*x**3
+    assert genocchi_poly(5, x) == -5*x + 10*x**3 - 5*x**4
+    assert genocchi_poly(6, x) == 3 - 15*x**2 + 15*x**4 - 6*x**5
+
+    assert genocchi_poly(2).dummy_eq(-2*x + 1)
+    assert genocchi_poly(2, polys=True) == Poly(-2*x + 1)
+
+    assert 2 * (bernoulli_poly(8, x) - bernoulli_c_poly(8, x)) == genocchi_poly(8, x)
+    assert 2 * (bernoulli_poly(9, x) - bernoulli_c_poly(9, x)) == genocchi_poly(9, x)
+
+def test_euler_poly():
+    raises(ValueError, lambda: euler_poly(-1, x))
+    assert euler_poly(1, x, polys=True) == Poly(x - Q(1,2))
+
+    assert euler_poly(0, x) == 1
+    assert euler_poly(1, x) == x - Q(1,2)
+    assert euler_poly(2, x) == x**2 - x
+    assert euler_poly(3, x) == x**3 - Q(3,2)*x**2 + Q(1,4)
+    assert euler_poly(4, x) == x**4 - 2*x**3 + x
+    assert euler_poly(5, x) == x**5 - Q(5,2)*x**4 + Q(5,2)*x**2 - Q(1,2)
+    assert euler_poly(6, x) == x**6 - 3*x**5 + 5*x**3 - 3*x
+
+    assert euler_poly(1).dummy_eq(x - Q(1,2))
+    assert euler_poly(1, polys=True) == Poly(x - Q(1,2))
+
+    assert genocchi_poly(9, x) == euler_poly(8, x) * -9
+    assert genocchi_poly(10, x) == euler_poly(9, x) * -10
+
+def test_andre_poly():
+    raises(ValueError, lambda: andre_poly(-1, x))
+    assert andre_poly(1, x, polys=True) == Poly(x)
+
+    assert andre_poly(0, x) == 1
+    assert andre_poly(1, x) == x
+    assert andre_poly(2, x) == x**2 - 1
+    assert andre_poly(3, x) == x**3 - 3*x
+    assert andre_poly(4, x) == x**4 - 6*x**2 + 5
+    assert andre_poly(5, x) == x**5 - 10*x**3 + 25*x
+    assert andre_poly(6, x) == x**6 - 15*x**4 + 75*x**2 - 61
+
+    assert andre_poly(1).dummy_eq(x)
+    assert andre_poly(1, polys=True) == Poly(x)

rl/Lib/site-packages/sympy/polys/tests/test_constructor.py

@@ -0,0 +1,208 @@
+"""Tests for tools for constructing domains for expressions. """
+
+from sympy.polys.constructor import construct_domain
+from sympy.polys.domains import ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX
+from sympy.polys.domains.realfield import RealField
+from sympy.polys.domains.complexfield import ComplexField
+
+from sympy.core import (Catalan, GoldenRatio)
+from sympy.core.numbers import (E, Float, I, Rational, pi)
+from sympy.core.singleton import S
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin
+from sympy.abc import x, y
+
+
+def test_construct_domain():
+
+    assert construct_domain([1, 2, 3]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
+    assert construct_domain([1, 2, 3], field=True) == (QQ, [QQ(1), QQ(2), QQ(3)])
+
+    assert construct_domain([S.One, S(2), S(3)]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
+    assert construct_domain([S.One, S(2), S(3)], field=True) == (QQ, [QQ(1), QQ(2), QQ(3)])
+
+    assert construct_domain([S.Half, S(2)]) == (QQ, [QQ(1, 2), QQ(2)])
+    result = construct_domain([3.14, 1, S.Half])
+    assert isinstance(result[0], RealField)
+    assert result[1] == [RR(3.14), RR(1.0), RR(0.5)]
+
+    result = construct_domain([3.14, I, S.Half])
+    assert isinstance(result[0], ComplexField)
+    assert result[1] == [CC(3.14), CC(1.0j), CC(0.5)]
+
+    assert construct_domain([1.0+I]) == (CC, [CC(1.0, 1.0)])
+    assert construct_domain([2.0+3.0*I]) == (CC, [CC(2.0, 3.0)])
+
+    assert construct_domain([1, I]) == (ZZ_I, [ZZ_I(1, 0), ZZ_I(0, 1)])
+    assert construct_domain([1, I/2]) == (QQ_I, [QQ_I(1, 0), QQ_I(0, S.Half)])
+
+    assert construct_domain([3.14, sqrt(2)], extension=None) == (EX, [EX(3.14), EX(sqrt(2))])
+    assert construct_domain([3.14, sqrt(2)], extension=True) == (EX, [EX(3.14), EX(sqrt(2))])
+
+    assert construct_domain([1, sqrt(2)], extension=None) == (EX, [EX(1), EX(sqrt(2))])
+
+    assert construct_domain([x, sqrt(x)]) == (EX, [EX(x), EX(sqrt(x))])
+    assert construct_domain([x, sqrt(x), sqrt(y)]) == (EX, [EX(x), EX(sqrt(x)), EX(sqrt(y))])
+
+    alg = QQ.algebraic_field(sqrt(2))
+
+    assert construct_domain([7, S.Half, sqrt(2)], extension=True) == \
+        (alg, [alg.convert(7), alg.convert(S.Half), alg.convert(sqrt(2))])
+
+    alg = QQ.algebraic_field(sqrt(2) + sqrt(3))
+
+    assert construct_domain([7, sqrt(2), sqrt(3)], extension=True) == \
+        (alg, [alg.convert(7), alg.convert(sqrt(2)), alg.convert(sqrt(3))])
+
+    dom = ZZ[x]
+
+    assert construct_domain([2*x, 3]) == \
+        (dom, [dom.convert(2*x), dom.convert(3)])
+
+    dom = ZZ[x, y]
+
+    assert construct_domain([2*x, 3*y]) == \
+        (dom, [dom.convert(2*x), dom.convert(3*y)])
+
+    dom = QQ[x]
+
+    assert construct_domain([x/2, 3]) == \
+        (dom, [dom.convert(x/2), dom.convert(3)])
+
+    dom = QQ[x, y]
+
+    assert construct_domain([x/2, 3*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(3*y)])
+
+    dom = ZZ_I[x]
+
+    assert construct_domain([2*x, I]) == \
+        (dom, [dom.convert(2*x), dom.convert(I)])
+
+    dom = ZZ_I[x, y]
+
+    assert construct_domain([2*x, I*y]) == \
+        (dom, [dom.convert(2*x), dom.convert(I*y)])
+
+    dom = QQ_I[x]
+
+    assert construct_domain([x/2, I]) == \
+        (dom, [dom.convert(x/2), dom.convert(I)])
+
+    dom = QQ_I[x, y]
+
+    assert construct_domain([x/2, I*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(I*y)])
+
+    dom = RR[x]
+
+    assert construct_domain([x/2, 3.5]) == \
+        (dom, [dom.convert(x/2), dom.convert(3.5)])
+
+    dom = RR[x, y]
+
+    assert construct_domain([x/2, 3.5*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(3.5*y)])
+
+    dom = CC[x]
+
+    assert construct_domain([I*x/2, 3.5]) == \
+        (dom, [dom.convert(I*x/2), dom.convert(3.5)])
+
+    dom = CC[x, y]
+
+    assert construct_domain([I*x/2, 3.5*y]) == \
+        (dom, [dom.convert(I*x/2), dom.convert(3.5*y)])
+
+    dom = CC[x]
+
+    assert construct_domain([x/2, I*3.5]) == \
+        (dom, [dom.convert(x/2), dom.convert(I*3.5)])
+
+    dom = CC[x, y]
+
+    assert construct_domain([x/2, I*3.5*y]) == \
+        (dom, [dom.convert(x/2), dom.convert(I*3.5*y)])
+
+    dom = ZZ.frac_field(x)
+
+    assert construct_domain([2/x, 3]) == \
+        (dom, [dom.convert(2/x), dom.convert(3)])
+
+    dom = ZZ.frac_field(x, y)
+
+    assert construct_domain([2/x, 3*y]) == \
+        (dom, [dom.convert(2/x), dom.convert(3*y)])
+
+    dom = RR.frac_field(x)
+
+    assert construct_domain([2/x, 3.5]) == \
+        (dom, [dom.convert(2/x), dom.convert(3.5)])
+
+    dom = RR.frac_field(x, y)
+
+    assert construct_domain([2/x, 3.5*y]) == \
+        (dom, [dom.convert(2/x), dom.convert(3.5*y)])
+
+    dom = RealField(prec=336)[x]
+
+    assert construct_domain([pi.evalf(100)*x]) == \
+        (dom, [dom.convert(pi.evalf(100)*x)])
+
+    assert construct_domain(2) == (ZZ, ZZ(2))
+    assert construct_domain(S(2)/3) == (QQ, QQ(2, 3))
+    assert construct_domain(Rational(2, 3)) == (QQ, QQ(2, 3))
+
+    assert construct_domain({}) == (ZZ, {})
+
+
+def test_complex_exponential():
+    w = exp(-I*2*pi/3, evaluate=False)
+    alg = QQ.algebraic_field(w)
+    assert construct_domain([w**2, w, 1], extension=True) == (
+        alg,
+        [alg.convert(w**2),
+         alg.convert(w),
+         alg.convert(1)]
+    )
+
+
+def test_composite_option():
+    assert construct_domain({(1,): sin(y)}, composite=False) == \
+        (EX, {(1,): EX(sin(y))})
+
+    assert construct_domain({(1,): y}, composite=False) == \
+        (EX, {(1,): EX(y)})
+
+    assert construct_domain({(1, 1): 1}, composite=False) == \
+        (ZZ, {(1, 1): 1})
+
+    assert construct_domain({(1, 0): y}, composite=False) == \
+        (EX, {(1, 0): EX(y)})
+
+
+def test_precision():
+    f1 = Float("1.01")
+    f2 = Float("1.0000000000000000000001")
+    for u in [1, 1e-2, 1e-6, 1e-13, 1e-14, 1e-16, 1e-20, 1e-100, 1e-300,
+            f1, f2]:
+        result = construct_domain([u])
+        v = float(result[1][0])
+        assert abs(u - v) / u < 1e-14  # Test relative accuracy
+
+    result = construct_domain([f1])
+    y = result[1][0]
+    assert y-1 > 1e-50
+
+    result = construct_domain([f2])
+    y = result[1][0]
+    assert y-1 > 1e-50
+
+
+def test_issue_11538():
+    for n in [E, pi, Catalan]:
+        assert construct_domain(n)[0] == ZZ[n]
+        assert construct_domain(x + n)[0] == ZZ[x, n]
+    assert construct_domain(GoldenRatio)[0] == EX
+    assert construct_domain(x + GoldenRatio)[0] == EX

rl/Lib/site-packages/sympy/polys/tests/test_densearith.py

@@ -0,0 +1,997 @@
+"""Tests for dense recursive polynomials' arithmetics. """
+
+from sympy.external.gmpy import GROUND_TYPES
+
+from sympy.polys.densebasic import (
+    dup_normal, dmp_normal,
+)
+
+from sympy.polys.densearith import (
+    dup_add_term, dmp_add_term,
+    dup_sub_term, dmp_sub_term,
+    dup_mul_term, dmp_mul_term,
+    dup_add_ground, dmp_add_ground,
+    dup_sub_ground, dmp_sub_ground,
+    dup_mul_ground, dmp_mul_ground,
+    dup_quo_ground, dmp_quo_ground,
+    dup_exquo_ground, dmp_exquo_ground,
+    dup_lshift, dup_rshift,
+    dup_abs, dmp_abs,
+    dup_neg, dmp_neg,
+    dup_add, dmp_add,
+    dup_sub, dmp_sub,
+    dup_mul, dmp_mul,
+    dup_sqr, dmp_sqr,
+    dup_pow, dmp_pow,
+    dup_add_mul, dmp_add_mul,
+    dup_sub_mul, dmp_sub_mul,
+    dup_pdiv, dup_prem, dup_pquo, dup_pexquo,
+    dmp_pdiv, dmp_prem, dmp_pquo, dmp_pexquo,
+    dup_rr_div, dmp_rr_div,
+    dup_ff_div, dmp_ff_div,
+    dup_div, dup_rem, dup_quo, dup_exquo,
+    dmp_div, dmp_rem, dmp_quo, dmp_exquo,
+    dup_max_norm, dmp_max_norm,
+    dup_l1_norm, dmp_l1_norm,
+    dup_l2_norm_squared, dmp_l2_norm_squared,
+    dup_expand, dmp_expand,
+)
+
+from sympy.polys.polyerrors import (
+    ExactQuotientFailed,
+)
+
+from sympy.polys.specialpolys import f_polys
+from sympy.polys.domains import FF, ZZ, QQ
+
+from sympy.testing.pytest import raises
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
+F_0 = dmp_mul_ground(dmp_normal(f_0, 2, QQ), QQ(1, 7), 2, QQ)
+
+def test_dup_add_term():
+    f = dup_normal([], ZZ)
+
+    assert dup_add_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ)
+
+    assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1], ZZ)
+    assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 0], ZZ)
+    assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 0, 0], ZZ)
+
+    f = dup_normal([1, 1, 1], ZZ)
+
+    assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1, 1, 2], ZZ)
+    assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 2, 1], ZZ)
+    assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([2, 1, 1], ZZ)
+
+    assert dup_add_term(f, ZZ(1), 3, ZZ) == dup_normal([1, 1, 1, 1], ZZ)
+    assert dup_add_term(f, ZZ(1), 4, ZZ) == dup_normal([1, 0, 1, 1, 1], ZZ)
+    assert dup_add_term(f, ZZ(1), 5, ZZ) == dup_normal([1, 0, 0, 1, 1, 1], ZZ)
+    assert dup_add_term(
+        f, ZZ(1), 6, ZZ) == dup_normal([1, 0, 0, 0, 1, 1, 1], ZZ)
+
+    assert dup_add_term(f, ZZ(-1), 2, ZZ) == dup_normal([1, 1], ZZ)
+
+
+def test_dmp_add_term():
+    assert dmp_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \
+        dup_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ)
+    assert dmp_add_term(f_0, [[]], 3, 2, ZZ) == f_0
+    assert dmp_add_term(F_0, [[]], 3, 2, QQ) == F_0
+
+
+def test_dup_sub_term():
+    f = dup_normal([], ZZ)
+
+    assert dup_sub_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ)
+
+    assert dup_sub_term(f, ZZ(1), 0, ZZ) == dup_normal([-1], ZZ)
+    assert dup_sub_term(f, ZZ(1), 1, ZZ) == dup_normal([-1, 0], ZZ)
+    assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([-1, 0, 0], ZZ)
+
+    f = dup_normal([1, 1, 1], ZZ)
+
+    assert dup_sub_term(f, ZZ(2), 0, ZZ) == dup_normal([ 1, 1, -1], ZZ)
+    assert dup_sub_term(f, ZZ(2), 1, ZZ) == dup_normal([ 1, -1, 1], ZZ)
+    assert dup_sub_term(f, ZZ(2), 2, ZZ) == dup_normal([-1, 1, 1], ZZ)
+
+    assert dup_sub_term(f, ZZ(1), 3, ZZ) == dup_normal([-1, 1, 1, 1], ZZ)
+    assert dup_sub_term(f, ZZ(1), 4, ZZ) == dup_normal([-1, 0, 1, 1, 1], ZZ)
+    assert dup_sub_term(f, ZZ(1), 5, ZZ) == dup_normal([-1, 0, 0, 1, 1, 1], ZZ)
+    assert dup_sub_term(
+        f, ZZ(1), 6, ZZ) == dup_normal([-1, 0, 0, 0, 1, 1, 1], ZZ)
+
+    assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 1], ZZ)
+
+
+def test_dmp_sub_term():
+    assert dmp_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \
+        dup_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ)
+    assert dmp_sub_term(f_0, [[]], 3, 2, ZZ) == f_0
+    assert dmp_sub_term(F_0, [[]], 3, 2, QQ) == F_0
+
+
+def test_dup_mul_term():
+    f = dup_normal([], ZZ)
+
+    assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([1, 1], ZZ)
+
+    assert dup_mul_term(f, ZZ(0), 3, ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([1, 2, 3], ZZ)
+
+    assert dup_mul_term(f, ZZ(2), 0, ZZ) == dup_normal([2, 4, 6], ZZ)
+    assert dup_mul_term(f, ZZ(2), 1, ZZ) == dup_normal([2, 4, 6, 0], ZZ)
+    assert dup_mul_term(f, ZZ(2), 2, ZZ) == dup_normal([2, 4, 6, 0, 0], ZZ)
+    assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([2, 4, 6, 0, 0, 0], ZZ)
+
+
+def test_dmp_mul_term():
+    assert dmp_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, 0, ZZ) == \
+        dup_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, ZZ)
+
+    assert dmp_mul_term([[]], [ZZ(2)], 3, 1, ZZ) == [[]]
+    assert dmp_mul_term([[ZZ(1)]], [], 3, 1, ZZ) == [[]]
+
+    assert dmp_mul_term([[ZZ(1), ZZ(2)], [ZZ(3)]], [ZZ(2)], 2, 1, ZZ) == \
+        [[ZZ(2), ZZ(4)], [ZZ(6)], [], []]
+
+    assert dmp_mul_term([[]], [QQ(2, 3)], 3, 1, QQ) == [[]]
+    assert dmp_mul_term([[QQ(1, 2)]], [], 3, 1, QQ) == [[]]
+
+    assert dmp_mul_term([[QQ(1, 5), QQ(2, 5)], [QQ(3, 5)]], [QQ(2, 3)], 2, 1, QQ) == \
+        [[QQ(2, 15), QQ(4, 15)], [QQ(6, 15)], [], []]
+
+
+def test_dup_add_ground():
+    f = ZZ.map([1, 2, 3, 4])
+    g = ZZ.map([1, 2, 3, 8])
+
+    assert dup_add_ground(f, ZZ(4), ZZ) == g
+
+
+def test_dmp_add_ground():
+    f = ZZ.map([[1], [2], [3], [4]])
+    g = ZZ.map([[1], [2], [3], [8]])
+
+    assert dmp_add_ground(f, ZZ(4), 1, ZZ) == g
+
+
+def test_dup_sub_ground():
+    f = ZZ.map([1, 2, 3, 4])
+    g = ZZ.map([1, 2, 3, 0])
+
+    assert dup_sub_ground(f, ZZ(4), ZZ) == g
+
+
+def test_dmp_sub_ground():
+    f = ZZ.map([[1], [2], [3], [4]])
+    g = ZZ.map([[1], [2], [3], []])
+
+    assert dmp_sub_ground(f, ZZ(4), 1, ZZ) == g
+
+
+def test_dup_mul_ground():
+    f = dup_normal([], ZZ)
+
+    assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([1, 2, 3], ZZ)
+
+    assert dup_mul_ground(f, ZZ(0), ZZ) == dup_normal([], ZZ)
+    assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([2, 4, 6], ZZ)
+
+
+def test_dmp_mul_ground():
+    assert dmp_mul_ground(f_0, ZZ(2), 2, ZZ) == [
+        [[ZZ(2), ZZ(4), ZZ(6)], [ZZ(4)]],
+        [[ZZ(6)]],
+        [[ZZ(8), ZZ(10), ZZ(12)], [ZZ(2), ZZ(4), ZZ(2)], [ZZ(2)]]
+    ]
+
+    assert dmp_mul_ground(F_0, QQ(1, 2), 2, QQ) == [
+        [[QQ(1, 14), QQ(2, 14), QQ(3, 14)], [QQ(2, 14)]],
+        [[QQ(3, 14)]],
+        [[QQ(4, 14), QQ(5, 14), QQ(6, 14)], [QQ(1, 14), QQ(2, 14),
+             QQ(1, 14)], [QQ(1, 14)]]
+    ]
+
+
+def test_dup_quo_ground():
+    raises(ZeroDivisionError, lambda: dup_quo_ground(dup_normal([1, 2,
+           3], ZZ), ZZ(0), ZZ))
+
+    f = dup_normal([], ZZ)
+
+    assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([6, 2, 8], ZZ)
+
+    assert dup_quo_ground(f, ZZ(1), ZZ) == f
+    assert dup_quo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ)
+
+    assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([2, 0, 2], ZZ)
+
+    f = dup_normal([6, 2, 8], QQ)
+
+    assert dup_quo_ground(f, QQ(1), QQ) == f
+    assert dup_quo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)]
+    assert dup_quo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)]
+
+
+def test_dup_exquo_ground():
+    raises(ZeroDivisionError, lambda: dup_exquo_ground(dup_normal([1,
+           2, 3], ZZ), ZZ(0), ZZ))
+    raises(ExactQuotientFailed, lambda: dup_exquo_ground(dup_normal([1,
+           2, 3], ZZ), ZZ(3), ZZ))
+
+    f = dup_normal([], ZZ)
+
+    assert dup_exquo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ)
+
+    f = dup_normal([6, 2, 8], ZZ)
+
+    assert dup_exquo_ground(f, ZZ(1), ZZ) == f
+    assert dup_exquo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ)
+
+    f = dup_normal([6, 2, 8], QQ)
+
+    assert dup_exquo_ground(f, QQ(1), QQ) == f
+    assert dup_exquo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)]
+    assert dup_exquo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)]
+
+
+def test_dmp_quo_ground():
+    f = dmp_normal([[6], [2], [8]], 1, ZZ)
+
+    assert dmp_quo_ground(f, ZZ(1), 1, ZZ) == f
+    assert dmp_quo_ground(
+        f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ)
+
+    assert dmp_normal(dmp_quo_ground(
+        f, ZZ(3), 1, ZZ), 1, ZZ) == dmp_normal([[2], [], [2]], 1, ZZ)
+
+
+def test_dmp_exquo_ground():
+    f = dmp_normal([[6], [2], [8]], 1, ZZ)
+
+    assert dmp_exquo_ground(f, ZZ(1), 1, ZZ) == f
+    assert dmp_exquo_ground(
+        f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ)
+
+
+def test_dup_lshift():
+    assert dup_lshift([], 3, ZZ) == []
+    assert dup_lshift([1], 3, ZZ) == [1, 0, 0, 0]
+
+
+def test_dup_rshift():
+    assert dup_rshift([], 3, ZZ) == []
+    assert dup_rshift([1, 0, 0, 0], 3, ZZ) == [1]
+
+
+def test_dup_abs():
+    assert dup_abs([], ZZ) == []
+    assert dup_abs([ZZ( 1)], ZZ) == [ZZ(1)]
+    assert dup_abs([ZZ(-7)], ZZ) == [ZZ(7)]
+    assert dup_abs([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(2), ZZ(3)]
+
+    assert dup_abs([], QQ) == []
+    assert dup_abs([QQ( 1, 2)], QQ) == [QQ(1, 2)]
+    assert dup_abs([QQ(-7, 3)], QQ) == [QQ(7, 3)]
+    assert dup_abs(
+        [QQ(-1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(2, 7), QQ(3, 7)]
+
+
+def test_dmp_abs():
+    assert dmp_abs([ZZ(-1)], 0, ZZ) == [ZZ(1)]
+    assert dmp_abs([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)]
+
+    assert dmp_abs([[[]]], 2, ZZ) == [[[]]]
+    assert dmp_abs([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_abs([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]]
+
+    assert dmp_abs([[[]]], 2, QQ) == [[[]]]
+    assert dmp_abs([[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_abs([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]]
+
+
+def test_dup_neg():
+    assert dup_neg([], ZZ) == []
+    assert dup_neg([ZZ(1)], ZZ) == [ZZ(-1)]
+    assert dup_neg([ZZ(-7)], ZZ) == [ZZ(7)]
+    assert dup_neg([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(-2), ZZ(-3)]
+
+    assert dup_neg([], QQ) == []
+    assert dup_neg([QQ(1, 2)], QQ) == [QQ(-1, 2)]
+    assert dup_neg([QQ(-7, 9)], QQ) == [QQ(7, 9)]
+    assert dup_neg([QQ(
+        -1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(-2, 7), QQ(-3, 7)]
+
+
+def test_dmp_neg():
+    assert dmp_neg([ZZ(-1)], 0, ZZ) == [ZZ(1)]
+    assert dmp_neg([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)]
+
+    assert dmp_neg([[[]]], 2, ZZ) == [[[]]]
+    assert dmp_neg([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
+    assert dmp_neg([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]]
+
+    assert dmp_neg([[[]]], 2, QQ) == [[[]]]
+    assert dmp_neg([[[QQ(1, 9)]]], 2, QQ) == [[[QQ(-1, 9)]]]
+    assert dmp_neg([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]]
+
+
+def test_dup_add():
+    assert dup_add([], [], ZZ) == []
+    assert dup_add([ZZ(1)], [], ZZ) == [ZZ(1)]
+    assert dup_add([], [ZZ(1)], ZZ) == [ZZ(1)]
+    assert dup_add([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(2)]
+    assert dup_add([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(3)]
+
+    assert dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(3)]
+    assert dup_add([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(3)]
+
+    assert dup_add([ZZ(1), ZZ(
+        2), ZZ(3)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(9), ZZ(11), ZZ(13)]
+
+    assert dup_add([], [], QQ) == []
+    assert dup_add([QQ(1, 2)], [], QQ) == [QQ(1, 2)]
+    assert dup_add([], [QQ(1, 2)], QQ) == [QQ(1, 2)]
+    assert dup_add([QQ(1, 4)], [QQ(1, 4)], QQ) == [QQ(1, 2)]
+    assert dup_add([QQ(1, 4)], [QQ(1, 2)], QQ) == [QQ(3, 4)]
+
+    assert dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ) == [QQ(1, 2), QQ(5, 3)]
+    assert dup_add([QQ(1)], [QQ(1, 2), QQ(2, 3)], QQ) == [QQ(1, 2), QQ(5, 3)]
+
+    assert dup_add([QQ(1, 7), QQ(2, 7), QQ(3, 7)], [QQ(
+        8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(9, 7), QQ(11, 7), QQ(13, 7)]
+
+
+def test_dmp_add():
+    assert dmp_add([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \
+        dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
+    assert dmp_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \
+        dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
+
+    assert dmp_add([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_add([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_add([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_add([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(3)]]]
+    assert dmp_add([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(3)]]]
+
+    assert dmp_add([[[]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_add([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_add([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_add([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
+    assert dmp_add([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
+
+
+def test_dup_sub():
+    assert dup_sub([], [], ZZ) == []
+    assert dup_sub([ZZ(1)], [], ZZ) == [ZZ(1)]
+    assert dup_sub([], [ZZ(1)], ZZ) == [ZZ(-1)]
+    assert dup_sub([ZZ(1)], [ZZ(1)], ZZ) == []
+    assert dup_sub([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(-1)]
+
+    assert dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(1)]
+    assert dup_sub([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(-1), ZZ(-1)]
+
+    assert dup_sub([ZZ(3), ZZ(
+        2), ZZ(1)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(-5), ZZ(-7), ZZ(-9)]
+
+    assert dup_sub([], [], QQ) == []
+    assert dup_sub([QQ(1, 2)], [], QQ) == [QQ(1, 2)]
+    assert dup_sub([], [QQ(1, 2)], QQ) == [QQ(-1, 2)]
+    assert dup_sub([QQ(1, 3)], [QQ(1, 3)], QQ) == []
+    assert dup_sub([QQ(1, 3)], [QQ(2, 3)], QQ) == [QQ(-1, 3)]
+
+    assert dup_sub([QQ(1, 7), QQ(2, 7)], [QQ(1)], QQ) == [QQ(1, 7), QQ(-5, 7)]
+    assert dup_sub([QQ(1)], [QQ(1, 7), QQ(2, 7)], QQ) == [QQ(-1, 7), QQ(5, 7)]
+
+    assert dup_sub([QQ(3, 7), QQ(2, 7), QQ(1, 7)], [QQ(
+        8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(-5, 7), QQ(-7, 7), QQ(-9, 7)]
+
+
+def test_dmp_sub():
+    assert dmp_sub([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \
+        dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
+    assert dmp_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \
+        dup_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
+
+    assert dmp_sub([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_sub([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_sub([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
+    assert dmp_sub([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
+    assert dmp_sub([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(-1)]]]
+
+    assert dmp_sub([[[]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_sub([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
+    assert dmp_sub([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(-1, 2)]]]
+    assert dmp_sub([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(1, 7)]]]
+    assert dmp_sub([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(-1, 7)]]]
+
+
+def test_dup_add_mul():
+    assert dup_add_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)],
+               [ZZ(1), ZZ(2)], ZZ) == [ZZ(3), ZZ(9), ZZ(7), ZZ(5)]
+    assert dmp_add_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]],
+               [[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(3)], [ZZ(3), ZZ(9)], [ZZ(4), ZZ(5)]]
+
+
+def test_dup_sub_mul():
+    assert dup_sub_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)],
+               [ZZ(1), ZZ(2)], ZZ) == [ZZ(-3), ZZ(-7), ZZ(-3), ZZ(1)]
+    assert dmp_sub_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]],
+               [[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(-3)], [ZZ(-1), ZZ(-5)], [ZZ(-4), ZZ(1)]]
+
+
+def test_dup_mul():
+    assert dup_mul([], [], ZZ) == []
+    assert dup_mul([], [ZZ(1)], ZZ) == []
+    assert dup_mul([ZZ(1)], [], ZZ) == []
+    assert dup_mul([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(1)]
+    assert dup_mul([ZZ(5)], [ZZ(7)], ZZ) == [ZZ(35)]
+
+    assert dup_mul([], [], QQ) == []
+    assert dup_mul([], [QQ(1, 2)], QQ) == []
+    assert dup_mul([QQ(1, 2)], [], QQ) == []
+    assert dup_mul([QQ(1, 2)], [QQ(4, 7)], QQ) == [QQ(2, 7)]
+    assert dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ) == [QQ(15, 49)]
+
+    f = dup_normal([3, 0, 0, 6, 1, 2], ZZ)
+    g = dup_normal([4, 0, 1, 0], ZZ)
+    h = dup_normal([12, 0, 3, 24, 4, 14, 1, 2, 0], ZZ)
+
+    assert dup_mul(f, g, ZZ) == h
+    assert dup_mul(g, f, ZZ) == h
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+    h = dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
+
+    assert dup_mul(f, f, ZZ) == h
+
+    K = FF(6)
+
+    assert dup_mul([K(2), K(1)], [K(3), K(4)], K) == [K(5), K(4)]
+
+    p1 = dup_normal([79, -1, 78, -94, -10, 11, 32, -19, 78, 2, -89, 30, 73, 42,
+        85, 77, 83, -30, -34, -2, 95, -81, 37, -49, -46, -58, -16, 37, 35, -11,
+        -57, -15, -31, 67, -20, 27, 76, 2, 70, 67, -65, 65, -26, -93, -44, -12,
+        -92, 57, -90, -57, -11, -67, -98, -69, 97, -41, 89, 33, 89, -50, 81,
+        -31, 60, -27, 43, 29, -77, 44, 21, -91, 32, -57, 33, 3, 53, -51, -38,
+        -99, -84, 23, -50, 66, -100, 1, -75, -25, 27, -60, 98, -51, -87, 6, 8,
+        78, -28, -95, -88, 12, -35, 26, -9, 16, -92, 55, -7, -86, 68, -39, -46,
+        84, 94, 45, 60, 92, 68, -75, -74, -19, 8, 75, 78, 91, 57, 34, 14, -3,
+        -49, 65, 78, -18, 6, -29, -80, -98, 17, 13, 58, 21, 20, 9, 37, 7, -30,
+        -53, -20, 34, 67, -42, 89, -22, 73, 43, -6, 5, 51, -8, -15, -52, -22,
+        -58, -72, -3, 43, -92, 82, 83, -2, -13, -23, -60, 16, -94, -8, -28,
+        -95, -72, 63, -90, 76, 6, -43, -100, -59, 76, 3, 3, 46, -85, 75, 62,
+        -71, -76, 88, 97, -72, -1, 30, -64, 72, -48, 14, -78, 58, 63, -91, 24,
+        -87, -27, -80, -100, -44, 98, 70, 100, -29, -38, 11, 77, 100, 52, 86,
+        65, -5, -42, -81, -38, -42, 43, -2, -70, -63, -52], ZZ)
+    p2 = dup_normal([65, -19, -47, 1, 90, 81, -15, -34, 25, -75, 9, -83, 50, -5,
+        -44, 31, 1, 70, -7, 78, 74, 80, 85, 65, 21, 41, 66, 19, -40, 63, -21,
+        -27, 32, 69, 83, 34, -35, 14, 81, 57, -75, 32, -67, -89, -100, -61, 46,
+        84, -78, -29, -50, -94, -24, -32, -68, -16, 100, -7, -72, -89, 35, 82,
+        58, 81, -92, 62, 5, -47, -39, -58, -72, -13, 84, 44, 55, -25, 48, -54,
+        -31, -56, -11, -50, -84, 10, 67, 17, 13, -14, 61, 76, -64, -44, -40,
+        -96, 11, -11, -94, 2, 6, 27, -6, 68, -54, 66, -74, -14, -1, -24, -73,
+        96, 89, -11, -89, 56, -53, 72, -43, 96, 25, 63, -31, 29, 68, 83, 91,
+        -93, -19, -38, -40, 40, -12, -19, -79, 44, 100, -66, -29, -77, 62, 39,
+        -8, 11, -97, 14, 87, 64, 21, -18, 13, 15, -59, -75, -99, -88, 57, 54,
+        56, -67, 6, -63, -59, -14, 28, 87, -20, -39, 84, -91, -2, 49, -75, 11,
+        -24, -95, 36, 66, 5, 25, -72, -40, 86, 90, 37, -33, 57, -35, 29, -18,
+        4, -79, 64, -17, -27, 21, 29, -5, -44, -87, -24, 52, 78, 11, -23, -53,
+        36, 42, 21, -68, 94, -91, -51, -21, 51, -76, 72, 31, 24, -48, -80, -9,
+        37, -47, -6, -8, -63, -91, 79, -79, -100, 38, -20, 38, 100, 83, -90,
+        87, 63, -36, 82, -19, 18, -98, -38, 26, 98, -70, 79, 92, 12, 12, 70,
+        74, 36, 48, -13, 31, 31, -47, -71, -12, -64, 36, -42, 32, -86, 60, 83,
+        70, 55, 0, 1, 29, -35, 8, -82, 8, -73, -46, -50, 43, 48, -5, -86, -72,
+        44, -90, 19, 19, 5, -20, 97, -13, -66, -5, 5, -69, 64, -30, 41, 51, 36,
+        13, -99, -61, 94, -12, 74, 98, 68, 24, 46, -97, -87, -6, -27, 82, 62,
+        -11, -77, 86, 66, -47, -49, -50, 13, 18, 89, -89, 46, -80, 13, 98, -35,
+        -36, -25, 12, 20, 26, -52, 79, 27, 79, 100, 8, 62, -58, -28, 37], ZZ)
+    res = dup_normal([5135, -1566, 1376, -7466, 4579, 11710, 8001, -7183,
+        -3737, -7439, 345, -10084, 24522, -1201, 1070, -10245, 9582, 9264,
+        1903, 23312, 18953, 10037, -15268, -5450, 6442, -6243, -3777, 5110,
+        10936, -16649, -6022, 16255, 31300, 24818, 31922, 32760, 7854, 27080,
+        15766, 29596, 7139, 31945, -19810, 465, -38026, -3971, 9641, 465,
+        -19375, 5524, -30112, -11960, -12813, 13535, 30670, 5925, -43725,
+        -14089, 11503, -22782, 6371, 43881, 37465, -33529, -33590, -39798,
+        -37854, -18466, -7908, -35825, -26020, -36923, -11332, -5699, 25166,
+        -3147, 19885, 12962, -20659, -1642, 27723, -56331, -24580, -11010,
+        -20206, 20087, -23772, -16038, 38580, 20901, -50731, 32037, -4299,
+        26508, 18038, -28357, 31846, -7405, -20172, -15894, 2096, 25110,
+        -45786, 45918, -55333, -31928, -49428, -29824, -58796, -24609, -15408,
+        69, -35415, -18439, 10123, -20360, -65949, 33356, -20333, 26476,
+        -32073, 33621, 930, 28803, -42791, 44716, 38164, 12302, -1739, 11421,
+        73385, -7613, 14297, 38155, -414, 77587, 24338, -21415, 29367, 42639,
+        13901, -288, 51027, -11827, 91260, 43407, 88521, -15186, 70572, -12049,
+        5090, -12208, -56374, 15520, -623, -7742, 50825, 11199, -14894, 40892,
+        59591, -31356, -28696, -57842, -87751, -33744, -28436, -28945, -40287,
+        37957, -35638, 33401, -61534, 14870, 40292, 70366, -10803, 102290,
+        -71719, -85251, 7902, -22409, 75009, 99927, 35298, -1175, -762, -34744,
+        -10587, -47574, -62629, -19581, -43659, -54369, -32250, -39545, 15225,
+        -24454, 11241, -67308, -30148, 39929, 37639, 14383, -73475, -77636,
+        -81048, -35992, 41601, -90143, 76937, -8112, 56588, 9124, -40094,
+        -32340, 13253, 10898, -51639, 36390, 12086, -1885, 100714, -28561,
+        -23784, -18735, 18916, 16286, 10742, -87360, -13697, 10689, -19477,
+        -29770, 5060, 20189, -8297, 112407, 47071, 47743, 45519, -4109, 17468,
+        -68831, 78325, -6481, -21641, -19459, 30919, 96115, 8607, 53341, 32105,
+        -16211, 23538, 57259, -76272, -40583, 62093, 38511, -34255, -40665,
+        -40604, -37606, -15274, 33156, -13885, 103636, 118678, -14101, -92682,
+        -100791, 2634, 63791, 98266, 19286, -34590, -21067, -71130, 25380,
+        -40839, -27614, -26060, 52358, -15537, 27138, -6749, 36269, -33306,
+        13207, -91084, -5540, -57116, 69548, 44169, -57742, -41234, -103327,
+        -62904, -8566, 41149, -12866, 71188, 23980, 1838, 58230, 73950, 5594,
+        43113, -8159, -15925, 6911, 85598, -75016, -16214, -62726, -39016,
+        8618, -63882, -4299, 23182, 49959, 49342, -3238, -24913, -37138, 78361,
+        32451, 6337, -11438, -36241, -37737, 8169, -3077, -24829, 57953, 53016,
+        -31511, -91168, 12599, -41849, 41576, 55275, -62539, 47814, -62319,
+        12300, -32076, -55137, -84881, -27546, 4312, -3433, -54382, 113288,
+        -30157, 74469, 18219, 79880, -2124, 98911, 17655, -33499, -32861,
+        47242, -37393, 99765, 14831, -44483, 10800, -31617, -52710, 37406,
+        22105, 29704, -20050, 13778, 43683, 36628, 8494, 60964, -22644, 31550,
+        -17693, 33805, -124879, -12302, 19343, 20400, -30937, -21574, -34037,
+        -33380, 56539, -24993, -75513, -1527, 53563, 65407, -101, 53577, 37991,
+        18717, -23795, -8090, -47987, -94717, 41967, 5170, -14815, -94311,
+        17896, -17734, -57718, -774, -38410, 24830, 29682, 76480, 58802,
+        -46416, -20348, -61353, -68225, -68306, 23822, -31598, 42972, 36327,
+        28968, -65638, -21638, 24354, -8356, 26777, 52982, -11783, -44051,
+        -26467, -44721, -28435, -53265, -25574, -2669, 44155, 22946, -18454,
+        -30718, -11252, 58420, 8711, 67447, 4425, 41749, 67543, 43162, 11793,
+        -41907, 20477, -13080, 6559, -6104, -13244, 42853, 42935, 29793, 36730,
+        -28087, 28657, 17946, 7503, 7204, 21491, -27450, -24241, -98156,
+        -18082, -42613, -24928, 10775, -14842, -44127, 55910, 14777, 31151, -2194,
+        39206, -2100, -4211, 11827, -8918, -19471, 72567, 36447, -65590, -34861,
+        -17147, -45303, 9025, -7333, -35473, 11101, 11638, 3441, 6626, -41800,
+        9416, 13679, 33508, 40502, -60542, 16358, 8392, -43242, -35864, -34127,
+        -48721, 35878, 30598, 28630, 20279, -19983, -14638, -24455, -1851, -11344,
+        45150, 42051, 26034, -28889, -32382, -3527, -14532, 22564, -22346, 477,
+        11706, 28338, -25972, -9185, -22867, -12522, 32120, -4424, 11339, -33913,
+        -7184, 5101, -23552, -17115, -31401, -6104, 21906, 25708, 8406, 6317,
+        -7525, 5014, 20750, 20179, 22724, 11692, 13297, 2493, -253, -16841, -17339,
+        -6753, -4808, 2976, -10881, -10228, -13816, -12686, 1385, 2316, 2190, -875,
+        -1924], ZZ)
+
+    assert dup_mul(p1, p2, ZZ) == res
+
+    p1 = dup_normal([83, -61, -86, -24, 12, 43, -88, -9, 42, 55, -66, 74, 95,
+        -25, -12, 68, -99, 4, 45, 6, -15, -19, 78, 65, -55, 47, -13, 17, 86,
+        81, -58, -27, 50, -40, -24, 39, -41, -92, 75, 90, -1, 40, -15, -27,
+        -35, 68, 70, -64, -40, 78, -88, -58, -39, 69, 46, 12, 28, -94, -37,
+        -50, -80, -96, -61, 25, 1, 71, 4, 12, 48, 4, 34, -47, -75, 5, 48, 82,
+        88, 23, 98, 35, 17, -10, 48, -61, -95, 47, 65, -19, -66, -57, -6, -51,
+        -42, -89, 66, -13, 18, 37, 90, -23, 72, 96, -53, 0, 40, -73, -52, -68,
+        32, -25, -53, 79, -52, 18, 44, 73, -81, 31, -90, 70, 3, 36, 48, 76,
+        -24, -44, 23, 98, -4, 73, 69, 88, -70, 14, -68, 94, -78, -15, -64, -97,
+        -70, -35, 65, 88, 49, -53, -7, 12, -45, -7, 59, -94, 99, -2, 67, -60,
+        -71, 29, -62, -77, 1, 51, 17, 80, -20, -47, -19, 24, -9, 39, -23, 21,
+        -84, 10, 84, 56, -17, -21, -66, 85, 70, 46, -51, -22, -95, 78, -60,
+        -96, -97, -45, 72, 35, 30, -61, -92, -93, -60, -61, 4, -4, -81, -73,
+        46, 53, -11, 26, 94, 45, 14, -78, 55, 84, -68, 98, 60, 23, 100, -63,
+        68, 96, -16, 3, 56, 21, -58, 62, -67, 66, 85, 41, -79, -22, 97, -67,
+        82, 82, -96, -20, -7, 48, -67, 48, -9, -39, 78], ZZ)
+    p2 = dup_normal([52, 88, 76, 66, 9, -64, 46, -20, -28, 69, 60, 96, -36,
+        -92, -30, -11, -35, 35, 55, 63, -92, -7, 25, -58, 74, 55, -6, 4, 47,
+        -92, -65, 67, -45, 74, -76, 59, -6, 69, 39, 24, -71, -7, 39, -45, 60,
+        -68, 98, 97, -79, 17, 4, 94, -64, 68, -100, -96, -2, 3, 22, 96, 54,
+        -77, -86, 67, 6, 57, 37, 40, 89, -78, 64, -94, -45, -92, 57, 87, -26,
+        36, 19, 97, 25, 77, -87, 24, 43, -5, 35, 57, 83, 71, 35, 63, 61, 96,
+        -22, 8, -1, 96, 43, 45, 94, -93, 36, 71, -41, -99, 85, -48, 59, 52,
+        -17, 5, 87, -16, -68, -54, 76, -18, 100, 91, -42, -70, -66, -88, -12,
+        1, 95, -82, 52, 43, -29, 3, 12, 72, -99, -43, -32, -93, -51, 16, -20,
+        -12, -11, 5, 33, -38, 93, -5, -74, 25, 74, -58, 93, 59, -63, -86, 63,
+        -20, -4, -74, -73, -95, 29, -28, 93, -91, -2, -38, -62, 77, -58, -85,
+        -28, 95, 38, 19, -69, 86, 94, 25, -2, -4, 47, 34, -59, 35, -48, 29,
+        -63, -53, 34, 29, 66, 73, 6, 92, -84, 89, 15, 81, 93, 97, 51, -72, -78,
+        25, 60, 90, -45, 39, 67, -84, -62, 57, 26, -32, -56, -14, -83, 76, 5,
+        -2, 99, -100, 28, 46, 94, -7, 53, -25, 16, -23, -36, 89, -78, -63, 31,
+        1, 84, -99, -52, 76, 48, 90, -76, 44, -19, 54, -36, -9, -73, -100, -69,
+        31, 42, 25, -39, 76, -26, -8, -14, 51, 3, 37, 45, 2, -54, 13, -34, -92,
+        17, -25, -65, 53, -63, 30, 4, -70, -67, 90, 52, 51, 18, -3, 31, -45,
+        -9, 59, 63, -87, 22, -32, 29, -38, 21, 36, -82, 27, -11], ZZ)
+    res = dup_normal([4316, 4132, -3532, -7974, -11303, -10069, 5484, -3330,
+        -5874, 7734, 4673, 11327, -9884, -8031, 17343, 21035, -10570, -9285,
+        15893, 3780, -14083, 8819, 17592, 10159, 7174, -11587, 8598, -16479,
+        3602, 25596, 9781, 12163, 150, 18749, -21782, -12307, 27578, -2757,
+        -12573, 12565, 6345, -18956, 19503, -15617, 1443, -16778, 36851, 23588,
+        -28474, 5749, 40695, -7521, -53669, -2497, -18530, 6770, 57038, 3926,
+        -6927, -15399, 1848, -64649, -27728, 3644, 49608, 15187, -8902, -9480,
+        -7398, -40425, 4824, 23767, -7594, -6905, 33089, 18786, 12192, 24670,
+        31114, 35334, -4501, -14676, 7107, -59018, -21352, 20777, 19661, 20653,
+        33754, -885, -43758, 6269, 51897, -28719, -97488, -9527, 13746, 11644,
+        17644, -21720, 23782, -10481, 47867, 20752, 33810, -1875, 39918, -7710,
+        -40840, 19808, -47075, 23066, 46616, 25201, 9287, 35436, -1602, 9645,
+        -11978, 13273, 15544, 33465, 20063, 44539, 11687, 27314, -6538, -37467,
+        14031, 32970, -27086, 41323, 29551, 65910, -39027, -37800, -22232,
+        8212, 46316, -28981, -55282, 50417, -44929, -44062, 73879, 37573,
+        -2596, -10877, -21893, -133218, -33707, -25753, -9531, 17530, 61126,
+        2748, -56235, 43874, -10872, -90459, -30387, 115267, -7264, -44452,
+        122626, 14839, -599, 10337, 57166, -67467, -54957, 63669, 1202, 18488,
+        52594, 7205, -97822, 612, 78069, -5403, -63562, 47236, 36873, -154827,
+        -26188, 82427, -39521, 5628, 7416, 5276, -53095, 47050, 26121, -42207,
+        79021, -13035, 2499, -66943, 29040, -72355, -23480, 23416, -12885,
+        -44225, -42688, -4224, 19858, 55299, 15735, 11465, 101876, -39169,
+        51786, 14723, 43280, -68697, 16410, 92295, 56767, 7183, 111850, 4550,
+        115451, -38443, -19642, -35058, 10230, 93829, 8925, 63047, 3146, 29250,
+        8530, 5255, -98117, -115517, -76817, -8724, 41044, 1312, -35974, 79333,
+        -28567, 7547, -10580, -24559, -16238, 10794, -3867, 24848, 57770,
+        -51536, -35040, 71033, 29853, 62029, -7125, -125585, -32169, -47907,
+        156811, -65176, -58006, -15757, -57861, 11963, 30225, -41901, -41681,
+        31310, 27982, 18613, 61760, 60746, -59096, 33499, 30097, -17997, 24032,
+        56442, -83042, 23747, -20931, -21978, -158752, -9883, -73598, -7987,
+        -7333, -125403, -116329, 30585, 53281, 51018, -29193, 88575, 8264,
+        -40147, -16289, 113088, 12810, -6508, 101552, -13037, 34440, -41840,
+        101643, 24263, 80532, 61748, 65574, 6423, -20672, 6591, -10834, -71716,
+        86919, -92626, 39161, 28490, 81319, 46676, 106720, 43530, 26998, 57456,
+        -8862, 60989, 13982, 3119, -2224, 14743, 55415, -49093, -29303, 28999,
+        1789, 55953, -84043, -7780, -65013, 57129, -47251, 61484, 61994,
+        -78361, -82778, 22487, -26894, 9756, -74637, -15519, -4360, 30115,
+        42433, 35475, 15286, 69768, 21509, -20214, 78675, -21163, 13596, 11443,
+        -10698, -53621, -53867, -24155, 64500, -42784, -33077, -16500, 873,
+        -52788, 14546, -38011, 36974, -39849, -34029, -94311, 83068, -50437,
+        -26169, -46746, 59185, 42259, -101379, -12943, 30089, -59086, 36271,
+        22723, -30253, -52472, -70826, -23289, 3331, -31687, 14183, -857,
+        -28627, 35246, -51284, 5636, -6933, 66539, 36654, 50927, 24783, 3457,
+        33276, 45281, 45650, -4938, -9968, -22590, 47995, 69229, 5214, -58365,
+        -17907, -14651, 18668, 18009, 12649, -11851, -13387, 20339, 52472,
+        -1087, -21458, -68647, 52295, 15849, 40608, 15323, 25164, -29368,
+        10352, -7055, 7159, 21695, -5373, -54849, 101103, -24963, -10511,
+        33227, 7659, 41042, -69588, 26718, -20515, 6441, 38135, -63, 24088,
+        -35364, -12785, -18709, 47843, 48533, -48575, 17251, -19394, 32878,
+        -9010, -9050, 504, -12407, 28076, -3429, 25324, -4210, -26119, 752,
+        -29203, 28251, -11324, -32140, -3366, -25135, 18702, -31588, -7047,
+        -24267, 49987, -14975, -33169, 37744, -7720, -9035, 16964, -2807, -421,
+        14114, -17097, -13662, 40628, -12139, -9427, 5369, 17551, -13232, -16211,
+        9804, -7422, 2677, 28635, -8280, -4906, 2908, -22558, 5604, 12459, 8756,
+        -3980, -4745, -18525, 7913, 5970, -16457, 20230, -6247, -13812, 2505,
+        11899, 1409, -15094, 22540, -18863, 137, 11123, -4516, 2290, -8594, 12150,
+        -10380, 3005, 5235, -7350, 2535, -858], ZZ)
+
+    assert dup_mul(p1, p2, ZZ) == res
+
+
+def test_dmp_mul():
+    assert dmp_mul([ZZ(5)], [ZZ(7)], 0, ZZ) == \
+        dup_mul([ZZ(5)], [ZZ(7)], ZZ)
+    assert dmp_mul([QQ(5, 7)], [QQ(3, 7)], 0, QQ) == \
+        dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ)
+
+    assert dmp_mul([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_mul([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_mul([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[]]]
+    assert dmp_mul([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(2)]]]
+    assert dmp_mul([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(2)]]]
+
+    assert dmp_mul([[[]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_mul([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[]]]
+    assert dmp_mul([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[]]]
+    assert dmp_mul([[[QQ(2, 7)]]], [[[QQ(1, 3)]]], 2, QQ) == [[[QQ(2, 21)]]]
+    assert dmp_mul([[[QQ(1, 7)]]], [[[QQ(2, 3)]]], 2, QQ) == [[[QQ(2, 21)]]]
+
+    K = FF(6)
+
+    assert dmp_mul(
+        [[K(2)], [K(1)]], [[K(3)], [K(4)]], 1, K) == [[K(5)], [K(4)]]
+
+
+def test_dup_sqr():
+    assert dup_sqr([], ZZ) == []
+    assert dup_sqr([ZZ(2)], ZZ) == [ZZ(4)]
+    assert dup_sqr([ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(4), ZZ(4)]
+
+    assert dup_sqr([], QQ) == []
+    assert dup_sqr([QQ(2, 3)], QQ) == [QQ(4, 9)]
+    assert dup_sqr([QQ(1, 3), QQ(2, 3)], QQ) == [QQ(1, 9), QQ(4, 9), QQ(4, 9)]
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+
+    assert dup_sqr(f, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
+
+    K = FF(9)
+
+    assert dup_sqr([K(3), K(4)], K) == [K(6), K(7)]
+
+
+def test_dmp_sqr():
+    assert dmp_sqr([ZZ(1), ZZ(2)], 0, ZZ) == \
+        dup_sqr([ZZ(1), ZZ(2)], ZZ)
+
+    assert dmp_sqr([[[]]], 2, ZZ) == [[[]]]
+    assert dmp_sqr([[[ZZ(2)]]], 2, ZZ) == [[[ZZ(4)]]]
+
+    assert dmp_sqr([[[]]], 2, QQ) == [[[]]]
+    assert dmp_sqr([[[QQ(2, 3)]]], 2, QQ) == [[[QQ(4, 9)]]]
+
+    K = FF(9)
+
+    assert dmp_sqr([[K(3)], [K(4)]], 1, K) == [[K(6)], [K(7)]]
+
+
+def test_dup_pow():
+    assert dup_pow([], 0, ZZ) == [ZZ(1)]
+    assert dup_pow([], 0, QQ) == [QQ(1)]
+
+    assert dup_pow([], 1, ZZ) == []
+    assert dup_pow([], 7, ZZ) == []
+
+    assert dup_pow([ZZ(1)], 0, ZZ) == [ZZ(1)]
+    assert dup_pow([ZZ(1)], 1, ZZ) == [ZZ(1)]
+    assert dup_pow([ZZ(1)], 7, ZZ) == [ZZ(1)]
+
+    assert dup_pow([ZZ(3)], 0, ZZ) == [ZZ(1)]
+    assert dup_pow([ZZ(3)], 1, ZZ) == [ZZ(3)]
+    assert dup_pow([ZZ(3)], 7, ZZ) == [ZZ(2187)]
+
+    assert dup_pow([QQ(1, 1)], 0, QQ) == [QQ(1, 1)]
+    assert dup_pow([QQ(1, 1)], 1, QQ) == [QQ(1, 1)]
+    assert dup_pow([QQ(1, 1)], 7, QQ) == [QQ(1, 1)]
+
+    assert dup_pow([QQ(3, 7)], 0, QQ) == [QQ(1, 1)]
+    assert dup_pow([QQ(3, 7)], 1, QQ) == [QQ(3, 7)]
+    assert dup_pow([QQ(3, 7)], 7, QQ) == [QQ(2187, 823543)]
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+
+    assert dup_pow(f, 0, ZZ) == dup_normal([1], ZZ)
+    assert dup_pow(f, 1, ZZ) == dup_normal([2, 0, 0, 1, 7], ZZ)
+    assert dup_pow(f, 2, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
+    assert dup_pow(f, 3, ZZ) == dup_normal(
+        [8, 0, 0, 12, 84, 0, 6, 84, 294, 1, 21, 147, 343], ZZ)
+
+
+def test_dmp_pow():
+    assert dmp_pow([[]], 0, 1, ZZ) == [[ZZ(1)]]
+    assert dmp_pow([[]], 0, 1, QQ) == [[QQ(1)]]
+
+    assert dmp_pow([[]], 1, 1, ZZ) == [[]]
+    assert dmp_pow([[]], 7, 1, ZZ) == [[]]
+
+    assert dmp_pow([[ZZ(1)]], 0, 1, ZZ) == [[ZZ(1)]]
+    assert dmp_pow([[ZZ(1)]], 1, 1, ZZ) == [[ZZ(1)]]
+    assert dmp_pow([[ZZ(1)]], 7, 1, ZZ) == [[ZZ(1)]]
+
+    assert dmp_pow([[QQ(3, 7)]], 0, 1, QQ) == [[QQ(1, 1)]]
+    assert dmp_pow([[QQ(3, 7)]], 1, 1, QQ) == [[QQ(3, 7)]]
+    assert dmp_pow([[QQ(3, 7)]], 7, 1, QQ) == [[QQ(2187, 823543)]]
+
+    f = dup_normal([2, 0, 0, 1, 7], ZZ)
+
+    assert dmp_pow(f, 2, 0, ZZ) == dup_pow(f, 2, ZZ)
+
+
+def test_dup_pdiv():
+    f = dup_normal([3, 1, 1, 5], ZZ)
+    g = dup_normal([5, -3, 1], ZZ)
+
+    q = dup_normal([15, 14], ZZ)
+    r = dup_normal([52, 111], ZZ)
+
+    assert dup_pdiv(f, g, ZZ) == (q, r)
+    assert dup_pquo(f, g, ZZ) == q
+    assert dup_prem(f, g, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, ZZ))
+
+    f = dup_normal([3, 1, 1, 5], QQ)
+    g = dup_normal([5, -3, 1], QQ)
+
+    q = dup_normal([15, 14], QQ)
+    r = dup_normal([52, 111], QQ)
+
+    assert dup_pdiv(f, g, QQ) == (q, r)
+    assert dup_pquo(f, g, QQ) == q
+    assert dup_prem(f, g, QQ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, QQ))
+
+
+def test_dmp_pdiv():
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[1], [-1, 0]], 1, ZZ)
+
+    q = dmp_normal([[1], [1, 0]], 1, ZZ)
+    r = dmp_normal([[2, 0, 0]], 1, ZZ)
+
+    assert dmp_pdiv(f, g, 1, ZZ) == (q, r)
+    assert dmp_pquo(f, g, 1, ZZ) == q
+    assert dmp_prem(f, g, 1, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ))
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[2], [-2, 0]], 1, ZZ)
+
+    q = dmp_normal([[2], [2, 0]], 1, ZZ)
+    r = dmp_normal([[8, 0, 0]], 1, ZZ)
+
+    assert dmp_pdiv(f, g, 1, ZZ) == (q, r)
+    assert dmp_pquo(f, g, 1, ZZ) == q
+    assert dmp_prem(f, g, 1, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ))
+
+
+def test_dup_rr_div():
+    raises(ZeroDivisionError, lambda: dup_rr_div([1, 2, 3], [], ZZ))
+
+    f = dup_normal([3, 1, 1, 5], ZZ)
+    g = dup_normal([5, -3, 1], ZZ)
+
+    q, r = [], f
+
+    assert dup_rr_div(f, g, ZZ) == (q, r)
+
+
+def test_dmp_rr_div():
+    raises(ZeroDivisionError, lambda: dmp_rr_div([[1, 2], [3]], [[]], 1, ZZ))
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[1], [-1, 0]], 1, ZZ)
+
+    q = dmp_normal([[1], [1, 0]], 1, ZZ)
+    r = dmp_normal([[2, 0, 0]], 1, ZZ)
+
+    assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[-1], [1, 0]], 1, ZZ)
+
+    q = dmp_normal([[-1], [-1, 0]], 1, ZZ)
+    r = dmp_normal([[2, 0, 0]], 1, ZZ)
+
+    assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
+    g = dmp_normal([[2], [-2, 0]], 1, ZZ)
+
+    q, r = [[]], f
+
+    assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
+
+
+def test_dup_ff_div():
+    raises(ZeroDivisionError, lambda: dup_ff_div([1, 2, 3], [], QQ))
+
+    f = dup_normal([3, 1, 1, 5], QQ)
+    g = dup_normal([5, -3, 1], QQ)
+
+    q = [QQ(3, 5), QQ(14, 25)]
+    r = [QQ(52, 25), QQ(111, 25)]
+
+    assert dup_ff_div(f, g, QQ) == (q, r)
+
+def test_dup_ff_div_gmpy2():
+    if GROUND_TYPES != 'gmpy2':
+        return
+
+    from gmpy2 import mpq
+    from sympy.polys.domains import GMPYRationalField
+    K = GMPYRationalField()
+
+    f = [mpq(1,3), mpq(3,2)]
+    g = [mpq(2,1)]
+    assert dmp_ff_div(f, g, 0, K) == ([mpq(1,6), mpq(3,4)], [])
+
+    f = [mpq(1,2), mpq(1,3), mpq(1,4), mpq(1,5)]
+    g = [mpq(-1,1), mpq(1,1), mpq(-1,1)]
+    assert dmp_ff_div(f, g, 0, K) == ([mpq(-1,2), mpq(-5,6)], [mpq(7,12), mpq(-19,30)])
+
+def test_dmp_ff_div():
+    raises(ZeroDivisionError, lambda: dmp_ff_div([[1, 2], [3]], [[]], 1, QQ))
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
+    g = dmp_normal([[1], [-1, 0]], 1, QQ)
+
+    q = [[QQ(1, 1)], [QQ(1, 1), QQ(0, 1)]]
+    r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
+
+    assert dmp_ff_div(f, g, 1, QQ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
+    g = dmp_normal([[-1], [1, 0]], 1, QQ)
+
+    q = [[QQ(-1, 1)], [QQ(-1, 1), QQ(0, 1)]]
+    r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
+
+    assert dmp_ff_div(f, g, 1, QQ) == (q, r)
+
+    f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
+    g = dmp_normal([[2], [-2, 0]], 1, QQ)
+
+    q = [[QQ(1, 2)], [QQ(1, 2), QQ(0, 1)]]
+    r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
+
+    assert dmp_ff_div(f, g, 1, QQ) == (q, r)
+
+
+def test_dup_div():
+    f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1]
+
+    assert dup_div(f, g, ZZ) == (q, r)
+    assert dup_quo(f, g, ZZ) == q
+    assert dup_rem(f, g, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
+
+    f, g, q, r = [5, 4, 3, 2, 1, 0], [1, 2, 0, 0, 9], [5, -6], [15, 2, -44, 54]
+
+    assert dup_div(f, g, ZZ) == (q, r)
+    assert dup_quo(f, g, ZZ) == q
+    assert dup_rem(f, g, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
+
+
+def test_dmp_div():
+    f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1]
+
+    assert dmp_div(f, g, 0, ZZ) == (q, r)
+    assert dmp_quo(f, g, 0, ZZ) == q
+    assert dmp_rem(f, g, 0, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 0, ZZ))
+
+    f, g, q, r = [[[1]]], [[[2]], [1]], [[[]]], [[[1]]]
+
+    assert dmp_div(f, g, 2, ZZ) == (q, r)
+    assert dmp_quo(f, g, 2, ZZ) == q
+    assert dmp_rem(f, g, 2, ZZ) == r
+
+    raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 2, ZZ))
+
+
+def test_dup_max_norm():
+    assert dup_max_norm([], ZZ) == 0
+    assert dup_max_norm([1], ZZ) == 1
+
+    assert dup_max_norm([1, 4, 2, 3], ZZ) == 4
+
+
+def test_dmp_max_norm():
+    assert dmp_max_norm([[[]]], 2, ZZ) == 0
+    assert dmp_max_norm([[[1]]], 2, ZZ) == 1
+
+    assert dmp_max_norm(f_0, 2, ZZ) == 6
+
+
+def test_dup_l1_norm():
+    assert dup_l1_norm([], ZZ) == 0
+    assert dup_l1_norm([1], ZZ) == 1
+    assert dup_l1_norm([1, 4, 2, 3], ZZ) == 10
+
+
+def test_dmp_l1_norm():
+    assert dmp_l1_norm([[[]]], 2, ZZ) == 0
+    assert dmp_l1_norm([[[1]]], 2, ZZ) == 1
+
+    assert dmp_l1_norm(f_0, 2, ZZ) == 31
+
+
+def test_dup_l2_norm_squared():
+    assert dup_l2_norm_squared([], ZZ) == 0
+    assert dup_l2_norm_squared([1], ZZ) == 1
+    assert dup_l2_norm_squared([1, 4, 2, 3], ZZ) == 30
+
+
+def test_dmp_l2_norm_squared():
+    assert dmp_l2_norm_squared([[[]]], 2, ZZ) == 0
+    assert dmp_l2_norm_squared([[[1]]], 2, ZZ) == 1
+    assert dmp_l2_norm_squared(f_0, 2, ZZ) == 111
+
+
+def test_dup_expand():
+    assert dup_expand((), ZZ) == [1]
+    assert dup_expand(([1, 2, 3], [1, 2], [7, 5, 4, 3]), ZZ) == \
+        dup_mul([1, 2, 3], dup_mul([1, 2], [7, 5, 4, 3], ZZ), ZZ)
+
+
+def test_dmp_expand():
+    assert dmp_expand((), 1, ZZ) == [[1]]
+    assert dmp_expand(([[1], [2], [3]], [[1], [2]], [[7], [5], [4], [3]]), 1, ZZ) == \
+        dmp_mul([[1], [2], [3]], dmp_mul([[1], [2]], [[7], [5], [
+                4], [3]], 1, ZZ), 1, ZZ)

rl/Lib/site-packages/sympy/polys/tests/test_densebasic.py

@@ -0,0 +1,730 @@
+"""Tests for dense recursive polynomials' basic tools. """
+
+from sympy.polys.densebasic import (
+    ninf,
+    dup_LC, dmp_LC,
+    dup_TC, dmp_TC,
+    dmp_ground_LC, dmp_ground_TC,
+    dmp_true_LT,
+    dup_degree, dmp_degree,
+    dmp_degree_in, dmp_degree_list,
+    dup_strip, dmp_strip,
+    dmp_validate,
+    dup_reverse,
+    dup_copy, dmp_copy,
+    dup_normal, dmp_normal,
+    dup_convert, dmp_convert,
+    dup_from_sympy, dmp_from_sympy,
+    dup_nth, dmp_nth, dmp_ground_nth,
+    dmp_zero_p, dmp_zero,
+    dmp_one_p, dmp_one,
+    dmp_ground_p, dmp_ground,
+    dmp_negative_p, dmp_positive_p,
+    dmp_zeros, dmp_grounds,
+    dup_from_dict, dup_from_raw_dict,
+    dup_to_dict, dup_to_raw_dict,
+    dmp_from_dict, dmp_to_dict,
+    dmp_swap, dmp_permute,
+    dmp_nest, dmp_raise,
+    dup_deflate, dmp_deflate,
+    dup_multi_deflate, dmp_multi_deflate,
+    dup_inflate, dmp_inflate,
+    dmp_exclude, dmp_include,
+    dmp_inject, dmp_eject,
+    dup_terms_gcd, dmp_terms_gcd,
+    dmp_list_terms, dmp_apply_pairs,
+    dup_slice,
+    dup_random,
+)
+
+from sympy.polys.specialpolys import f_polys
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.rings import ring
+
+from sympy.core.singleton import S
+from sympy.testing.pytest import raises
+
+from sympy.core.numbers import oo
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
+
+def test_dup_LC():
+    assert dup_LC([], ZZ) == 0
+    assert dup_LC([2, 3, 4, 5], ZZ) == 2
+
+
+def test_dup_TC():
+    assert dup_TC([], ZZ) == 0
+    assert dup_TC([2, 3, 4, 5], ZZ) == 5
+
+
+def test_dmp_LC():
+    assert dmp_LC([[]], ZZ) == []
+    assert dmp_LC([[2, 3, 4], [5]], ZZ) == [2, 3, 4]
+    assert dmp_LC([[[]]], ZZ) == [[]]
+    assert dmp_LC([[[2], [3, 4]], [[5]]], ZZ) == [[2], [3, 4]]
+
+
+def test_dmp_TC():
+    assert dmp_TC([[]], ZZ) == []
+    assert dmp_TC([[2, 3, 4], [5]], ZZ) == [5]
+    assert dmp_TC([[[]]], ZZ) == [[]]
+    assert dmp_TC([[[2], [3, 4]], [[5]]], ZZ) == [[5]]
+
+
+def test_dmp_ground_LC():
+    assert dmp_ground_LC([[]], 1, ZZ) == 0
+    assert dmp_ground_LC([[2, 3, 4], [5]], 1, ZZ) == 2
+    assert dmp_ground_LC([[[]]], 2, ZZ) == 0
+    assert dmp_ground_LC([[[2], [3, 4]], [[5]]], 2, ZZ) == 2
+
+
+def test_dmp_ground_TC():
+    assert dmp_ground_TC([[]], 1, ZZ) == 0
+    assert dmp_ground_TC([[2, 3, 4], [5]], 1, ZZ) == 5
+    assert dmp_ground_TC([[[]]], 2, ZZ) == 0
+    assert dmp_ground_TC([[[2], [3, 4]], [[5]]], 2, ZZ) == 5
+
+
+def test_dmp_true_LT():
+    assert dmp_true_LT([[]], 1, ZZ) == ((0, 0), 0)
+    assert dmp_true_LT([[7]], 1, ZZ) == ((0, 0), 7)
+
+    assert dmp_true_LT([[1, 0]], 1, ZZ) == ((0, 1), 1)
+    assert dmp_true_LT([[1], []], 1, ZZ) == ((1, 0), 1)
+    assert dmp_true_LT([[1, 0], []], 1, ZZ) == ((1, 1), 1)
+
+
+def test_dup_degree():
+    assert ninf == float('-inf')
+    assert dup_degree([]) is ninf
+    assert dup_degree([1]) == 0
+    assert dup_degree([1, 0]) == 1
+    assert dup_degree([1, 0, 0, 0, 1]) == 4
+
+
+def test_dmp_degree():
+    assert dmp_degree([[]], 1) is ninf
+    assert dmp_degree([[[]]], 2) is ninf
+
+    assert dmp_degree([[1]], 1) == 0
+    assert dmp_degree([[2], [1]], 1) == 1
+
+
+def test_dmp_degree_in():
+    assert dmp_degree_in([[[]]], 0, 2) is ninf
+    assert dmp_degree_in([[[]]], 1, 2) is ninf
+    assert dmp_degree_in([[[]]], 2, 2) is ninf
+
+    assert dmp_degree_in([[[1]]], 0, 2) == 0
+    assert dmp_degree_in([[[1]]], 1, 2) == 0
+    assert dmp_degree_in([[[1]]], 2, 2) == 0
+
+    assert dmp_degree_in(f_4, 0, 2) == 9
+    assert dmp_degree_in(f_4, 1, 2) == 12
+    assert dmp_degree_in(f_4, 2, 2) == 8
+
+    assert dmp_degree_in(f_6, 0, 2) == 4
+    assert dmp_degree_in(f_6, 1, 2) == 4
+    assert dmp_degree_in(f_6, 2, 2) == 6
+    assert dmp_degree_in(f_6, 3, 3) == 3
+
+    raises(IndexError, lambda: dmp_degree_in([[1]], -5, 1))
+
+
+def test_dmp_degree_list():
+    assert dmp_degree_list([[[[ ]]]], 3) == (-oo, -oo, -oo, -oo)
+    assert dmp_degree_list([[[[1]]]], 3) == ( 0, 0, 0, 0)
+
+    assert dmp_degree_list(f_0, 2) == (2, 2, 2)
+    assert dmp_degree_list(f_1, 2) == (3, 3, 3)
+    assert dmp_degree_list(f_2, 2) == (5, 3, 3)
+    assert dmp_degree_list(f_3, 2) == (5, 4, 7)
+    assert dmp_degree_list(f_4, 2) == (9, 12, 8)
+    assert dmp_degree_list(f_5, 2) == (3, 3, 3)
+    assert dmp_degree_list(f_6, 3) == (4, 4, 6, 3)
+
+
+def test_dup_strip():
+    assert dup_strip([]) == []
+    assert dup_strip([0]) == []
+    assert dup_strip([0, 0, 0]) == []
+
+    assert dup_strip([1]) == [1]
+    assert dup_strip([0, 1]) == [1]
+    assert dup_strip([0, 0, 0, 1]) == [1]
+
+    assert dup_strip([1, 2, 0]) == [1, 2, 0]
+    assert dup_strip([0, 1, 2, 0]) == [1, 2, 0]
+    assert dup_strip([0, 0, 0, 1, 2, 0]) == [1, 2, 0]
+
+
+def test_dmp_strip():
+    assert dmp_strip([0, 1, 0], 0) == [1, 0]
+
+    assert dmp_strip([[]], 1) == [[]]
+    assert dmp_strip([[], []], 1) == [[]]
+    assert dmp_strip([[], [], []], 1) == [[]]
+
+    assert dmp_strip([[[]]], 2) == [[[]]]
+    assert dmp_strip([[[]], [[]]], 2) == [[[]]]
+    assert dmp_strip([[[]], [[]], [[]]], 2) == [[[]]]
+
+    assert dmp_strip([[[1]]], 2) == [[[1]]]
+    assert dmp_strip([[[]], [[1]]], 2) == [[[1]]]
+    assert dmp_strip([[[]], [[1]], [[]]], 2) == [[[1]], [[]]]
+
+
+def test_dmp_validate():
+    assert dmp_validate([]) == ([], 0)
+    assert dmp_validate([0, 0, 0, 1, 0]) == ([1, 0], 0)
+
+    assert dmp_validate([[[]]]) == ([[[]]], 2)
+    assert dmp_validate([[0], [], [0], [1], [0]]) == ([[1], []], 1)
+
+    raises(ValueError, lambda: dmp_validate([[0], 0, [0], [1], [0]]))
+
+
+def test_dup_reverse():
+    assert dup_reverse([1, 2, 0, 3]) == [3, 0, 2, 1]
+    assert dup_reverse([1, 2, 3, 0]) == [3, 2, 1]
+
+
+def test_dup_copy():
+    f = [ZZ(1), ZZ(0), ZZ(2)]
+    g = dup_copy(f)
+
+    g[0], g[2] = ZZ(7), ZZ(0)
+
+    assert f != g
+
+
+def test_dmp_copy():
+    f = [[ZZ(1)], [ZZ(2), ZZ(0)]]
+    g = dmp_copy(f, 1)
+
+    g[0][0], g[1][1] = ZZ(7), ZZ(1)
+
+    assert f != g
+
+
+def test_dup_normal():
+    assert dup_normal([0, 0, 2, 1, 0, 11, 0], ZZ) == \
+        [ZZ(2), ZZ(1), ZZ(0), ZZ(11), ZZ(0)]
+
+
+def test_dmp_normal():
+    assert dmp_normal([[0], [], [0, 2, 1], [0], [11], []], 1, ZZ) == \
+        [[ZZ(2), ZZ(1)], [], [ZZ(11)], []]
+
+
+def test_dup_convert():
+    K0, K1 = ZZ['x'], ZZ
+
+    f = [K0(1), K0(2), K0(0), K0(3)]
+
+    assert dup_convert(f, K0, K1) == \
+        [ZZ(1), ZZ(2), ZZ(0), ZZ(3)]
+
+
+def test_dmp_convert():
+    K0, K1 = ZZ['x'], ZZ
+
+    f = [[K0(1)], [K0(2)], [], [K0(3)]]
+
+    assert dmp_convert(f, 1, K0, K1) == \
+        [[ZZ(1)], [ZZ(2)], [], [ZZ(3)]]
+
+
+def test_dup_from_sympy():
+    assert dup_from_sympy([S.One, S(2)], ZZ) == \
+        [ZZ(1), ZZ(2)]
+    assert dup_from_sympy([S.Half, S(3)], QQ) == \
+        [QQ(1, 2), QQ(3, 1)]
+
+
+def test_dmp_from_sympy():
+    assert dmp_from_sympy([[S.One, S(2)], [S.Zero]], 1, ZZ) == \
+        [[ZZ(1), ZZ(2)], []]
+    assert dmp_from_sympy([[S.Half, S(2)]], 1, QQ) == \
+        [[QQ(1, 2), QQ(2, 1)]]
+
+
+def test_dup_nth():
+    assert dup_nth([1, 2, 3], 0, ZZ) == 3
+    assert dup_nth([1, 2, 3], 1, ZZ) == 2
+    assert dup_nth([1, 2, 3], 2, ZZ) == 1
+
+    assert dup_nth([1, 2, 3], 9, ZZ) == 0
+
+    raises(IndexError, lambda: dup_nth([3, 4, 5], -1, ZZ))
+
+
+def test_dmp_nth():
+    assert dmp_nth([[1], [2], [3]], 0, 1, ZZ) == [3]
+    assert dmp_nth([[1], [2], [3]], 1, 1, ZZ) == [2]
+    assert dmp_nth([[1], [2], [3]], 2, 1, ZZ) == [1]
+
+    assert dmp_nth([[1], [2], [3]], 9, 1, ZZ) == []
+
+    raises(IndexError, lambda: dmp_nth([[3], [4], [5]], -1, 1, ZZ))
+
+
+def test_dmp_ground_nth():
+    assert dmp_ground_nth([[]], (0, 0), 1, ZZ) == 0
+    assert dmp_ground_nth([[1], [2], [3]], (0, 0), 1, ZZ) == 3
+    assert dmp_ground_nth([[1], [2], [3]], (1, 0), 1, ZZ) == 2
+    assert dmp_ground_nth([[1], [2], [3]], (2, 0), 1, ZZ) == 1
+
+    assert dmp_ground_nth([[1], [2], [3]], (2, 1), 1, ZZ) == 0
+    assert dmp_ground_nth([[1], [2], [3]], (3, 0), 1, ZZ) == 0
+
+    raises(IndexError, lambda: dmp_ground_nth([[3], [4], [5]], (2, -1), 1, ZZ))
+
+
+def test_dmp_zero_p():
+    assert dmp_zero_p([], 0) is True
+    assert dmp_zero_p([[]], 1) is True
+
+    assert dmp_zero_p([[[]]], 2) is True
+    assert dmp_zero_p([[[1]]], 2) is False
+
+
+def test_dmp_zero():
+    assert dmp_zero(0) == []
+    assert dmp_zero(2) == [[[]]]
+
+
+def test_dmp_one_p():
+    assert dmp_one_p([1], 0, ZZ) is True
+    assert dmp_one_p([[1]], 1, ZZ) is True
+    assert dmp_one_p([[[1]]], 2, ZZ) is True
+    assert dmp_one_p([[[12]]], 2, ZZ) is False
+
+
+def test_dmp_one():
+    assert dmp_one(0, ZZ) == [ZZ(1)]
+    assert dmp_one(2, ZZ) == [[[ZZ(1)]]]
+
+
+def test_dmp_ground_p():
+    assert dmp_ground_p([], 0, 0) is True
+    assert dmp_ground_p([[]], 0, 1) is True
+    assert dmp_ground_p([[]], 1, 1) is False
+
+    assert dmp_ground_p([[ZZ(1)]], 1, 1) is True
+    assert dmp_ground_p([[[ZZ(2)]]], 2, 2) is True
+
+    assert dmp_ground_p([[[ZZ(2)]]], 3, 2) is False
+    assert dmp_ground_p([[[ZZ(3)], []]], 3, 2) is False
+
+    assert dmp_ground_p([], None, 0) is True
+    assert dmp_ground_p([[]], None, 1) is True
+
+    assert dmp_ground_p([ZZ(1)], None, 0) is True
+    assert dmp_ground_p([[[ZZ(1)]]], None, 2) is True
+
+    assert dmp_ground_p([[[ZZ(3)], []]], None, 2) is False
+
+
+def test_dmp_ground():
+    assert dmp_ground(ZZ(0), 2) == [[[]]]
+
+    assert dmp_ground(ZZ(7), -1) == ZZ(7)
+    assert dmp_ground(ZZ(7), 0) == [ZZ(7)]
+    assert dmp_ground(ZZ(7), 2) == [[[ZZ(7)]]]
+
+
+def test_dmp_zeros():
+    assert dmp_zeros(4, 0, ZZ) == [[], [], [], []]
+
+    assert dmp_zeros(0, 2, ZZ) == []
+    assert dmp_zeros(1, 2, ZZ) == [[[[]]]]
+    assert dmp_zeros(2, 2, ZZ) == [[[[]]], [[[]]]]
+    assert dmp_zeros(3, 2, ZZ) == [[[[]]], [[[]]], [[[]]]]
+
+    assert dmp_zeros(3, -1, ZZ) == [0, 0, 0]
+
+
+def test_dmp_grounds():
+    assert dmp_grounds(ZZ(7), 0, 2) == []
+
+    assert dmp_grounds(ZZ(7), 1, 2) == [[[[7]]]]
+    assert dmp_grounds(ZZ(7), 2, 2) == [[[[7]]], [[[7]]]]
+    assert dmp_grounds(ZZ(7), 3, 2) == [[[[7]]], [[[7]]], [[[7]]]]
+
+    assert dmp_grounds(ZZ(7), 3, -1) == [7, 7, 7]
+
+
+def test_dmp_negative_p():
+    assert dmp_negative_p([[[]]], 2, ZZ) is False
+    assert dmp_negative_p([[[1], [2]]], 2, ZZ) is False
+    assert dmp_negative_p([[[-1], [2]]], 2, ZZ) is True
+
+
+def test_dmp_positive_p():
+    assert dmp_positive_p([[[]]], 2, ZZ) is False
+    assert dmp_positive_p([[[1], [2]]], 2, ZZ) is True
+    assert dmp_positive_p([[[-1], [2]]], 2, ZZ) is False
+
+
+def test_dup_from_to_dict():
+    assert dup_from_raw_dict({}, ZZ) == []
+    assert dup_from_dict({}, ZZ) == []
+
+    assert dup_to_raw_dict([]) == {}
+    assert dup_to_dict([]) == {}
+
+    assert dup_to_raw_dict([], ZZ, zero=True) == {0: ZZ(0)}
+    assert dup_to_dict([], ZZ, zero=True) == {(0,): ZZ(0)}
+
+    f = [3, 0, 0, 2, 0, 0, 0, 0, 8]
+    g = {8: 3, 5: 2, 0: 8}
+    h = {(8,): 3, (5,): 2, (0,): 8}
+
+    assert dup_from_raw_dict(g, ZZ) == f
+    assert dup_from_dict(h, ZZ) == f
+
+    assert dup_to_raw_dict(f) == g
+    assert dup_to_dict(f) == h
+
+    R, x,y = ring("x,y", ZZ)
+    K = R.to_domain()
+
+    f = [R(3), R(0), R(2), R(0), R(0), R(8)]
+    g = {5: R(3), 3: R(2), 0: R(8)}
+    h = {(5,): R(3), (3,): R(2), (0,): R(8)}
+
+    assert dup_from_raw_dict(g, K) == f
+    assert dup_from_dict(h, K) == f
+
+    assert dup_to_raw_dict(f) == g
+    assert dup_to_dict(f) == h
+
+
+def test_dmp_from_to_dict():
+    assert dmp_from_dict({}, 1, ZZ) == [[]]
+    assert dmp_to_dict([[]], 1) == {}
+
+    assert dmp_to_dict([], 0, ZZ, zero=True) == {(0,): ZZ(0)}
+    assert dmp_to_dict([[]], 1, ZZ, zero=True) == {(0, 0): ZZ(0)}
+
+    f = [[3], [], [], [2], [], [], [], [], [8]]
+    g = {(8, 0): 3, (5, 0): 2, (0, 0): 8}
+
+    assert dmp_from_dict(g, 1, ZZ) == f
+    assert dmp_to_dict(f, 1) == g
+
+
+def test_dmp_swap():
+    f = dmp_normal([[1, 0, 0], [], [1, 0], [], [1]], 1, ZZ)
+    g = dmp_normal([[1, 0, 0, 0, 0], [1, 0, 0], [1]], 1, ZZ)
+
+    assert dmp_swap(f, 1, 1, 1, ZZ) == f
+
+    assert dmp_swap(f, 0, 1, 1, ZZ) == g
+    assert dmp_swap(g, 0, 1, 1, ZZ) == f
+
+    raises(IndexError, lambda: dmp_swap(f, -1, -7, 1, ZZ))
+
+
+def test_dmp_permute():
+    f = dmp_normal([[1, 0, 0], [], [1, 0], [], [1]], 1, ZZ)
+    g = dmp_normal([[1, 0, 0, 0, 0], [1, 0, 0], [1]], 1, ZZ)
+
+    assert dmp_permute(f, [0, 1], 1, ZZ) == f
+    assert dmp_permute(g, [0, 1], 1, ZZ) == g
+
+    assert dmp_permute(f, [1, 0], 1, ZZ) == g
+    assert dmp_permute(g, [1, 0], 1, ZZ) == f
+
+
+def test_dmp_nest():
+    assert dmp_nest(ZZ(1), 2, ZZ) == [[[1]]]
+
+    assert dmp_nest([[1]], 0, ZZ) == [[1]]
+    assert dmp_nest([[1]], 1, ZZ) == [[[1]]]
+    assert dmp_nest([[1]], 2, ZZ) == [[[[1]]]]
+
+
+def test_dmp_raise():
+    assert dmp_raise([], 2, 0, ZZ) == [[[]]]
+    assert dmp_raise([[1]], 0, 1, ZZ) == [[1]]
+
+    assert dmp_raise([[1, 2, 3], [], [2, 3]], 2, 1, ZZ) == \
+        [[[[1]], [[2]], [[3]]], [[[]]], [[[2]], [[3]]]]
+
+
+def test_dup_deflate():
+    assert dup_deflate([], ZZ) == (1, [])
+    assert dup_deflate([2], ZZ) == (1, [2])
+    assert dup_deflate([1, 2, 3], ZZ) == (1, [1, 2, 3])
+    assert dup_deflate([1, 0, 2, 0, 3], ZZ) == (2, [1, 2, 3])
+
+    assert dup_deflate(dup_from_raw_dict({7: 1, 1: 1}, ZZ), ZZ) == \
+        (1, [1, 0, 0, 0, 0, 0, 1, 0])
+    assert dup_deflate(dup_from_raw_dict({7: 1, 0: 1}, ZZ), ZZ) == \
+        (7, [1, 1])
+    assert dup_deflate(dup_from_raw_dict({7: 1, 3: 1}, ZZ), ZZ) == \
+        (1, [1, 0, 0, 0, 1, 0, 0, 0])
+
+    assert dup_deflate(dup_from_raw_dict({7: 1, 4: 1}, ZZ), ZZ) == \
+        (1, [1, 0, 0, 1, 0, 0, 0, 0])
+    assert dup_deflate(dup_from_raw_dict({8: 1, 4: 1}, ZZ), ZZ) == \
+        (4, [1, 1, 0])
+
+    assert dup_deflate(dup_from_raw_dict({8: 1}, ZZ), ZZ) == \
+        (8, [1, 0])
+    assert dup_deflate(dup_from_raw_dict({7: 1}, ZZ), ZZ) == \
+        (7, [1, 0])
+    assert dup_deflate(dup_from_raw_dict({1: 1}, ZZ), ZZ) == \
+        (1, [1, 0])
+
+
+def test_dmp_deflate():
+    assert dmp_deflate([[]], 1, ZZ) == ((1, 1), [[]])
+    assert dmp_deflate([[2]], 1, ZZ) == ((1, 1), [[2]])
+
+    f = [[1, 0, 0], [], [1, 0], [], [1]]
+
+    assert dmp_deflate(f, 1, ZZ) == ((2, 1), [[1, 0, 0], [1, 0], [1]])
+
+
+def test_dup_multi_deflate():
+    assert dup_multi_deflate(([2],), ZZ) == (1, ([2],))
+    assert dup_multi_deflate(([], []), ZZ) == (1, ([], []))
+
+    assert dup_multi_deflate(([1, 2, 3],), ZZ) == (1, ([1, 2, 3],))
+    assert dup_multi_deflate(([1, 0, 2, 0, 3],), ZZ) == (2, ([1, 2, 3],))
+
+    assert dup_multi_deflate(([1, 0, 2, 0, 3], [2, 0, 0]), ZZ) == \
+        (2, ([1, 2, 3], [2, 0]))
+    assert dup_multi_deflate(([1, 0, 2, 0, 3], [2, 1, 0]), ZZ) == \
+        (1, ([1, 0, 2, 0, 3], [2, 1, 0]))
+
+
+def test_dmp_multi_deflate():
+    assert dmp_multi_deflate(([[]],), 1, ZZ) == \
+        ((1, 1), ([[]],))
+    assert dmp_multi_deflate(([[]], [[]]), 1, ZZ) == \
+        ((1, 1), ([[]], [[]]))
+
+    assert dmp_multi_deflate(([[1]], [[]]), 1, ZZ) == \
+        ((1, 1), ([[1]], [[]]))
+    assert dmp_multi_deflate(([[1]], [[2]]), 1, ZZ) == \
+        ((1, 1), ([[1]], [[2]]))
+    assert dmp_multi_deflate(([[1]], [[2, 0]]), 1, ZZ) == \
+        ((1, 1), ([[1]], [[2, 0]]))
+
+    assert dmp_multi_deflate(([[2, 0]], [[2, 0]]), 1, ZZ) == \
+        ((1, 1), ([[2, 0]], [[2, 0]]))
+
+    assert dmp_multi_deflate(
+        ([[2]], [[2, 0, 0]]), 1, ZZ) == ((1, 2), ([[2]], [[2, 0]]))
+    assert dmp_multi_deflate(
+        ([[2, 0, 0]], [[2, 0, 0]]), 1, ZZ) == ((1, 2), ([[2, 0]], [[2, 0]]))
+
+    assert dmp_multi_deflate(([2, 0, 0], [1, 0, 4, 0, 1]), 0, ZZ) == \
+        ((2,), ([2, 0], [1, 4, 1]))
+
+    f = [[1, 0, 0], [], [1, 0], [], [1]]
+    g = [[1, 0, 1, 0], [], [1]]
+
+    assert dmp_multi_deflate((f,), 1, ZZ) == \
+        ((2, 1), ([[1, 0, 0], [1, 0], [1]],))
+
+    assert dmp_multi_deflate((f, g), 1, ZZ) == \
+        ((2, 1), ([[1, 0, 0], [1, 0], [1]],
+                  [[1, 0, 1, 0], [1]]))
+
+
+def test_dup_inflate():
+    assert dup_inflate([], 17, ZZ) == []
+
+    assert dup_inflate([1, 2, 3], 1, ZZ) == [1, 2, 3]
+    assert dup_inflate([1, 2, 3], 2, ZZ) == [1, 0, 2, 0, 3]
+    assert dup_inflate([1, 2, 3], 3, ZZ) == [1, 0, 0, 2, 0, 0, 3]
+    assert dup_inflate([1, 2, 3], 4, ZZ) == [1, 0, 0, 0, 2, 0, 0, 0, 3]
+
+    raises(IndexError, lambda: dup_inflate([1, 2, 3], 0, ZZ))
+
+
+def test_dmp_inflate():
+    assert dmp_inflate([1], (3,), 0, ZZ) == [1]
+
+    assert dmp_inflate([[]], (3, 7), 1, ZZ) == [[]]
+    assert dmp_inflate([[2]], (1, 2), 1, ZZ) == [[2]]
+
+    assert dmp_inflate([[2, 0]], (1, 1), 1, ZZ) == [[2, 0]]
+    assert dmp_inflate([[2, 0]], (1, 2), 1, ZZ) == [[2, 0, 0]]
+    assert dmp_inflate([[2, 0]], (1, 3), 1, ZZ) == [[2, 0, 0, 0]]
+
+    assert dmp_inflate([[1, 0, 0], [1], [1, 0]], (2, 1), 1, ZZ) == \
+        [[1, 0, 0], [], [1], [], [1, 0]]
+
+    raises(IndexError, lambda: dmp_inflate([[]], (-3, 7), 1, ZZ))
+
+
+def test_dmp_exclude():
+    assert dmp_exclude([[[]]], 2, ZZ) == ([], [[[]]], 2)
+    assert dmp_exclude([[[7]]], 2, ZZ) == ([], [[[7]]], 2)
+
+    assert dmp_exclude([1, 2, 3], 0, ZZ) == ([], [1, 2, 3], 0)
+    assert dmp_exclude([[1], [2, 3]], 1, ZZ) == ([], [[1], [2, 3]], 1)
+
+    assert dmp_exclude([[1, 2, 3]], 1, ZZ) == ([0], [1, 2, 3], 0)
+    assert dmp_exclude([[1], [2], [3]], 1, ZZ) == ([1], [1, 2, 3], 0)
+
+    assert dmp_exclude([[[1, 2, 3]]], 2, ZZ) == ([0, 1], [1, 2, 3], 0)
+    assert dmp_exclude([[[1]], [[2]], [[3]]], 2, ZZ) == ([1, 2], [1, 2, 3], 0)
+
+
+def test_dmp_include():
+    assert dmp_include([1, 2, 3], [], 0, ZZ) == [1, 2, 3]
+
+    assert dmp_include([1, 2, 3], [0], 0, ZZ) == [[1, 2, 3]]
+    assert dmp_include([1, 2, 3], [1], 0, ZZ) == [[1], [2], [3]]
+
+    assert dmp_include([1, 2, 3], [0, 1], 0, ZZ) == [[[1, 2, 3]]]
+    assert dmp_include([1, 2, 3], [1, 2], 0, ZZ) == [[[1]], [[2]], [[3]]]
+
+
+def test_dmp_inject():
+    R, x,y = ring("x,y", ZZ)
+    K = R.to_domain()
+
+    assert dmp_inject([], 0, K) == ([[[]]], 2)
+    assert dmp_inject([[]], 1, K) == ([[[[]]]], 3)
+
+    assert dmp_inject([R(1)], 0, K) == ([[[1]]], 2)
+    assert dmp_inject([[R(1)]], 1, K) == ([[[[1]]]], 3)
+
+    assert dmp_inject([R(1), 2*x + 3*y + 4], 0, K) == ([[[1]], [[2], [3, 4]]], 2)
+
+    f = [3*x**2 + 7*x*y + 5*y**2, 2*x, R(0), x*y**2 + 11]
+    g = [[[3], [7, 0], [5, 0, 0]], [[2], []], [[]], [[1, 0, 0], [11]]]
+
+    assert dmp_inject(f, 0, K) == (g, 2)
+
+
+def test_dmp_eject():
+    R, x,y = ring("x,y", ZZ)
+    K = R.to_domain()
+
+    assert dmp_eject([[[]]], 2, K) == []
+    assert dmp_eject([[[[]]]], 3, K) == [[]]
+
+    assert dmp_eject([[[1]]], 2, K) == [R(1)]
+    assert dmp_eject([[[[1]]]], 3, K) == [[R(1)]]
+
+    assert dmp_eject([[[1]], [[2], [3, 4]]], 2, K) == [R(1), 2*x + 3*y + 4]
+
+    f = [3*x**2 + 7*x*y + 5*y**2, 2*x, R(0), x*y**2 + 11]
+    g = [[[3], [7, 0], [5, 0, 0]], [[2], []], [[]], [[1, 0, 0], [11]]]
+
+    assert dmp_eject(g, 2, K) == f
+
+
+def test_dup_terms_gcd():
+    assert dup_terms_gcd([], ZZ) == (0, [])
+    assert dup_terms_gcd([1, 0, 1], ZZ) == (0, [1, 0, 1])
+    assert dup_terms_gcd([1, 0, 1, 0], ZZ) == (1, [1, 0, 1])
+
+
+def test_dmp_terms_gcd():
+    assert dmp_terms_gcd([[]], 1, ZZ) == ((0, 0), [[]])
+
+    assert dmp_terms_gcd([1, 0, 1, 0], 0, ZZ) == ((1,), [1, 0, 1])
+    assert dmp_terms_gcd([[1], [], [1], []], 1, ZZ) == ((1, 0), [[1], [], [1]])
+
+    assert dmp_terms_gcd(
+        [[1, 0], [], [1]], 1, ZZ) == ((0, 0), [[1, 0], [], [1]])
+    assert dmp_terms_gcd(
+        [[1, 0], [1, 0, 0], [], []], 1, ZZ) == ((2, 1), [[1], [1, 0]])
+
+
+def test_dmp_list_terms():
+    assert dmp_list_terms([[[]]], 2, ZZ) == [((0, 0, 0), 0)]
+    assert dmp_list_terms([[[1]]], 2, ZZ) == [((0, 0, 0), 1)]
+
+    assert dmp_list_terms([1, 2, 4, 3, 5], 0, ZZ) == \
+        [((4,), 1), ((3,), 2), ((2,), 4), ((1,), 3), ((0,), 5)]
+
+    assert dmp_list_terms([[1], [2, 4], [3, 5, 0]], 1, ZZ) == \
+        [((2, 0), 1), ((1, 1), 2), ((1, 0), 4), ((0, 2), 3), ((0, 1), 5)]
+
+    f = [[2, 0, 0, 0], [1, 0, 0], []]
+
+    assert dmp_list_terms(f, 1, ZZ, order='lex') == [((2, 3), 2), ((1, 2), 1)]
+    assert dmp_list_terms(
+        f, 1, ZZ, order='grlex') == [((2, 3), 2), ((1, 2), 1)]
+
+    f = [[2, 0, 0, 0], [1, 0, 0, 0, 0, 0], []]
+
+    assert dmp_list_terms(f, 1, ZZ, order='lex') == [((2, 3), 2), ((1, 5), 1)]
+    assert dmp_list_terms(
+        f, 1, ZZ, order='grlex') == [((1, 5), 1), ((2, 3), 2)]
+
+
+def test_dmp_apply_pairs():
+    h = lambda a, b: a*b
+
+    assert dmp_apply_pairs([1, 2, 3], [4, 5, 6], h, [], 0, ZZ) == [4, 10, 18]
+
+    assert dmp_apply_pairs([2, 3], [4, 5, 6], h, [], 0, ZZ) == [10, 18]
+    assert dmp_apply_pairs([1, 2, 3], [5, 6], h, [], 0, ZZ) == [10, 18]
+
+    assert dmp_apply_pairs(
+        [[1, 2], [3]], [[4, 5], [6]], h, [], 1, ZZ) == [[4, 10], [18]]
+
+    assert dmp_apply_pairs(
+        [[1, 2], [3]], [[4], [5, 6]], h, [], 1, ZZ) == [[8], [18]]
+    assert dmp_apply_pairs(
+        [[1], [2, 3]], [[4, 5], [6]], h, [], 1, ZZ) == [[5], [18]]
+
+
+def test_dup_slice():
+    f = [1, 2, 3, 4]
+
+    assert dup_slice(f, 0, 0, ZZ) == []
+    assert dup_slice(f, 0, 1, ZZ) == [4]
+    assert dup_slice(f, 0, 2, ZZ) == [3, 4]
+    assert dup_slice(f, 0, 3, ZZ) == [2, 3, 4]
+    assert dup_slice(f, 0, 4, ZZ) == [1, 2, 3, 4]
+
+    assert dup_slice(f, 0, 4, ZZ) == f
+    assert dup_slice(f, 0, 9, ZZ) == f
+
+    assert dup_slice(f, 1, 0, ZZ) == []
+    assert dup_slice(f, 1, 1, ZZ) == []
+    assert dup_slice(f, 1, 2, ZZ) == [3, 0]
+    assert dup_slice(f, 1, 3, ZZ) == [2, 3, 0]
+    assert dup_slice(f, 1, 4, ZZ) == [1, 2, 3, 0]
+
+    assert dup_slice([1, 2], 0, 3, ZZ) == [1, 2]
+
+    g = [1, 0, 0, 2]
+
+    assert dup_slice(g, 0, 3, ZZ) == [2]
+
+
+def test_dup_random():
+    f = dup_random(0, -10, 10, ZZ)
+
+    assert dup_degree(f) == 0
+    assert all(-10 <= c <= 10 for c in f)
+
+    f = dup_random(1, -20, 20, ZZ)
+
+    assert dup_degree(f) == 1
+    assert all(-20 <= c <= 20 for c in f)
+
+    f = dup_random(2, -30, 30, ZZ)
+
+    assert dup_degree(f) == 2
+    assert all(-30 <= c <= 30 for c in f)
+
+    f = dup_random(3, -40, 40, ZZ)
+
+    assert dup_degree(f) == 3
+    assert all(-40 <= c <= 40 for c in f)

rl/Lib/site-packages/sympy/polys/tests/test_densetools.py

@@ -0,0 +1,715 @@
+"""Tests for dense recursive polynomials' tools. """
+
+from sympy.polys.densebasic import (
+    dup_normal, dmp_normal,
+    dup_from_raw_dict,
+    dmp_convert, dmp_swap,
+)
+
+from sympy.polys.densearith import dmp_mul_ground
+
+from sympy.polys.densetools import (
+    dup_clear_denoms, dmp_clear_denoms,
+    dup_integrate, dmp_integrate, dmp_integrate_in,
+    dup_diff, dmp_diff, dmp_diff_in,
+    dup_eval, dmp_eval, dmp_eval_in,
+    dmp_eval_tail, dmp_diff_eval_in,
+    dup_trunc, dmp_trunc, dmp_ground_trunc,
+    dup_monic, dmp_ground_monic,
+    dup_content, dmp_ground_content,
+    dup_primitive, dmp_ground_primitive,
+    dup_extract, dmp_ground_extract,
+    dup_real_imag,
+    dup_mirror, dup_scale, dup_shift, dmp_shift,
+    dup_transform,
+    dup_compose, dmp_compose,
+    dup_decompose,
+    dmp_lift,
+    dup_sign_variations,
+    dup_revert, dmp_revert,
+)
+from sympy.polys.polyclasses import ANP
+
+from sympy.polys.polyerrors import (
+    MultivariatePolynomialError,
+    ExactQuotientFailed,
+    NotReversible,
+    DomainError,
+)
+
+from sympy.polys.specialpolys import f_polys
+
+from sympy.polys.domains import FF, ZZ, QQ, ZZ_I, QQ_I, EX, RR
+from sympy.polys.rings import ring
+
+from sympy.core.numbers import I
+from sympy.core.singleton import S
+from sympy.functions.elementary.trigonometric import sin
+
+from sympy.abc import x
+from sympy.testing.pytest import raises
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
+
+def test_dup_integrate():
+    assert dup_integrate([], 1, QQ) == []
+    assert dup_integrate([], 2, QQ) == []
+
+    assert dup_integrate([QQ(1)], 1, QQ) == [QQ(1), QQ(0)]
+    assert dup_integrate([QQ(1)], 2, QQ) == [QQ(1, 2), QQ(0), QQ(0)]
+
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 0, QQ) == \
+        [QQ(1), QQ(2), QQ(3)]
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 1, QQ) == \
+        [QQ(1, 3), QQ(1), QQ(3), QQ(0)]
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 2, QQ) == \
+        [QQ(1, 12), QQ(1, 3), QQ(3, 2), QQ(0), QQ(0)]
+    assert dup_integrate([QQ(1), QQ(2), QQ(3)], 3, QQ) == \
+        [QQ(1, 60), QQ(1, 12), QQ(1, 2), QQ(0), QQ(0), QQ(0)]
+
+    assert dup_integrate(dup_from_raw_dict({29: QQ(17)}, QQ), 3, QQ) == \
+        dup_from_raw_dict({32: QQ(17, 29760)}, QQ)
+
+    assert dup_integrate(dup_from_raw_dict({29: QQ(17), 5: QQ(1, 2)}, QQ), 3, QQ) == \
+        dup_from_raw_dict({32: QQ(17, 29760), 8: QQ(1, 672)}, QQ)
+
+
+def test_dmp_integrate():
+    assert dmp_integrate([QQ(1)], 2, 0, QQ) == [QQ(1, 2), QQ(0), QQ(0)]
+
+    assert dmp_integrate([[[]]], 1, 2, QQ) == [[[]]]
+    assert dmp_integrate([[[]]], 2, 2, QQ) == [[[]]]
+
+    assert dmp_integrate([[[QQ(1)]]], 1, 2, QQ) == [[[QQ(1)]], [[]]]
+    assert dmp_integrate([[[QQ(1)]]], 2, 2, QQ) == [[[QQ(1, 2)]], [[]], [[]]]
+
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 0, 1, QQ) == \
+        [[QQ(1)], [QQ(2)], [QQ(3)]]
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 1, 1, QQ) == \
+        [[QQ(1, 3)], [QQ(1)], [QQ(3)], []]
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 2, 1, QQ) == \
+        [[QQ(1, 12)], [QQ(1, 3)], [QQ(3, 2)], [], []]
+    assert dmp_integrate([[QQ(1)], [QQ(2)], [QQ(3)]], 3, 1, QQ) == \
+        [[QQ(1, 60)], [QQ(1, 12)], [QQ(1, 2)], [], [], []]
+
+
+def test_dmp_integrate_in():
+    f = dmp_convert(f_6, 3, ZZ, QQ)
+
+    assert dmp_integrate_in(f, 2, 1, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 2, 3, QQ), 0, 1, 3, QQ)
+    assert dmp_integrate_in(f, 3, 1, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 1, 3, QQ), 3, 3, QQ), 0, 1, 3, QQ)
+    assert dmp_integrate_in(f, 2, 2, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 2, 3, QQ), 0, 2, 3, QQ)
+    assert dmp_integrate_in(f, 3, 2, 3, QQ) == \
+        dmp_swap(
+            dmp_integrate(dmp_swap(f, 0, 2, 3, QQ), 3, 3, QQ), 0, 2, 3, QQ)
+
+    raises(IndexError, lambda: dmp_integrate_in(f, 1, -1, 3, QQ))
+    raises(IndexError, lambda: dmp_integrate_in(f, 1, 4, 3, QQ))
+
+
+def test_dup_diff():
+    assert dup_diff([], 1, ZZ) == []
+    assert dup_diff([7], 1, ZZ) == []
+    assert dup_diff([2, 7], 1, ZZ) == [2]
+    assert dup_diff([1, 2, 1], 1, ZZ) == [2, 2]
+    assert dup_diff([1, 2, 3, 4], 1, ZZ) == [3, 4, 3]
+    assert dup_diff([1, -1, 0, 0, 2], 1, ZZ) == [4, -3, 0, 0]
+
+    f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], ZZ)
+
+    assert dup_diff(f, 0, ZZ) == f
+    assert dup_diff(f, 1, ZZ) == [170, 306, 448, -2415, 138, 380, 0, 0, 24, 3]
+    assert dup_diff(f, 2, ZZ) == dup_diff(dup_diff(f, 1, ZZ), 1, ZZ)
+    assert dup_diff(
+        f, 3, ZZ) == dup_diff(dup_diff(dup_diff(f, 1, ZZ), 1, ZZ), 1, ZZ)
+
+    K = FF(3)
+    f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], K)
+
+    assert dup_diff(f, 1, K) == dup_normal([2, 0, 1, 0, 0, 2, 0, 0, 0, 0], K)
+    assert dup_diff(f, 2, K) == dup_normal([1, 0, 0, 2, 0, 0, 0], K)
+    assert dup_diff(f, 3, K) == dup_normal([], K)
+
+    assert dup_diff(f, 0, K) == f
+    assert dup_diff(f, 2, K) == dup_diff(dup_diff(f, 1, K), 1, K)
+    assert dup_diff(
+        f, 3, K) == dup_diff(dup_diff(dup_diff(f, 1, K), 1, K), 1, K)
+
+
+def test_dmp_diff():
+    assert dmp_diff([], 1, 0, ZZ) == []
+    assert dmp_diff([[]], 1, 1, ZZ) == [[]]
+    assert dmp_diff([[[]]], 1, 2, ZZ) == [[[]]]
+
+    assert dmp_diff([[[1], [2]]], 1, 2, ZZ) == [[[]]]
+
+    assert dmp_diff([[[1]], [[]]], 1, 2, ZZ) == [[[1]]]
+    assert dmp_diff([[[3]], [[1]], [[]]], 1, 2, ZZ) == [[[6]], [[1]]]
+
+    assert dmp_diff([1, -1, 0, 0, 2], 1, 0, ZZ) == \
+        dup_diff([1, -1, 0, 0, 2], 1, ZZ)
+
+    assert dmp_diff(f_6, 0, 3, ZZ) == f_6
+    assert dmp_diff(f_6, 1, 3, ZZ) == [[[[8460]], [[]]],
+                       [[[135, 0, 0], [], [], [-135, 0, 0]]],
+                       [[[]]],
+                       [[[-423]], [[-47]], [[]], [[141], [], [94, 0], []], [[]]]]
+    assert dmp_diff(
+        f_6, 2, 3, ZZ) == dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ)
+    assert dmp_diff(f_6, 3, 3, ZZ) == dmp_diff(
+        dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ), 1, 3, ZZ)
+
+    K = FF(23)
+    F_6 = dmp_normal(f_6, 3, K)
+
+    assert dmp_diff(F_6, 0, 3, K) == F_6
+    assert dmp_diff(F_6, 1, 3, K) == dmp_diff(F_6, 1, 3, K)
+    assert dmp_diff(F_6, 2, 3, K) == dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K)
+    assert dmp_diff(F_6, 3, 3, K) == dmp_diff(
+        dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K), 1, 3, K)
+
+
+def test_dmp_diff_in():
+    assert dmp_diff_in(f_6, 2, 1, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 0, 1, 3, ZZ)
+    assert dmp_diff_in(f_6, 3, 1, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 3, 3, ZZ), 0, 1, 3, ZZ)
+    assert dmp_diff_in(f_6, 2, 2, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 2, 3, ZZ), 0, 2, 3, ZZ)
+    assert dmp_diff_in(f_6, 3, 2, 3, ZZ) == \
+        dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 3, 3, ZZ), 0, 2, 3, ZZ)
+
+    raises(IndexError, lambda: dmp_diff_in(f_6, 1, -1, 3, ZZ))
+    raises(IndexError, lambda: dmp_diff_in(f_6, 1, 4, 3, ZZ))
+
+def test_dup_eval():
+    assert dup_eval([], 7, ZZ) == 0
+    assert dup_eval([1, 2], 0, ZZ) == 2
+    assert dup_eval([1, 2, 3], 7, ZZ) == 66
+
+
+def test_dmp_eval():
+    assert dmp_eval([], 3, 0, ZZ) == 0
+
+    assert dmp_eval([[]], 3, 1, ZZ) == []
+    assert dmp_eval([[[]]], 3, 2, ZZ) == [[]]
+
+    assert dmp_eval([[1, 2]], 0, 1, ZZ) == [1, 2]
+
+    assert dmp_eval([[[1]]], 3, 2, ZZ) == [[1]]
+    assert dmp_eval([[[1, 2]]], 3, 2, ZZ) == [[1, 2]]
+
+    assert dmp_eval([[3, 2], [1, 2]], 3, 1, ZZ) == [10, 8]
+    assert dmp_eval([[[3, 2]], [[1, 2]]], 3, 2, ZZ) == [[10, 8]]
+
+
+def test_dmp_eval_in():
+    assert dmp_eval_in(
+        f_6, -2, 1, 3, ZZ) == dmp_eval(dmp_swap(f_6, 0, 1, 3, ZZ), -2, 3, ZZ)
+    assert dmp_eval_in(
+        f_6, 7, 1, 3, ZZ) == dmp_eval(dmp_swap(f_6, 0, 1, 3, ZZ), 7, 3, ZZ)
+    assert dmp_eval_in(f_6, -2, 2, 3, ZZ) == dmp_swap(
+        dmp_eval(dmp_swap(f_6, 0, 2, 3, ZZ), -2, 3, ZZ), 0, 1, 2, ZZ)
+    assert dmp_eval_in(f_6, 7, 2, 3, ZZ) == dmp_swap(
+        dmp_eval(dmp_swap(f_6, 0, 2, 3, ZZ), 7, 3, ZZ), 0, 1, 2, ZZ)
+
+    f = [[[int(45)]], [[]], [[]], [[int(-9)], [-1], [], [int(3), int(0), int(10), int(0)]]]
+
+    assert dmp_eval_in(f, -2, 2, 2, ZZ) == \
+        [[45], [], [], [-9, -1, 0, -44]]
+
+    raises(IndexError, lambda: dmp_eval_in(f_6, ZZ(1), -1, 3, ZZ))
+    raises(IndexError, lambda: dmp_eval_in(f_6, ZZ(1), 4, 3, ZZ))
+
+
+def test_dmp_eval_tail():
+    assert dmp_eval_tail([[]], [1], 1, ZZ) == []
+    assert dmp_eval_tail([[[]]], [1], 2, ZZ) == [[]]
+    assert dmp_eval_tail([[[]]], [1, 2], 2, ZZ) == []
+
+    assert dmp_eval_tail(f_0, [], 2, ZZ) == f_0
+
+    assert dmp_eval_tail(f_0, [1, -17, 8], 2, ZZ) == 84496
+    assert dmp_eval_tail(f_0, [-17, 8], 2, ZZ) == [-1409, 3, 85902]
+    assert dmp_eval_tail(f_0, [8], 2, ZZ) == [[83, 2], [3], [302, 81, 1]]
+
+    assert dmp_eval_tail(f_1, [-17, 8], 2, ZZ) == [-136, 15699, 9166, -27144]
+
+    assert dmp_eval_tail(
+        f_2, [-12, 3], 2, ZZ) == [-1377, 0, -702, -1224, 0, -624]
+    assert dmp_eval_tail(
+        f_3, [-12, 3], 2, ZZ) == [144, 82, -5181, -28872, -14868, -540]
+
+    assert dmp_eval_tail(
+        f_4, [25, -1], 2, ZZ) == [152587890625, 9765625, -59605407714843750,
+        -3839159765625, -1562475, 9536712644531250, 610349546750, -4, 24414375000, 1562520]
+    assert dmp_eval_tail(f_5, [25, -1], 2, ZZ) == [-1, -78, -2028, -17576]
+
+    assert dmp_eval_tail(f_6, [0, 2, 4], 3, ZZ) == [5040, 0, 0, 4480]
+
+
+def test_dmp_diff_eval_in():
+    assert dmp_diff_eval_in(f_6, 2, 7, 1, 3, ZZ) == \
+        dmp_eval(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 7, 3, ZZ)
+
+    assert dmp_diff_eval_in(f_6, 2, 7, 0, 3, ZZ) == \
+        dmp_eval(dmp_diff(f_6, 2, 3, ZZ), 7, 3, ZZ)
+
+    raises(IndexError, lambda: dmp_diff_eval_in(f_6, 1, ZZ(1), 4, 3, ZZ))
+
+
+def test_dup_revert():
+    f = [-QQ(1, 720), QQ(0), QQ(1, 24), QQ(0), -QQ(1, 2), QQ(0), QQ(1)]
+    g = [QQ(61, 720), QQ(0), QQ(5, 24), QQ(0), QQ(1, 2), QQ(0), QQ(1)]
+
+    assert dup_revert(f, 8, QQ) == g
+
+    raises(NotReversible, lambda: dup_revert([QQ(1), QQ(0)], 3, QQ))
+
+
+def test_dmp_revert():
+    f = [-QQ(1, 720), QQ(0), QQ(1, 24), QQ(0), -QQ(1, 2), QQ(0), QQ(1)]
+    g = [QQ(61, 720), QQ(0), QQ(5, 24), QQ(0), QQ(1, 2), QQ(0), QQ(1)]
+
+    assert dmp_revert(f, 8, 0, QQ) == g
+
+    raises(MultivariatePolynomialError, lambda: dmp_revert([[1]], 2, 1, QQ))
+
+
+def test_dup_trunc():
+    assert dup_trunc([1, 2, 3, 4, 5, 6], ZZ(3), ZZ) == [1, -1, 0, 1, -1, 0]
+    assert dup_trunc([6, 5, 4, 3, 2, 1], ZZ(3), ZZ) == [-1, 1, 0, -1, 1]
+
+    R = ZZ_I
+    assert dup_trunc([R(3), R(4), R(5)], R(3), R) == [R(1), R(-1)]
+
+    K = FF(5)
+    assert dup_trunc([K(3), K(4), K(5)], K(3), K) == [K(1), K(0)]
+
+
+def test_dmp_trunc():
+    assert dmp_trunc([[]], [1, 2], 2, ZZ) == [[]]
+    assert dmp_trunc([[1, 2], [1, 4, 1], [1]], [1, 2], 1, ZZ) == [[-3], [1]]
+
+
+def test_dmp_ground_trunc():
+    assert dmp_ground_trunc(f_0, ZZ(3), 2, ZZ) == \
+        dmp_normal(
+            [[[1, -1, 0], [-1]], [[]], [[1, -1, 0], [1, -1, 1], [1]]], 2, ZZ)
+
+
+def test_dup_monic():
+    assert dup_monic([3, 6, 9], ZZ) == [1, 2, 3]
+
+    raises(ExactQuotientFailed, lambda: dup_monic([3, 4, 5], ZZ))
+
+    assert dup_monic([], QQ) == []
+    assert dup_monic([QQ(1)], QQ) == [QQ(1)]
+    assert dup_monic([QQ(7), QQ(1), QQ(21)], QQ) == [QQ(1), QQ(1, 7), QQ(3)]
+
+
+def test_dmp_ground_monic():
+    assert dmp_ground_monic([3, 6, 9], 0, ZZ) == [1, 2, 3]
+
+    assert dmp_ground_monic([[3], [6], [9]], 1, ZZ) == [[1], [2], [3]]
+
+    raises(
+        ExactQuotientFailed, lambda: dmp_ground_monic([[3], [4], [5]], 1, ZZ))
+
+    assert dmp_ground_monic([[]], 1, QQ) == [[]]
+    assert dmp_ground_monic([[QQ(1)]], 1, QQ) == [[QQ(1)]]
+    assert dmp_ground_monic(
+        [[QQ(7)], [QQ(1)], [QQ(21)]], 1, QQ) == [[QQ(1)], [QQ(1, 7)], [QQ(3)]]
+
+
+def test_dup_content():
+    assert dup_content([], ZZ) == ZZ(0)
+    assert dup_content([1], ZZ) == ZZ(1)
+    assert dup_content([-1], ZZ) == ZZ(1)
+    assert dup_content([1, 1], ZZ) == ZZ(1)
+    assert dup_content([2, 2], ZZ) == ZZ(2)
+    assert dup_content([1, 2, 1], ZZ) == ZZ(1)
+    assert dup_content([2, 4, 2], ZZ) == ZZ(2)
+
+    assert dup_content([QQ(2, 3), QQ(4, 9)], QQ) == QQ(2, 9)
+    assert dup_content([QQ(2, 3), QQ(4, 5)], QQ) == QQ(2, 15)
+
+
+def test_dmp_ground_content():
+    assert dmp_ground_content([[]], 1, ZZ) == ZZ(0)
+    assert dmp_ground_content([[]], 1, QQ) == QQ(0)
+    assert dmp_ground_content([[1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[-1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[1], [1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[2], [2]], 1, ZZ) == ZZ(2)
+    assert dmp_ground_content([[1], [2], [1]], 1, ZZ) == ZZ(1)
+    assert dmp_ground_content([[2], [4], [2]], 1, ZZ) == ZZ(2)
+
+    assert dmp_ground_content([[QQ(2, 3)], [QQ(4, 9)]], 1, QQ) == QQ(2, 9)
+    assert dmp_ground_content([[QQ(2, 3)], [QQ(4, 5)]], 1, QQ) == QQ(2, 15)
+
+    assert dmp_ground_content(f_0, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == ZZ(2)
+
+    assert dmp_ground_content(f_1, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == ZZ(3)
+
+    assert dmp_ground_content(f_2, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == ZZ(4)
+
+    assert dmp_ground_content(f_3, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == ZZ(5)
+
+    assert dmp_ground_content(f_4, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == ZZ(6)
+
+    assert dmp_ground_content(f_5, 2, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == ZZ(7)
+
+    assert dmp_ground_content(f_6, 3, ZZ) == ZZ(1)
+    assert dmp_ground_content(
+        dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == ZZ(8)
+
+
+def test_dup_primitive():
+    assert dup_primitive([], ZZ) == (ZZ(0), [])
+    assert dup_primitive([ZZ(1)], ZZ) == (ZZ(1), [ZZ(1)])
+    assert dup_primitive([ZZ(1), ZZ(1)], ZZ) == (ZZ(1), [ZZ(1), ZZ(1)])
+    assert dup_primitive([ZZ(2), ZZ(2)], ZZ) == (ZZ(2), [ZZ(1), ZZ(1)])
+    assert dup_primitive(
+        [ZZ(1), ZZ(2), ZZ(1)], ZZ) == (ZZ(1), [ZZ(1), ZZ(2), ZZ(1)])
+    assert dup_primitive(
+        [ZZ(2), ZZ(4), ZZ(2)], ZZ) == (ZZ(2), [ZZ(1), ZZ(2), ZZ(1)])
+
+    assert dup_primitive([], QQ) == (QQ(0), [])
+    assert dup_primitive([QQ(1)], QQ) == (QQ(1), [QQ(1)])
+    assert dup_primitive([QQ(1), QQ(1)], QQ) == (QQ(1), [QQ(1), QQ(1)])
+    assert dup_primitive([QQ(2), QQ(2)], QQ) == (QQ(2), [QQ(1), QQ(1)])
+    assert dup_primitive(
+        [QQ(1), QQ(2), QQ(1)], QQ) == (QQ(1), [QQ(1), QQ(2), QQ(1)])
+    assert dup_primitive(
+        [QQ(2), QQ(4), QQ(2)], QQ) == (QQ(2), [QQ(1), QQ(2), QQ(1)])
+
+    assert dup_primitive(
+        [QQ(2, 3), QQ(4, 9)], QQ) == (QQ(2, 9), [QQ(3), QQ(2)])
+    assert dup_primitive(
+        [QQ(2, 3), QQ(4, 5)], QQ) == (QQ(2, 15), [QQ(5), QQ(6)])
+
+
+def test_dmp_ground_primitive():
+    assert dmp_ground_primitive([ZZ(1)], 0, ZZ) == (ZZ(1), [ZZ(1)])
+
+    assert dmp_ground_primitive([[]], 1, ZZ) == (ZZ(0), [[]])
+
+    assert dmp_ground_primitive(f_0, 2, ZZ) == (ZZ(1), f_0)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_0, ZZ(2), 2, ZZ), 2, ZZ) == (ZZ(2), f_0)
+
+    assert dmp_ground_primitive(f_1, 2, ZZ) == (ZZ(1), f_1)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_1, ZZ(3), 2, ZZ), 2, ZZ) == (ZZ(3), f_1)
+
+    assert dmp_ground_primitive(f_2, 2, ZZ) == (ZZ(1), f_2)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_2, ZZ(4), 2, ZZ), 2, ZZ) == (ZZ(4), f_2)
+
+    assert dmp_ground_primitive(f_3, 2, ZZ) == (ZZ(1), f_3)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_3, ZZ(5), 2, ZZ), 2, ZZ) == (ZZ(5), f_3)
+
+    assert dmp_ground_primitive(f_4, 2, ZZ) == (ZZ(1), f_4)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_4, ZZ(6), 2, ZZ), 2, ZZ) == (ZZ(6), f_4)
+
+    assert dmp_ground_primitive(f_5, 2, ZZ) == (ZZ(1), f_5)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_5, ZZ(7), 2, ZZ), 2, ZZ) == (ZZ(7), f_5)
+
+    assert dmp_ground_primitive(f_6, 3, ZZ) == (ZZ(1), f_6)
+    assert dmp_ground_primitive(
+        dmp_mul_ground(f_6, ZZ(8), 3, ZZ), 3, ZZ) == (ZZ(8), f_6)
+
+    assert dmp_ground_primitive([[ZZ(2)]], 1, ZZ) == (ZZ(2), [[ZZ(1)]])
+    assert dmp_ground_primitive([[QQ(2)]], 1, QQ) == (QQ(2), [[QQ(1)]])
+
+    assert dmp_ground_primitive(
+        [[QQ(2, 3)], [QQ(4, 9)]], 1, QQ) == (QQ(2, 9), [[QQ(3)], [QQ(2)]])
+    assert dmp_ground_primitive(
+        [[QQ(2, 3)], [QQ(4, 5)]], 1, QQ) == (QQ(2, 15), [[QQ(5)], [QQ(6)]])
+
+
+def test_dup_extract():
+    f = dup_normal([2930944, 0, 2198208, 0, 549552, 0, 45796], ZZ)
+    g = dup_normal([17585664, 0, 8792832, 0, 1099104, 0], ZZ)
+
+    F = dup_normal([64, 0, 48, 0, 12, 0, 1], ZZ)
+    G = dup_normal([384, 0, 192, 0, 24, 0], ZZ)
+
+    assert dup_extract(f, g, ZZ) == (45796, F, G)
+
+
+def test_dmp_ground_extract():
+    f = dmp_normal(
+        [[2930944], [], [2198208], [], [549552], [], [45796]], 1, ZZ)
+    g = dmp_normal([[17585664], [], [8792832], [], [1099104], []], 1, ZZ)
+
+    F = dmp_normal([[64], [], [48], [], [12], [], [1]], 1, ZZ)
+    G = dmp_normal([[384], [], [192], [], [24], []], 1, ZZ)
+
+    assert dmp_ground_extract(f, g, 1, ZZ) == (45796, F, G)
+
+
+def test_dup_real_imag():
+    assert dup_real_imag([], ZZ) == ([[]], [[]])
+    assert dup_real_imag([1], ZZ) == ([[1]], [[]])
+
+    assert dup_real_imag([1, 1], ZZ) == ([[1], [1]], [[1, 0]])
+    assert dup_real_imag([1, 2], ZZ) == ([[1], [2]], [[1, 0]])
+
+    assert dup_real_imag(
+        [1, 2, 3], ZZ) == ([[1], [2], [-1, 0, 3]], [[2, 0], [2, 0]])
+
+    assert dup_real_imag([ZZ(1), ZZ(0), ZZ(1), ZZ(3)], ZZ) == (
+        [[ZZ(1)], [], [ZZ(-3), ZZ(0), ZZ(1)], [ZZ(3)]],
+        [[ZZ(3), ZZ(0)], [], [ZZ(-1), ZZ(0), ZZ(1), ZZ(0)]]
+    )
+
+    raises(DomainError, lambda: dup_real_imag([EX(1), EX(2)], EX))
+
+
+
+def test_dup_mirror():
+    assert dup_mirror([], ZZ) == []
+    assert dup_mirror([1], ZZ) == [1]
+
+    assert dup_mirror([1, 2, 3, 4, 5], ZZ) == [1, -2, 3, -4, 5]
+    assert dup_mirror([1, 2, 3, 4, 5, 6], ZZ) == [-1, 2, -3, 4, -5, 6]
+
+
+def test_dup_scale():
+    assert dup_scale([], -1, ZZ) == []
+    assert dup_scale([1], -1, ZZ) == [1]
+
+    assert dup_scale([1, 2, 3, 4, 5], -1, ZZ) == [1, -2, 3, -4, 5]
+    assert dup_scale([1, 2, 3, 4, 5], -7, ZZ) == [2401, -686, 147, -28, 5]
+
+
+def test_dup_shift():
+    assert dup_shift([], 1, ZZ) == []
+    assert dup_shift([1], 1, ZZ) == [1]
+
+    assert dup_shift([1, 2, 3, 4, 5], 1, ZZ) == [1, 6, 15, 20, 15]
+    assert dup_shift([1, 2, 3, 4, 5], 7, ZZ) == [1, 30, 339, 1712, 3267]
+
+
+def test_dmp_shift():
+    assert dmp_shift([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == [ZZ(1), ZZ(3)]
+
+    assert dmp_shift([[]], [ZZ(1), ZZ(2)], 1, ZZ) == [[]]
+
+    xy = [[ZZ(1), ZZ(0)], []]               # x*y
+    x1y2 = [[ZZ(1), ZZ(2)], [ZZ(1), ZZ(2)]] # (x+1)*(y+2)
+    assert dmp_shift(xy, [ZZ(1), ZZ(2)], 1, ZZ) == x1y2
+
+
+def test_dup_transform():
+    assert dup_transform([], [], [1, 1], ZZ) == []
+    assert dup_transform([], [1], [1, 1], ZZ) == []
+    assert dup_transform([], [1, 2], [1, 1], ZZ) == []
+
+    assert dup_transform([6, -5, 4, -3, 17], [1, -3, 4], [2, -3], ZZ) == \
+        [6, -82, 541, -2205, 6277, -12723, 17191, -13603, 4773]
+
+
+def test_dup_compose():
+    assert dup_compose([], [], ZZ) == []
+    assert dup_compose([], [1], ZZ) == []
+    assert dup_compose([], [1, 2], ZZ) == []
+
+    assert dup_compose([1], [], ZZ) == [1]
+
+    assert dup_compose([1, 2, 0], [], ZZ) == []
+    assert dup_compose([1, 2, 1], [], ZZ) == [1]
+
+    assert dup_compose([1, 2, 1], [1], ZZ) == [4]
+    assert dup_compose([1, 2, 1], [7], ZZ) == [64]
+
+    assert dup_compose([1, 2, 1], [1, -1], ZZ) == [1, 0, 0]
+    assert dup_compose([1, 2, 1], [1, 1], ZZ) == [1, 4, 4]
+    assert dup_compose([1, 2, 1], [1, 2, 1], ZZ) == [1, 4, 8, 8, 4]
+
+
+def test_dmp_compose():
+    assert dmp_compose([1, 2, 1], [1, 2, 1], 0, ZZ) == [1, 4, 8, 8, 4]
+
+    assert dmp_compose([[[]]], [[[]]], 2, ZZ) == [[[]]]
+    assert dmp_compose([[[]]], [[[1]]], 2, ZZ) == [[[]]]
+    assert dmp_compose([[[]]], [[[1]], [[2]]], 2, ZZ) == [[[]]]
+
+    assert dmp_compose([[[1]]], [], 2, ZZ) == [[[1]]]
+
+    assert dmp_compose([[1], [2], [ ]], [[]], 1, ZZ) == [[]]
+    assert dmp_compose([[1], [2], [1]], [[]], 1, ZZ) == [[1]]
+
+    assert dmp_compose([[1], [2], [1]], [[1]], 1, ZZ) == [[4]]
+    assert dmp_compose([[1], [2], [1]], [[7]], 1, ZZ) == [[64]]
+
+    assert dmp_compose([[1], [2], [1]], [[1], [-1]], 1, ZZ) == [[1], [ ], [ ]]
+    assert dmp_compose([[1], [2], [1]], [[1], [ 1]], 1, ZZ) == [[1], [4], [4]]
+
+    assert dmp_compose(
+        [[1], [2], [1]], [[1], [2], [1]], 1, ZZ) == [[1], [4], [8], [8], [4]]
+
+
+def test_dup_decompose():
+    assert dup_decompose([1], ZZ) == [[1]]
+
+    assert dup_decompose([1, 0], ZZ) == [[1, 0]]
+    assert dup_decompose([1, 0, 0, 0], ZZ) == [[1, 0, 0, 0]]
+
+    assert dup_decompose([1, 0, 0, 0, 0], ZZ) == [[1, 0, 0], [1, 0, 0]]
+    assert dup_decompose(
+        [1, 0, 0, 0, 0, 0, 0], ZZ) == [[1, 0, 0, 0], [1, 0, 0]]
+
+    assert dup_decompose([7, 0, 0, 0, 1], ZZ) == [[7, 0, 1], [1, 0, 0]]
+    assert dup_decompose([4, 0, 3, 0, 2], ZZ) == [[4, 3, 2], [1, 0, 0]]
+
+    f = [1, 0, 20, 0, 150, 0, 500, 0, 625, -2, 0, -10, 9]
+
+    assert dup_decompose(f, ZZ) == [[1, 0, 0, -2, 9], [1, 0, 5, 0]]
+
+    f = [2, 0, 40, 0, 300, 0, 1000, 0, 1250, -4, 0, -20, 18]
+
+    assert dup_decompose(f, ZZ) == [[2, 0, 0, -4, 18], [1, 0, 5, 0]]
+
+    f = [1, 0, 20, -8, 150, -120, 524, -600, 865, -1034, 600, -170, 29]
+
+    assert dup_decompose(f, ZZ) == [[1, -8, 24, -34, 29], [1, 0, 5, 0]]
+
+    R, t = ring("t", ZZ)
+    f = [6*t**2 - 42,
+         48*t**2 + 96,
+         144*t**2 + 648*t + 288,
+         624*t**2 + 864*t + 384,
+         108*t**3 + 312*t**2 + 432*t + 192]
+
+    assert dup_decompose(f, R.to_domain()) == [f]
+
+
+def test_dmp_lift():
+    q = [QQ(1, 1), QQ(0, 1), QQ(1, 1)]
+
+    f_a = [ANP([QQ(1, 1)], q, QQ), ANP([], q, QQ), ANP([], q, QQ),
+         ANP([QQ(1, 1), QQ(0, 1)], q, QQ), ANP([QQ(17, 1), QQ(0, 1)], q, QQ)]
+
+    f_lift = [QQ(1), QQ(0), QQ(0), QQ(0), QQ(0), QQ(0), QQ(2), QQ(0), QQ(578),
+              QQ(0), QQ(0), QQ(0), QQ(1), QQ(0), QQ(-578), QQ(0), QQ(83521)]
+
+    assert dmp_lift(f_a, 0, QQ.algebraic_field(I)) == f_lift
+
+    f_g = [QQ_I(1), QQ_I(0), QQ_I(0), QQ_I(0, 1), QQ_I(0, 17)]
+
+    assert dmp_lift(f_g, 0, QQ_I) == f_lift
+
+    raises(DomainError, lambda: dmp_lift([EX(1), EX(2)], 0, EX))
+
+
+def test_dup_sign_variations():
+    assert dup_sign_variations([], ZZ) == 0
+    assert dup_sign_variations([1, 0], ZZ) == 0
+    assert dup_sign_variations([1, 0, 2], ZZ) == 0
+    assert dup_sign_variations([1, 0, 3, 0], ZZ) == 0
+    assert dup_sign_variations([1, 0, 4, 0, 5], ZZ) == 0
+
+    assert dup_sign_variations([-1, 0, 2], ZZ) == 1
+    assert dup_sign_variations([-1, 0, 3, 0], ZZ) == 1
+    assert dup_sign_variations([-1, 0, 4, 0, 5], ZZ) == 1
+
+    assert dup_sign_variations([-1, -4, -5], ZZ) == 0
+    assert dup_sign_variations([ 1, -4, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, 4, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, -4, 5], ZZ) == 2
+    assert dup_sign_variations([-1, 4, -5], ZZ) == 2
+    assert dup_sign_variations([-1, 4, 5], ZZ) == 1
+    assert dup_sign_variations([-1, -4, 5], ZZ) == 1
+    assert dup_sign_variations([ 1, 4, 5], ZZ) == 0
+
+    assert dup_sign_variations([-1, 0, -4, 0, -5], ZZ) == 0
+    assert dup_sign_variations([ 1, 0, -4, 0, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, 0, 4, 0, -5], ZZ) == 1
+    assert dup_sign_variations([ 1, 0, -4, 0, 5], ZZ) == 2
+    assert dup_sign_variations([-1, 0, 4, 0, -5], ZZ) == 2
+    assert dup_sign_variations([-1, 0, 4, 0, 5], ZZ) == 1
+    assert dup_sign_variations([-1, 0, -4, 0, 5], ZZ) == 1
+    assert dup_sign_variations([ 1, 0, 4, 0, 5], ZZ) == 0
+
+
+def test_dup_clear_denoms():
+    assert dup_clear_denoms([], QQ, ZZ) == (ZZ(1), [])
+
+    assert dup_clear_denoms([QQ(1)], QQ, ZZ) == (ZZ(1), [QQ(1)])
+    assert dup_clear_denoms([QQ(7)], QQ, ZZ) == (ZZ(1), [QQ(7)])
+
+    assert dup_clear_denoms([QQ(7, 3)], QQ) == (ZZ(3), [QQ(7)])
+    assert dup_clear_denoms([QQ(7, 3)], QQ, ZZ) == (ZZ(3), [QQ(7)])
+
+    assert dup_clear_denoms(
+        [QQ(3), QQ(1), QQ(0)], QQ, ZZ) == (ZZ(1), [QQ(3), QQ(1), QQ(0)])
+    assert dup_clear_denoms(
+        [QQ(1), QQ(1, 2), QQ(0)], QQ, ZZ) == (ZZ(2), [QQ(2), QQ(1), QQ(0)])
+
+    assert dup_clear_denoms([QQ(3), QQ(
+        1), QQ(0)], QQ, ZZ, convert=True) == (ZZ(1), [ZZ(3), ZZ(1), ZZ(0)])
+    assert dup_clear_denoms([QQ(1), QQ(
+        1, 2), QQ(0)], QQ, ZZ, convert=True) == (ZZ(2), [ZZ(2), ZZ(1), ZZ(0)])
+
+    assert dup_clear_denoms(
+        [EX(S(3)/2), EX(S(9)/4)], EX) == (EX(4), [EX(6), EX(9)])
+
+    assert dup_clear_denoms([EX(7)], EX) == (EX(1), [EX(7)])
+    assert dup_clear_denoms([EX(sin(x)/x), EX(0)], EX) == (EX(x), [EX(sin(x)), EX(0)])
+
+    F = RR.frac_field(x)
+    result = dup_clear_denoms([F(8.48717/(8.0089*x + 2.83)), F(0.0)], F)
+    assert str(result) == "(x + 0.353356890459364, [1.05971731448763, 0.0])"
+
+def test_dmp_clear_denoms():
+    assert dmp_clear_denoms([[]], 1, QQ, ZZ) == (ZZ(1), [[]])
+
+    assert dmp_clear_denoms([[QQ(1)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(1)]])
+    assert dmp_clear_denoms([[QQ(7)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(7)]])
+
+    assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ) == (ZZ(3), [[QQ(7)]])
+    assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ, ZZ) == (ZZ(3), [[QQ(7)]])
+
+    assert dmp_clear_denoms(
+        [[QQ(3)], [QQ(1)], []], 1, QQ, ZZ) == (ZZ(1), [[QQ(3)], [QQ(1)], []])
+    assert dmp_clear_denoms([[QQ(
+        1)], [QQ(1, 2)], []], 1, QQ, ZZ) == (ZZ(2), [[QQ(2)], [QQ(1)], []])
+
+    assert dmp_clear_denoms([QQ(3), QQ(
+        1), QQ(0)], 0, QQ, ZZ, convert=True) == (ZZ(1), [ZZ(3), ZZ(1), ZZ(0)])
+    assert dmp_clear_denoms([QQ(1), QQ(1, 2), QQ(
+        0)], 0, QQ, ZZ, convert=True) == (ZZ(2), [ZZ(2), ZZ(1), ZZ(0)])
+
+    assert dmp_clear_denoms([[QQ(3)], [QQ(
+        1)], []], 1, QQ, ZZ, convert=True) == (ZZ(1), [[QQ(3)], [QQ(1)], []])
+    assert dmp_clear_denoms([[QQ(1)], [QQ(1, 2)], []], 1, QQ, ZZ,
+                            convert=True) == (ZZ(2), [[QQ(2)], [QQ(1)], []])
+
+    assert dmp_clear_denoms(
+        [[EX(S(3)/2)], [EX(S(9)/4)]], 1, EX) == (EX(4), [[EX(6)], [EX(9)]])
+    assert dmp_clear_denoms([[EX(7)]], 1, EX) == (EX(1), [[EX(7)]])
+    assert dmp_clear_denoms([[EX(sin(x)/x), EX(0)]], 1, EX) == (EX(x), [[EX(sin(x)), EX(0)]])

rl/Lib/site-packages/sympy/polys/tests/test_dispersion.py

@@ -0,0 +1,95 @@
+from sympy.core import Symbol, S, oo
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.polys import poly
+from sympy.polys.dispersion import dispersion, dispersionset
+
+
+def test_dispersion():
+    x = Symbol("x")
+    a = Symbol("a")
+
+    fp = poly(S.Zero, x)
+    assert sorted(dispersionset(fp)) == [0]
+
+    fp = poly(S(2), x)
+    assert sorted(dispersionset(fp)) == [0]
+
+    fp = poly(x + 1, x)
+    assert sorted(dispersionset(fp)) == [0]
+    assert dispersion(fp) == 0
+
+    fp = poly((x + 1)*(x + 2), x)
+    assert sorted(dispersionset(fp)) == [0, 1]
+    assert dispersion(fp) == 1
+
+    fp = poly(x*(x + 3), x)
+    assert sorted(dispersionset(fp)) == [0, 3]
+    assert dispersion(fp) == 3
+
+    fp = poly((x - 3)*(x + 3), x)
+    assert sorted(dispersionset(fp)) == [0, 6]
+    assert dispersion(fp) == 6
+
+    fp = poly(x**4 - 3*x**2 + 1, x)
+    gp = fp.shift(-3)
+    assert sorted(dispersionset(fp, gp)) == [2, 3, 4]
+    assert dispersion(fp, gp) == 4
+    assert sorted(dispersionset(gp, fp)) == []
+    assert dispersion(gp, fp) is -oo
+
+    fp = poly(x*(3*x**2+a)*(x-2536)*(x**3+a), x)
+    gp = fp.as_expr().subs(x, x-345).as_poly(x)
+    assert sorted(dispersionset(fp, gp)) == [345, 2881]
+    assert sorted(dispersionset(gp, fp)) == [2191]
+
+    gp = poly((x-2)**2*(x-3)**3*(x-5)**3, x)
+    assert sorted(dispersionset(gp)) == [0, 1, 2, 3]
+    assert sorted(dispersionset(gp, (gp+4)**2)) == [1, 2]
+
+    fp = poly(x*(x+2)*(x-1), x)
+    assert sorted(dispersionset(fp)) == [0, 1, 2, 3]
+
+    fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
+    gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
+    assert sorted(dispersionset(fp, gp)) == [2]
+    assert sorted(dispersionset(gp, fp)) == [1, 4]
+
+    # There are some difficulties if we compute over Z[a]
+    # and alpha happenes to lie in Z[a] instead of simply Z.
+    # Hence we can not decide if alpha is indeed integral
+    # in general.
+
+    fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
+    assert sorted(dispersionset(fp)) == [0, 1]
+
+    # For any specific value of a, the dispersion is 3*a
+    # but the algorithm can not find this in general.
+    # This is the point where the resultant based Ansatz
+    # is superior to the current one.
+    fp = poly(a**2*x**3 + (a**3 + a**2 + a + 1)*x, x)
+    gp = fp.as_expr().subs(x, x - 3*a).as_poly(x)
+    assert sorted(dispersionset(fp, gp)) == []
+
+    fpa = fp.as_expr().subs(a, 2).as_poly(x)
+    gpa = gp.as_expr().subs(a, 2).as_poly(x)
+    assert sorted(dispersionset(fpa, gpa)) == [6]
+
+    # Work with Expr instead of Poly
+    f = (x + 1)*(x + 2)
+    assert sorted(dispersionset(f)) == [0, 1]
+    assert dispersion(f) == 1
+
+    f = x**4 - 3*x**2 + 1
+    g = x**4 - 12*x**3 + 51*x**2 - 90*x + 55
+    assert sorted(dispersionset(f, g)) == [2, 3, 4]
+    assert dispersion(f, g) == 4
+
+    # Work with Expr and specify a generator
+    f = (x + 1)*(x + 2)
+    assert sorted(dispersionset(f, None, x)) == [0, 1]
+    assert dispersion(f, None, x) == 1
+
+    f = x**4 - 3*x**2 + 1
+    g = x**4 - 12*x**3 + 51*x**2 - 90*x + 55
+    assert sorted(dispersionset(f, g, x)) == [2, 3, 4]
+    assert dispersion(f, g, x) == 4

rl/Lib/site-packages/sympy/polys/tests/test_distributedmodules.py

@@ -0,0 +1,208 @@
+"""Tests for sparse distributed modules. """
+
+from sympy.polys.distributedmodules import (
+    sdm_monomial_mul, sdm_monomial_deg, sdm_monomial_divides,
+    sdm_add, sdm_LM, sdm_LT, sdm_mul_term, sdm_zero, sdm_deg,
+    sdm_LC, sdm_from_dict,
+    sdm_spoly, sdm_ecart, sdm_nf_mora, sdm_groebner,
+    sdm_from_vector, sdm_to_vector, sdm_monomial_lcm
+)
+
+from sympy.polys.orderings import lex, grlex, InverseOrder
+from sympy.polys.domains import QQ
+
+from sympy.abc import x, y, z
+
+
+def test_sdm_monomial_mul():
+    assert sdm_monomial_mul((1, 1, 0), (1, 3)) == (1, 2, 3)
+
+
+def test_sdm_monomial_deg():
+    assert sdm_monomial_deg((5, 2, 1)) == 3
+
+
+def test_sdm_monomial_lcm():
+    assert sdm_monomial_lcm((1, 2, 3), (1, 5, 0)) == (1, 5, 3)
+
+
+def test_sdm_monomial_divides():
+    assert sdm_monomial_divides((1, 0, 0), (1, 0, 0)) is True
+    assert sdm_monomial_divides((1, 0, 0), (1, 2, 1)) is True
+    assert sdm_monomial_divides((5, 1, 1), (5, 2, 1)) is True
+
+    assert sdm_monomial_divides((1, 0, 0), (2, 0, 0)) is False
+    assert sdm_monomial_divides((1, 1, 0), (1, 0, 0)) is False
+    assert sdm_monomial_divides((5, 1, 2), (5, 0, 1)) is False
+
+
+def test_sdm_LC():
+    assert sdm_LC([((1, 2, 3), QQ(5))], QQ) == QQ(5)
+
+
+def test_sdm_from_dict():
+    dic = {(1, 2, 1, 1): QQ(1), (1, 1, 2, 1): QQ(1), (1, 0, 2, 1): QQ(1),
+           (1, 0, 0, 3): QQ(1), (1, 1, 1, 0): QQ(1)}
+    assert sdm_from_dict(dic, grlex) == \
+        [((1, 2, 1, 1), QQ(1)), ((1, 1, 2, 1), QQ(1)),
+         ((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1))]
+
+# TODO test to_dict?
+
+
+def test_sdm_add():
+    assert sdm_add([((1, 1, 1), QQ(1))], [((2, 0, 0), QQ(1))], lex, QQ) == \
+        [((2, 0, 0), QQ(1)), ((1, 1, 1), QQ(1))]
+    assert sdm_add([((1, 1, 1), QQ(1))], [((1, 1, 1), QQ(-1))], lex, QQ) == []
+    assert sdm_add([((1, 0, 0), QQ(1))], [((1, 0, 0), QQ(2))], lex, QQ) == \
+        [((1, 0, 0), QQ(3))]
+    assert sdm_add([((1, 0, 1), QQ(1))], [((1, 1, 0), QQ(1))], lex, QQ) == \
+        [((1, 1, 0), QQ(1)), ((1, 0, 1), QQ(1))]
+
+
+def test_sdm_LM():
+    dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(1), (4, 0, 1): QQ(1)}
+    assert sdm_LM(sdm_from_dict(dic, lex)) == (4, 0, 1)
+
+
+def test_sdm_LT():
+    dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(2), (4, 0, 1): QQ(3)}
+    assert sdm_LT(sdm_from_dict(dic, lex)) == ((4, 0, 1), QQ(3))
+
+
+def test_sdm_mul_term():
+    assert sdm_mul_term([((1, 0, 0), QQ(1))], ((0, 0), QQ(0)), lex, QQ) == []
+    assert sdm_mul_term([], ((1, 0), QQ(1)), lex, QQ) == []
+    assert sdm_mul_term([((1, 0, 0), QQ(1))], ((1, 0), QQ(1)), lex, QQ) == \
+        [((1, 1, 0), QQ(1))]
+    f = [((2, 0, 1), QQ(4)), ((1, 1, 0), QQ(3))]
+    assert sdm_mul_term(f, ((1, 1), QQ(2)), lex, QQ) == \
+        [((2, 1, 2), QQ(8)), ((1, 2, 1), QQ(6))]
+
+
+def test_sdm_zero():
+    assert sdm_zero() == []
+
+
+def test_sdm_deg():
+    assert sdm_deg([((1, 2, 3), 1), ((10, 0, 1), 1), ((2, 3, 4), 4)]) == 7
+
+
+def test_sdm_spoly():
+    f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))]
+    g = [((2, 3, 0), QQ(1))]
+    h = [((1, 2, 3), QQ(1))]
+    assert sdm_spoly(f, h, lex, QQ) == []
+    assert sdm_spoly(f, g, lex, QQ) == [((1, 2, 1), QQ(1))]
+
+
+def test_sdm_ecart():
+    assert sdm_ecart([((1, 2, 3), 1), ((1, 0, 1), 1)]) == 0
+    assert sdm_ecart([((2, 2, 1), 1), ((1, 5, 1), 1)]) == 3
+
+
+def test_sdm_nf_mora():
+    f = sdm_from_dict({(1, 2, 1, 1): QQ(1), (1, 1, 2, 1): QQ(1),
+                (1, 0, 2, 1): QQ(1), (1, 0, 0, 3): QQ(1), (1, 1, 1, 0): QQ(1)},
+        grlex)
+    f1 = sdm_from_dict({(1, 1, 1, 0): QQ(1), (1, 0, 2, 0): QQ(1),
+                        (1, 0, 0, 0): QQ(-1)}, grlex)
+    f2 = sdm_from_dict({(1, 1, 1, 0): QQ(1)}, grlex)
+    (id0, id1, id2) = [sdm_from_dict({(i, 0, 0, 0): QQ(1)}, grlex)
+                       for i in range(3)]
+
+    assert sdm_nf_mora(f, [f1, f2], grlex, QQ, phantom=(id0, [id1, id2])) == \
+        ([((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1)),
+          ((1, 1, 0, 1), QQ(1))],
+         [((1, 1, 0, 1), QQ(-1)), ((0, 0, 0, 0), QQ(1))])
+    assert sdm_nf_mora(f, [f2, f1], grlex, QQ, phantom=(id0, [id2, id1])) == \
+        ([((1, 0, 2, 1), QQ(1)), ((1, 0, 0, 3), QQ(1)), ((1, 1, 1, 0), QQ(1))],
+         [((2, 1, 0, 1), QQ(-1)), ((2, 0, 1, 1), QQ(-1)), ((0, 0, 0, 0), QQ(1))])
+
+    f = sdm_from_vector([x*z, y**2 + y*z - z, y], lex, QQ, gens=[x, y, z])
+    f1 = sdm_from_vector([x, y, 1], lex, QQ, gens=[x, y, z])
+    f2 = sdm_from_vector([x*y, z, z**2], lex, QQ, gens=[x, y, z])
+    assert sdm_nf_mora(f, [f1, f2], lex, QQ) == \
+        sdm_nf_mora(f, [f2, f1], lex, QQ) == \
+        [((1, 0, 1, 1), QQ(1)), ((1, 0, 0, 1), QQ(-1)), ((0, 1, 1, 0), QQ(-1)),
+         ((0, 1, 0, 1), QQ(1))]
+
+
+def test_conversion():
+    f = [x**2 + y**2, 2*z]
+    g = [((1, 0, 0, 1), QQ(2)), ((0, 2, 0, 0), QQ(1)), ((0, 0, 2, 0), QQ(1))]
+    assert sdm_to_vector(g, [x, y, z], QQ) == f
+    assert sdm_from_vector(f, lex, QQ) == g
+    assert sdm_from_vector(
+        [x, 1], lex, QQ) == [((1, 0), QQ(1)), ((0, 1), QQ(1))]
+    assert sdm_to_vector([((1, 1, 0, 0), 1)], [x, y, z], QQ, n=3) == [0, x, 0]
+    assert sdm_from_vector([0, 0], lex, QQ, gens=[x, y]) == sdm_zero()
+
+
+def test_nontrivial():
+    gens = [x, y, z]
+
+    def contains(I, f):
+        S = [sdm_from_vector([g], lex, QQ, gens=gens) for g in I]
+        G = sdm_groebner(S, sdm_nf_mora, lex, QQ)
+        return sdm_nf_mora(sdm_from_vector([f], lex, QQ, gens=gens),
+                           G, lex, QQ) == sdm_zero()
+
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z)
+    assert contains([x, 1 + x + y, 5 - 7*y], 1)
+    assert contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**3)
+    assert not contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**2 + y**2)
+
+    # compare local order
+    assert not contains([x*(1 + x + y), y*(1 + z)], x)
+    assert not contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_local():
+    igrlex = InverseOrder(grlex)
+    gens = [x, y, z]
+
+    def contains(I, f):
+        S = [sdm_from_vector([g], igrlex, QQ, gens=gens) for g in I]
+        G = sdm_groebner(S, sdm_nf_mora, igrlex, QQ)
+        return sdm_nf_mora(sdm_from_vector([f], lex, QQ, gens=gens),
+                           G, lex, QQ) == sdm_zero()
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x*(1 + x + y), y*(1 + z)], x)
+    assert contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_uncovered_line():
+    gens = [x, y]
+    f1 = sdm_zero()
+    f2 = sdm_from_vector([x, 0], lex, QQ, gens=gens)
+    f3 = sdm_from_vector([0, y], lex, QQ, gens=gens)
+
+    assert sdm_spoly(f1, f2, lex, QQ) == sdm_zero()
+    assert sdm_spoly(f3, f2, lex, QQ) == sdm_zero()
+
+
+def test_chain_criterion():
+    gens = [x]
+    f1 = sdm_from_vector([1, x], grlex, QQ, gens=gens)
+    f2 = sdm_from_vector([0, x - 2], grlex, QQ, gens=gens)
+    assert len(sdm_groebner([f1, f2], sdm_nf_mora, grlex, QQ)) == 2

rl/Lib/site-packages/sympy/polys/tests/test_euclidtools.py

@@ -0,0 +1,712 @@
+"""Tests for Euclidean algorithms, GCDs, LCMs and polynomial remainder sequences. """
+
+from sympy.polys.rings import ring
+from sympy.polys.domains import ZZ, QQ, RR
+
+from sympy.polys.specialpolys import (
+    f_polys,
+    dmp_fateman_poly_F_1,
+    dmp_fateman_poly_F_2,
+    dmp_fateman_poly_F_3)
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys()
+
+def test_dup_gcdex():
+    R, x = ring("x", QQ)
+
+    f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
+    g = x**3 + x**2 - 4*x - 4
+
+    s = -QQ(1,5)*x + QQ(3,5)
+    t = QQ(1,5)*x**2 - QQ(6,5)*x + 2
+    h = x + 1
+
+    assert R.dup_half_gcdex(f, g) == (s, h)
+    assert R.dup_gcdex(f, g) == (s, t, h)
+
+    f = x**4 + 4*x**3 - x + 1
+    g = x**3 - x + 1
+
+    s, t, h = R.dup_gcdex(f, g)
+    S, T, H = R.dup_gcdex(g, f)
+
+    assert R.dup_add(R.dup_mul(s, f),
+                     R.dup_mul(t, g)) == h
+    assert R.dup_add(R.dup_mul(S, g),
+                     R.dup_mul(T, f)) == H
+
+    f = 2*x
+    g = x**2 - 16
+
+    s = QQ(1,32)*x
+    t = -QQ(1,16)
+    h = 1
+
+    assert R.dup_half_gcdex(f, g) == (s, h)
+    assert R.dup_gcdex(f, g) == (s, t, h)
+
+
+def test_dup_invert():
+    R, x = ring("x", QQ)
+    assert R.dup_invert(2*x, x**2 - 16) == QQ(1,32)*x
+
+
+def test_dup_euclidean_prs():
+    R, x = ring("x", QQ)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    assert R.dup_euclidean_prs(f, g) == [
+        f,
+        g,
+        -QQ(5,9)*x**4 + QQ(1,9)*x**2 - QQ(1,3),
+        -QQ(117,25)*x**2 - 9*x + QQ(441,25),
+        QQ(233150,19773)*x - QQ(102500,6591),
+        -QQ(1288744821,543589225)]
+
+
+def test_dup_primitive_prs():
+    R, x = ring("x", ZZ)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    assert R.dup_primitive_prs(f, g) == [
+        f,
+        g,
+        -5*x**4 + x**2 - 3,
+        13*x**2 + 25*x - 49,
+        4663*x - 6150,
+        1]
+
+
+def test_dup_subresultants():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_resultant(0, 0) == 0
+
+    assert R.dup_resultant(1, 0) == 0
+    assert R.dup_resultant(0, 1) == 0
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    a = 15*x**4 - 3*x**2 + 9
+    b = 65*x**2 + 125*x - 245
+    c = 9326*x - 12300
+    d = 260708
+
+    assert R.dup_subresultants(f, g) == [f, g, a, b, c, d]
+    assert R.dup_resultant(f, g) == R.dup_LC(d)
+
+    f = x**2 - 2*x + 1
+    g = x**2 - 1
+
+    a = 2*x - 2
+
+    assert R.dup_subresultants(f, g) == [f, g, a]
+    assert R.dup_resultant(f, g) == 0
+
+    f = x**2 + 1
+    g = x**2 - 1
+
+    a = -2
+
+    assert R.dup_subresultants(f, g) == [f, g, a]
+    assert R.dup_resultant(f, g) == 4
+
+    f = x**2 - 1
+    g = x**3 - x**2 + 2
+
+    assert R.dup_resultant(f, g) == 0
+
+    f = 3*x**3 - x
+    g = 5*x**2 + 1
+
+    assert R.dup_resultant(f, g) == 64
+
+    f = x**2 - 2*x + 7
+    g = x**3 - x + 5
+
+    assert R.dup_resultant(f, g) == 265
+
+    f = x**3 - 6*x**2 + 11*x - 6
+    g = x**3 - 15*x**2 + 74*x - 120
+
+    assert R.dup_resultant(f, g) == -8640
+
+    f = x**3 - 6*x**2 + 11*x - 6
+    g = x**3 - 10*x**2 + 29*x - 20
+
+    assert R.dup_resultant(f, g) == 0
+
+    f = x**3 - 1
+    g = x**3 + 2*x**2 + 2*x - 1
+
+    assert R.dup_resultant(f, g) == 16
+
+    f = x**8 - 2
+    g = x - 1
+
+    assert R.dup_resultant(f, g) == -1
+
+
+def test_dmp_subresultants():
+    R, x, y = ring("x,y", ZZ)
+
+    assert R.dmp_resultant(0, 0) == 0
+    assert R.dmp_prs_resultant(0, 0)[0] == 0
+    assert R.dmp_zz_collins_resultant(0, 0) == 0
+    assert R.dmp_qq_collins_resultant(0, 0) == 0
+
+    assert R.dmp_resultant(1, 0) == 0
+    assert R.dmp_resultant(1, 0) == 0
+    assert R.dmp_resultant(1, 0) == 0
+
+    assert R.dmp_resultant(0, 1) == 0
+    assert R.dmp_prs_resultant(0, 1)[0] == 0
+    assert R.dmp_zz_collins_resultant(0, 1) == 0
+    assert R.dmp_qq_collins_resultant(0, 1) == 0
+
+    f = 3*x**2*y - y**3 - 4
+    g = x**2 + x*y**3 - 9
+
+    a = 3*x*y**4 + y**3 - 27*y + 4
+    b = -3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
+
+    r = R.dmp_LC(b)
+
+    assert R.dmp_subresultants(f, g) == [f, g, a, b]
+
+    assert R.dmp_resultant(f, g) == r
+    assert R.dmp_prs_resultant(f, g)[0] == r
+    assert R.dmp_zz_collins_resultant(f, g) == r
+    assert R.dmp_qq_collins_resultant(f, g) == r
+
+    f = -x**3 + 5
+    g = 3*x**2*y + x**2
+
+    a = 45*y**2 + 30*y + 5
+    b = 675*y**3 + 675*y**2 + 225*y + 25
+
+    r = R.dmp_LC(b)
+
+    assert R.dmp_subresultants(f, g) == [f, g, a]
+    assert R.dmp_resultant(f, g) == r
+    assert R.dmp_prs_resultant(f, g)[0] == r
+    assert R.dmp_zz_collins_resultant(f, g) == r
+    assert R.dmp_qq_collins_resultant(f, g) == r
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f = 6*x**2 - 3*x*y - 2*x*z + y*z
+    g = x**2 - x*u - x*v + u*v
+
+    r = y**2*z**2 - 3*y**2*z*u - 3*y**2*z*v + 9*y**2*u*v - 2*y*z**2*u \
+      - 2*y*z**2*v + 6*y*z*u**2 + 12*y*z*u*v + 6*y*z*v**2 - 18*y*u**2*v \
+      - 18*y*u*v**2 + 4*z**2*u*v - 12*z*u**2*v - 12*z*u*v**2 + 36*u**2*v**2
+
+    assert R.dmp_zz_collins_resultant(f, g) == r.drop(x)
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", QQ)
+
+    f = x**2 - QQ(1,2)*x*y - QQ(1,3)*x*z + QQ(1,6)*y*z
+    g = x**2 - x*u - x*v + u*v
+
+    r = QQ(1,36)*y**2*z**2 - QQ(1,12)*y**2*z*u - QQ(1,12)*y**2*z*v + QQ(1,4)*y**2*u*v \
+      - QQ(1,18)*y*z**2*u - QQ(1,18)*y*z**2*v + QQ(1,6)*y*z*u**2 + QQ(1,3)*y*z*u*v \
+      + QQ(1,6)*y*z*v**2 - QQ(1,2)*y*u**2*v - QQ(1,2)*y*u*v**2 + QQ(1,9)*z**2*u*v \
+      - QQ(1,3)*z*u**2*v - QQ(1,3)*z*u*v**2 + u**2*v**2
+
+    assert R.dmp_qq_collins_resultant(f, g) == r.drop(x)
+
+    Rt, t = ring("t", ZZ)
+    Rx, x = ring("x", Rt)
+
+    f = x**6 - 5*x**4 + 5*x**2 + 4
+    g = -6*t*x**5 + x**4 + 20*t*x**3 - 3*x**2 - 10*t*x + 6
+
+    assert Rx.dup_resultant(f, g) == 2930944*t**6 + 2198208*t**4 + 549552*t**2 + 45796
+
+
+def test_dup_discriminant():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_discriminant(0) == 0
+    assert R.dup_discriminant(x) == 1
+
+    assert R.dup_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664
+    assert R.dup_discriminant(5*x**5 + x**3 + 2) == 31252160
+    assert R.dup_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0
+    assert R.dup_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112
+
+
+def test_dmp_discriminant():
+    R, x = ring("x", ZZ)
+
+    assert R.dmp_discriminant(0) == 0
+
+    R, x, y = ring("x,y", ZZ)
+
+    assert R.dmp_discriminant(0) == 0
+    assert R.dmp_discriminant(y) == 0
+
+    assert R.dmp_discriminant(x**3 + 3*x**2 + 9*x - 13) == -11664
+    assert R.dmp_discriminant(5*x**5 + x**3 + 2) == 31252160
+    assert R.dmp_discriminant(x**4 + 2*x**3 + 6*x**2 - 22*x + 13) == 0
+    assert R.dmp_discriminant(12*x**7 + 15*x**4 + 30*x**3 + x**2 + 1) == -220289699947514112
+
+    assert R.dmp_discriminant(x**2*y + 2*y) == (-8*y**2).drop(x)
+    assert R.dmp_discriminant(x*y**2 + 2*x) == 1
+
+    R, x, y, z = ring("x,y,z", ZZ)
+    assert R.dmp_discriminant(x*y + z) == 1
+
+    R, x, y, z, u = ring("x,y,z,u", ZZ)
+    assert R.dmp_discriminant(x**2*y + x*z + u) == (-4*y*u + z**2).drop(x)
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+    assert R.dmp_discriminant(x**3*y + x**2*z + x*u + v) == \
+        (-27*y**2*v**2 + 18*y*z*u*v - 4*y*u**3 - 4*z**3*v + z**2*u**2).drop(x)
+
+
+def test_dup_gcd():
+    R, x = ring("x", ZZ)
+
+    f, g = 0, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (0, 0, 0)
+
+    f, g = 2, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 0)
+
+    f, g = -2, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 0)
+
+    f, g = 0, -2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 0, -1)
+
+    f, g = 0, 2*x + 4
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
+
+    f, g = 2*x + 4, 0
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
+
+    f, g = 2, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 1)
+
+    f, g = -2, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 1)
+
+    f, g = 2, -2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, -1)
+
+    f, g = -2, -2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, -1)
+
+    f, g = x**2 + 2*x + 1, 1
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
+
+    f, g = x**2 + 2*x + 1, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
+
+    f, g = 2*x**2 + 4*x + 2, 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
+
+    f, g = 2, 2*x**2 + 4*x + 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
+
+    f, g = 2*x**2 + 4*x + 2, x + 1
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
+
+    f, g = x + 1, 2*x**2 + 4*x + 2
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
+
+    f, g = x - 31, x
+    assert R.dup_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, f, g)
+
+    f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
+    g = x**3 + 6*x**2 + 11*x + 6
+
+    h = x**2 + 3*x + 2
+
+    cff = x**2 + 5*x + 4
+    cfg = x + 3
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    f = x**4 - 4
+    g = x**4 + 4*x**2 + 4
+
+    h = x**2 + 2
+
+    cff = x**2 - 2
+    cfg = x**2 + 2
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    h = 1
+
+    cff = f
+    cfg = g
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    R, x = ring("x", QQ)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    h = 1
+
+    cff = f
+    cfg = g
+
+    assert R.dup_qq_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dup_ff_prs_gcd(f, g) == (h, cff, cfg)
+
+    R, x = ring("x", ZZ)
+
+    f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+        + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+        + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+        + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
+        - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+        + 289127344604779611146960547954288113529690984687482920704*x**14 \
+        + 19007977035740498977629742919480623972236450681*x**7 \
+        + 311973482284542371301330321821976049
+
+    g =   365431878023781158602430064717380211405897160759702125019136*x**21 \
+        + 197599133478719444145775798221171663643171734081650688*x**14 \
+        - 9504116979659010018253915765478924103928886144*x**7 \
+        - 311973482284542371301330321821976049
+
+    assert R.dup_zz_heu_gcd(f, R.dup_diff(f, 1))[0] == g
+    assert R.dup_rr_prs_gcd(f, R.dup_diff(f, 1))[0] == g
+
+    R, x = ring("x", QQ)
+
+    f = QQ(1,2)*x**2 + x + QQ(1,2)
+    g = QQ(1,2)*x + QQ(1,2)
+
+    h = x + 1
+
+    assert R.dup_qq_heu_gcd(f, g) == (h, g, QQ(1,2))
+    assert R.dup_ff_prs_gcd(f, g) == (h, g, QQ(1,2))
+
+    R, x = ring("x", ZZ)
+
+    f = 1317378933230047068160*x + 2945748836994210856960
+    g = 120352542776360960*x + 269116466014453760
+
+    h = 120352542776360960*x + 269116466014453760
+    cff = 10946
+    cfg = 1
+
+    assert R.dup_zz_heu_gcd(f, g) == (h, cff, cfg)
+
+
+def test_dmp_gcd():
+    R, x, y = ring("x,y", ZZ)
+
+    f, g = 0, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (0, 0, 0)
+
+    f, g = 2, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 0)
+
+    f, g = -2, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 0)
+
+    f, g = 0, -2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 0, -1)
+
+    f, g = 0, 2*x + 4
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
+
+    f, g = 2*x + 4, 0
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
+
+    f, g = 2, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 1)
+
+    f, g = -2, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 1)
+
+    f, g = 2, -2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, -1)
+
+    f, g = -2, -2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, -1)
+
+    f, g = x**2 + 2*x + 1, 1
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
+
+    f, g = x**2 + 2*x + 1, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
+
+    f, g = 2*x**2 + 4*x + 2, 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
+
+    f, g = 2, 2*x**2 + 4*x + 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
+
+    f, g = 2*x**2 + 4*x + 2, x + 1
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
+
+    f, g = x + 1, 2*x**2 + 4*x + 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
+
+    R, x, y, z, u = ring("x,y,z,u", ZZ)
+
+    f, g = u**2 + 2*u + 1, 2*u + 2
+    assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (u + 1, u + 1, 2)
+
+    f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1
+    h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1
+
+    assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
+    assert R.dmp_rr_prs_gcd(f, g) == (h, cff, cfg)
+
+    assert R.dmp_zz_heu_gcd(g, f) == (h, cfg, cff)
+    assert R.dmp_rr_prs_gcd(g, f) == (h, cfg, cff)
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(4, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ))
+    H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ))
+    H, cff, cfg = R.dmp_inner_gcd(f, g)
+
+    assert H == h and R.dmp_mul(H, cff) == f \
+                  and R.dmp_mul(H, cfg) == g
+
+    R, x, y = ring("x,y", QQ)
+
+    f = QQ(1,2)*x**2 + x + QQ(1,2)
+    g = QQ(1,2)*x + QQ(1,2)
+
+    h = x + 1
+
+    assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1,2))
+    assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1,2))
+
+    R, x, y = ring("x,y", RR)
+
+    f = 2.1*x*y**2 - 2.2*x*y + 2.1*x
+    g = 1.0*x**3
+
+    assert R.dmp_ff_prs_gcd(f, g) == \
+        (1.0*x, 2.1*y**2 - 2.2*y + 2.1, 1.0*x**2)
+
+
+def test_dup_lcm():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_lcm(2, 6) == 6
+
+    assert R.dup_lcm(2*x**3, 6*x) == 6*x**3
+    assert R.dup_lcm(2*x**3, 3*x) == 6*x**3
+
+    assert R.dup_lcm(x**2 + x, x) == x**2 + x
+    assert R.dup_lcm(x**2 + x, 2*x) == 2*x**2 + 2*x
+    assert R.dup_lcm(x**2 + 2*x, x) == x**2 + 2*x
+    assert R.dup_lcm(2*x**2 + x, x) == 2*x**2 + x
+    assert R.dup_lcm(2*x**2 + x, 2*x) == 4*x**2 + 2*x
+
+
+def test_dmp_lcm():
+    R, x, y = ring("x,y", ZZ)
+
+    assert R.dmp_lcm(2, 6) == 6
+    assert R.dmp_lcm(x, y) == x*y
+
+    assert R.dmp_lcm(2*x**3, 6*x*y**2) == 6*x**3*y**2
+    assert R.dmp_lcm(2*x**3, 3*x*y**2) == 6*x**3*y**2
+
+    assert R.dmp_lcm(x**2*y, x*y**2) == x**2*y**2
+
+    f = 2*x*y**5 - 3*x*y**4 - 2*x*y**3 + 3*x*y**2
+    g = y**5 - 2*y**3 + y
+    h = 2*x*y**7 - 3*x*y**6 - 4*x*y**5 + 6*x*y**4 + 2*x*y**3 - 3*x*y**2
+
+    assert R.dmp_lcm(f, g) == h
+
+    f = x**3 - 3*x**2*y - 9*x*y**2 - 5*y**3
+    g = x**4 + 6*x**3*y + 12*x**2*y**2 + 10*x*y**3 + 3*y**4
+    h = x**5 + x**4*y - 18*x**3*y**2 - 50*x**2*y**3 - 47*x*y**4 - 15*y**5
+
+    assert R.dmp_lcm(f, g) == h
+
+
+def test_dmp_content():
+    R, x,y = ring("x,y", ZZ)
+
+    assert R.dmp_content(-2) == 2
+
+    f, g, F = 3*y**2 + 2*y + 1, 1, 0
+
+    for i in range(0, 5):
+        g *= f
+        F += x**i*g
+
+    assert R.dmp_content(F) == f.drop(x)
+
+    R, x,y,z = ring("x,y,z", ZZ)
+
+    assert R.dmp_content(f_4) == 1
+    assert R.dmp_content(f_5) == 1
+
+    R, x,y,z,t = ring("x,y,z,t", ZZ)
+    assert R.dmp_content(f_6) == 1
+
+
+def test_dmp_primitive():
+    R, x,y = ring("x,y", ZZ)
+
+    assert R.dmp_primitive(0) == (0, 0)
+    assert R.dmp_primitive(1) == (1, 1)
+
+    f, g, F = 3*y**2 + 2*y + 1, 1, 0
+
+    for i in range(0, 5):
+        g *= f
+        F += x**i*g
+
+    assert R.dmp_primitive(F) == (f.drop(x), F / f)
+
+    R, x,y,z = ring("x,y,z", ZZ)
+
+    cont, f = R.dmp_primitive(f_4)
+    assert cont == 1 and f == f_4
+    cont, f = R.dmp_primitive(f_5)
+    assert cont == 1 and f == f_5
+
+    R, x,y,z,t = ring("x,y,z,t", ZZ)
+
+    cont, f = R.dmp_primitive(f_6)
+    assert cont == 1 and f == f_6
+
+
+def test_dup_cancel():
+    R, x = ring("x", ZZ)
+
+    f = 2*x**2 - 2
+    g = x**2 - 2*x + 1
+
+    p = 2*x + 2
+    q = x - 1
+
+    assert R.dup_cancel(f, g) == (p, q)
+    assert R.dup_cancel(f, g, include=False) == (1, 1, p, q)
+
+    f = -x - 2
+    g = 3*x - 4
+
+    F = x + 2
+    G = -3*x + 4
+
+    assert R.dup_cancel(f, g) == (f, g)
+    assert R.dup_cancel(F, G) == (f, g)
+
+    assert R.dup_cancel(0, 0) == (0, 0)
+    assert R.dup_cancel(0, 0, include=False) == (1, 1, 0, 0)
+
+    assert R.dup_cancel(x, 0) == (1, 0)
+    assert R.dup_cancel(x, 0, include=False) == (1, 1, 1, 0)
+
+    assert R.dup_cancel(0, x) == (0, 1)
+    assert R.dup_cancel(0, x, include=False) == (1, 1, 0, 1)
+
+    f = 0
+    g = x
+    one = 1
+
+    assert R.dup_cancel(f, g, include=True) == (f, one)
+
+
+def test_dmp_cancel():
+    R, x, y = ring("x,y", ZZ)
+
+    f = 2*x**2 - 2
+    g = x**2 - 2*x + 1
+
+    p = 2*x + 2
+    q = x - 1
+
+    assert R.dmp_cancel(f, g) == (p, q)
+    assert R.dmp_cancel(f, g, include=False) == (1, 1, p, q)
+
+    assert R.dmp_cancel(0, 0) == (0, 0)
+    assert R.dmp_cancel(0, 0, include=False) == (1, 1, 0, 0)
+
+    assert R.dmp_cancel(y, 0) == (1, 0)
+    assert R.dmp_cancel(y, 0, include=False) == (1, 1, 1, 0)
+
+    assert R.dmp_cancel(0, y) == (0, 1)
+    assert R.dmp_cancel(0, y, include=False) == (1, 1, 0, 1)

rl/Lib/site-packages/sympy/polys/tests/test_factortools.py

@@ -0,0 +1,784 @@
+"""Tools for polynomial factorization routines in characteristic zero. """
+
+from sympy.polys.rings import ring, xring
+from sympy.polys.domains import FF, ZZ, QQ, ZZ_I, QQ_I, RR, EX
+
+from sympy.polys import polyconfig as config
+from sympy.polys.polyerrors import DomainError
+from sympy.polys.polyclasses import ANP
+from sympy.polys.specialpolys import f_polys, w_polys
+
+from sympy.core.numbers import I
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin
+from sympy.ntheory.generate import nextprime
+from sympy.testing.pytest import raises, XFAIL
+
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys()
+w_1, w_2 = w_polys()
+
+def test_dup_trial_division():
+    R, x = ring("x", ZZ)
+    assert R.dup_trial_division(x**5 + 8*x**4 + 25*x**3 + 38*x**2 + 28*x + 8, (x + 1, x + 2)) == [(x + 1, 2), (x + 2, 3)]
+
+
+def test_dmp_trial_division():
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_trial_division(x**5 + 8*x**4 + 25*x**3 + 38*x**2 + 28*x + 8, (x + 1, x + 2)) == [(x + 1, 2), (x + 2, 3)]
+
+
+def test_dup_zz_mignotte_bound():
+    R, x = ring("x", ZZ)
+    assert R.dup_zz_mignotte_bound(2*x**2 + 3*x + 4) == 6
+    assert R.dup_zz_mignotte_bound(x**3 + 14*x**2 + 56*x + 64) == 152
+
+
+def test_dmp_zz_mignotte_bound():
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_zz_mignotte_bound(2*x**2 + 3*x + 4) == 32
+
+
+def test_dup_zz_hensel_step():
+    R, x = ring("x", ZZ)
+
+    f = x**4 - 1
+    g = x**3 + 2*x**2 - x - 2
+    h = x - 2
+    s = -2
+    t = 2*x**2 - 2*x - 1
+
+    G, H, S, T = R.dup_zz_hensel_step(5, f, g, h, s, t)
+
+    assert G == x**3 + 7*x**2 - x - 7
+    assert H == x - 7
+    assert S == 8
+    assert T == -8*x**2 - 12*x - 1
+
+
+def test_dup_zz_hensel_lift():
+    R, x = ring("x", ZZ)
+
+    f = x**4 - 1
+    F = [x - 1, x - 2, x + 2, x + 1]
+
+    assert R.dup_zz_hensel_lift(ZZ(5), f, F, 4) == \
+        [x - 1, x - 182, x + 182, x + 1]
+
+
+def test_dup_zz_irreducible_p():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 7) is None
+    assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 4) is None
+
+    assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 10) is True
+    assert R.dup_zz_irreducible_p(3*x**4 + 2*x**3 + 6*x**2 + 8*x + 14) is True
+
+
+def test_dup_cyclotomic_p():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_cyclotomic_p(x - 1) is True
+    assert R.dup_cyclotomic_p(x + 1) is True
+    assert R.dup_cyclotomic_p(x**2 + x + 1) is True
+    assert R.dup_cyclotomic_p(x**2 + 1) is True
+    assert R.dup_cyclotomic_p(x**4 + x**3 + x**2 + x + 1) is True
+    assert R.dup_cyclotomic_p(x**2 - x + 1) is True
+    assert R.dup_cyclotomic_p(x**6 + x**5 + x**4 + x**3 + x**2 + x + 1) is True
+    assert R.dup_cyclotomic_p(x**4 + 1) is True
+    assert R.dup_cyclotomic_p(x**6 + x**3 + 1) is True
+
+    assert R.dup_cyclotomic_p(0) is False
+    assert R.dup_cyclotomic_p(1) is False
+    assert R.dup_cyclotomic_p(x) is False
+    assert R.dup_cyclotomic_p(x + 2) is False
+    assert R.dup_cyclotomic_p(3*x + 1) is False
+    assert R.dup_cyclotomic_p(x**2 - 1) is False
+
+    f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1
+    assert R.dup_cyclotomic_p(f) is False
+
+    g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1
+    assert R.dup_cyclotomic_p(g) is True
+
+    R, x = ring("x", QQ)
+    assert R.dup_cyclotomic_p(x**2 + x + 1) is True
+    assert R.dup_cyclotomic_p(QQ(1,2)*x**2 + x + 1) is False
+
+    R, x = ring("x", ZZ["y"])
+    assert R.dup_cyclotomic_p(x**2 + x + 1) is False
+
+
+def test_dup_zz_cyclotomic_poly():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_zz_cyclotomic_poly(1) == x - 1
+    assert R.dup_zz_cyclotomic_poly(2) == x + 1
+    assert R.dup_zz_cyclotomic_poly(3) == x**2 + x + 1
+    assert R.dup_zz_cyclotomic_poly(4) == x**2 + 1
+    assert R.dup_zz_cyclotomic_poly(5) == x**4 + x**3 + x**2 + x + 1
+    assert R.dup_zz_cyclotomic_poly(6) == x**2 - x + 1
+    assert R.dup_zz_cyclotomic_poly(7) == x**6 + x**5 + x**4 + x**3 + x**2 + x + 1
+    assert R.dup_zz_cyclotomic_poly(8) == x**4 + 1
+    assert R.dup_zz_cyclotomic_poly(9) == x**6 + x**3 + 1
+
+
+def test_dup_zz_cyclotomic_factor():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_zz_cyclotomic_factor(0) is None
+    assert R.dup_zz_cyclotomic_factor(1) is None
+
+    assert R.dup_zz_cyclotomic_factor(2*x**10 - 1) is None
+    assert R.dup_zz_cyclotomic_factor(x**10 - 3) is None
+    assert R.dup_zz_cyclotomic_factor(x**10 + x**5 - 1) is None
+
+    assert R.dup_zz_cyclotomic_factor(x + 1) == [x + 1]
+    assert R.dup_zz_cyclotomic_factor(x - 1) == [x - 1]
+
+    assert R.dup_zz_cyclotomic_factor(x**2 + 1) == [x**2 + 1]
+    assert R.dup_zz_cyclotomic_factor(x**2 - 1) == [x - 1, x + 1]
+
+    assert R.dup_zz_cyclotomic_factor(x**27 + 1) == \
+        [x + 1, x**2 - x + 1, x**6 - x**3 + 1, x**18 - x**9 + 1]
+    assert R.dup_zz_cyclotomic_factor(x**27 - 1) == \
+        [x - 1, x**2 + x + 1, x**6 + x**3 + 1, x**18 + x**9 + 1]
+
+
+def test_dup_zz_factor():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_zz_factor(0) == (0, [])
+    assert R.dup_zz_factor(7) == (7, [])
+    assert R.dup_zz_factor(-7) == (-7, [])
+
+    assert R.dup_zz_factor_sqf(0) == (0, [])
+    assert R.dup_zz_factor_sqf(7) == (7, [])
+    assert R.dup_zz_factor_sqf(-7) == (-7, [])
+
+    assert R.dup_zz_factor(2*x + 4) == (2, [(x + 2, 1)])
+    assert R.dup_zz_factor_sqf(2*x + 4) == (2, [x + 2])
+
+    f = x**4 + x + 1
+
+    for i in range(0, 20):
+        assert R.dup_zz_factor(f) == (1, [(f, 1)])
+
+    assert R.dup_zz_factor(x**2 + 2*x + 2) == \
+        (1, [(x**2 + 2*x + 2, 1)])
+
+    assert R.dup_zz_factor(18*x**2 + 12*x + 2) == \
+        (2, [(3*x + 1, 2)])
+
+    assert R.dup_zz_factor(-9*x**2 + 1) == \
+        (-1, [(3*x - 1, 1),
+              (3*x + 1, 1)])
+
+    assert R.dup_zz_factor_sqf(-9*x**2 + 1) == \
+        (-1, [3*x - 1,
+              3*x + 1])
+
+    # The order of the factors will be different when the ground types are
+    # flint. At the higher level dup_factor_list will sort the factors.
+    c, factors = R.dup_zz_factor(x**3 - 6*x**2 + 11*x - 6)
+    assert c == 1
+    assert set(factors) == {(x - 3, 1), (x - 2, 1), (x - 1, 1)}
+
+    assert R.dup_zz_factor_sqf(x**3 - 6*x**2 + 11*x - 6) == \
+        (1, [x - 3,
+             x - 2,
+             x - 1])
+
+    assert R.dup_zz_factor(3*x**3 + 10*x**2 + 13*x + 10) == \
+        (1, [(x + 2, 1),
+             (3*x**2 + 4*x + 5, 1)])
+
+    assert R.dup_zz_factor_sqf(3*x**3 + 10*x**2 + 13*x + 10) == \
+        (1, [x + 2,
+             3*x**2 + 4*x + 5])
+
+    c, factors = R.dup_zz_factor(-x**6 + x**2)
+    assert c == -1
+    assert set(factors) == {(x, 2), (x - 1, 1), (x + 1, 1), (x**2 + 1, 1)}
+
+    f = 1080*x**8 + 5184*x**7 + 2099*x**6 + 744*x**5 + 2736*x**4 - 648*x**3 + 129*x**2 - 324
+
+    assert R.dup_zz_factor(f) == \
+        (1, [(5*x**4 + 24*x**3 + 9*x**2 + 12, 1),
+             (216*x**4 + 31*x**2 - 27, 1)])
+
+    f = -29802322387695312500000000000000000000*x**25 \
+      + 2980232238769531250000000000000000*x**20 \
+      + 1743435859680175781250000000000*x**15 \
+      + 114142894744873046875000000*x**10 \
+      - 210106372833251953125*x**5 \
+      + 95367431640625
+
+    c, factors = R.dup_zz_factor(f)
+    assert c == -95367431640625
+    assert set(factors) == {
+        (5*x - 1, 1),
+        (100*x**2 + 10*x - 1, 2),
+        (625*x**4 + 125*x**3 + 25*x**2 + 5*x + 1, 1),
+        (10000*x**4 - 3000*x**3 + 400*x**2 - 20*x + 1, 2),
+        (10000*x**4 + 2000*x**3 + 400*x**2 + 30*x + 1, 2),
+    }
+
+    f = x**10 - 1
+
+    config.setup('USE_CYCLOTOMIC_FACTOR', True)
+    c0, F_0 = R.dup_zz_factor(f)
+
+    config.setup('USE_CYCLOTOMIC_FACTOR', False)
+    c1, F_1 = R.dup_zz_factor(f)
+
+    assert c0 == c1 == 1
+    assert set(F_0) == set(F_1) == {
+        (x - 1, 1),
+        (x + 1, 1),
+        (x**4 - x**3 + x**2 - x + 1, 1),
+        (x**4 + x**3 + x**2 + x + 1, 1),
+    }
+
+    config.setup('USE_CYCLOTOMIC_FACTOR')
+
+    f = x**10 + 1
+
+    config.setup('USE_CYCLOTOMIC_FACTOR', True)
+    F_0 = R.dup_zz_factor(f)
+
+    config.setup('USE_CYCLOTOMIC_FACTOR', False)
+    F_1 = R.dup_zz_factor(f)
+
+    assert F_0 == F_1 == \
+        (1, [(x**2 + 1, 1),
+             (x**8 - x**6 + x**4 - x**2 + 1, 1)])
+
+    config.setup('USE_CYCLOTOMIC_FACTOR')
+
+def test_dmp_zz_wang():
+    R, x,y,z = ring("x,y,z", ZZ)
+    UV, _x = ring("x", ZZ)
+
+    p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1)))
+    assert p == 6291469
+
+    t_1, k_1, e_1 = y, 1, ZZ(-14)
+    t_2, k_2, e_2 = z, 2, ZZ(3)
+    t_3, k_3, e_3 = y + z, 2, ZZ(-11)
+    t_4, k_4, e_4 = y - z, 1, ZZ(-17)
+
+    T = [t_1, t_2, t_3, t_4]
+    K = [k_1, k_2, k_3, k_4]
+    E = [e_1, e_2, e_3, e_4]
+
+    T = zip([ t.drop(x) for t in T ], K)
+
+    A = [ZZ(-14), ZZ(3)]
+
+    S = R.dmp_eval_tail(w_1, A)
+    cs, s = UV.dup_primitive(S)
+
+    assert cs == 1 and s == S == \
+        1036728*_x**6 + 915552*_x**5 + 55748*_x**4 + 105621*_x**3 - 17304*_x**2 - 26841*_x - 644
+
+    assert R.dmp_zz_wang_non_divisors(E, cs, ZZ(4)) == [7, 3, 11, 17]
+    assert UV.dup_sqf_p(s) and UV.dup_degree(s) == R.dmp_degree(w_1)
+
+    _, H = UV.dup_zz_factor_sqf(s)
+
+    h_1 = 44*_x**2 + 42*_x + 1
+    h_2 = 126*_x**2 - 9*_x + 28
+    h_3 = 187*_x**2 - 23
+
+    assert H == [h_1, h_2, h_3]
+
+    LC = [ lc.drop(x) for lc in [-4*y - 4*z, -y*z**2, y**2 - z**2] ]
+
+    assert R.dmp_zz_wang_lead_coeffs(w_1, T, cs, E, H, A) == (w_1, H, LC)
+
+    factors = R.dmp_zz_wang_hensel_lifting(w_1, H, LC, A, p)
+    assert R.dmp_expand(factors) == w_1
+
+
+@XFAIL
+def test_dmp_zz_wang_fail():
+    R, x,y,z = ring("x,y,z", ZZ)
+    UV, _x = ring("x", ZZ)
+
+    p = ZZ(nextprime(R.dmp_zz_mignotte_bound(w_1)))
+    assert p == 6291469
+
+    H_1 = [44*x**2 + 42*x + 1, 126*x**2 - 9*x + 28, 187*x**2 - 23]
+    H_2 = [-4*x**2*y - 12*x**2 - 3*x*y + 1, -9*x**2*y - 9*x - 2*y, x**2*y**2 - 9*x**2 + y - 9]
+    H_3 = [-4*x**2*y - 12*x**2 - 3*x*y + 1, -9*x**2*y - 9*x - 2*y, x**2*y**2 - 9*x**2 + y - 9]
+
+    c_1 = -70686*x**5 - 5863*x**4 - 17826*x**3 + 2009*x**2 + 5031*x + 74
+    c_2 = 9*x**5*y**4 + 12*x**5*y**3 - 45*x**5*y**2 - 108*x**5*y - 324*x**5 + 18*x**4*y**3 - 216*x**4*y**2 - 810*x**4*y + 2*x**3*y**4 + 9*x**3*y**3 - 252*x**3*y**2 - 288*x**3*y - 945*x**3 - 30*x**2*y**2 - 414*x**2*y + 2*x*y**3 - 54*x*y**2 - 3*x*y + 81*x + 12*y
+    c_3 = -36*x**4*y**2 - 108*x**4*y - 27*x**3*y**2 - 36*x**3*y - 108*x**3 - 8*x**2*y**2 - 42*x**2*y - 6*x*y**2 + 9*x + 2*y
+
+    assert R.dmp_zz_diophantine(H_1, c_1, [], 5, p) == [-3*x, -2, 1]
+    assert R.dmp_zz_diophantine(H_2, c_2, [ZZ(-14)], 5, p) == [-x*y, -3*x, -6]
+    assert R.dmp_zz_diophantine(H_3, c_3, [ZZ(-14)], 5, p) == [0, 0, -1]
+
+
+def test_issue_6355():
+    # This tests a bug in the Wang algorithm that occurred only with a very
+    # specific set of random numbers.
+    random_sequence = [-1, -1, 0, 0, 0, 0, -1, -1, 0, -1, 3, -1, 3, 3, 3, 3, -1, 3]
+
+    R, x, y, z = ring("x,y,z", ZZ)
+    f = 2*x**2 + y*z - y - z**2 + z
+
+    assert R.dmp_zz_wang(f, seed=random_sequence) == [f]
+
+
+def test_dmp_zz_factor():
+    R, x = ring("x", ZZ)
+    assert R.dmp_zz_factor(0) == (0, [])
+    assert R.dmp_zz_factor(7) == (7, [])
+    assert R.dmp_zz_factor(-7) == (-7, [])
+
+    assert R.dmp_zz_factor(x**2 - 9) == (1, [(x - 3, 1), (x + 3, 1)])
+
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_zz_factor(0) == (0, [])
+    assert R.dmp_zz_factor(7) == (7, [])
+    assert R.dmp_zz_factor(-7) == (-7, [])
+
+    assert R.dmp_zz_factor(x) == (1, [(x, 1)])
+    assert R.dmp_zz_factor(4*x) == (4, [(x, 1)])
+    assert R.dmp_zz_factor(4*x + 2) == (2, [(2*x + 1, 1)])
+    assert R.dmp_zz_factor(x*y + 1) == (1, [(x*y + 1, 1)])
+    assert R.dmp_zz_factor(y**2 + 1) == (1, [(y**2 + 1, 1)])
+    assert R.dmp_zz_factor(y**2 - 1) == (1, [(y - 1, 1), (y + 1, 1)])
+
+    assert R.dmp_zz_factor(x**2*y**2 + 6*x**2*y + 9*x**2 - 1) == (1, [(x*y + 3*x - 1, 1), (x*y + 3*x + 1, 1)])
+    assert R.dmp_zz_factor(x**2*y**2 - 9) == (1, [(x*y - 3, 1), (x*y + 3, 1)])
+
+    R, x, y, z = ring("x,y,z", ZZ)
+    assert R.dmp_zz_factor(x**2*y**2*z**2 - 9) == \
+        (1, [(x*y*z - 3, 1),
+             (x*y*z + 3, 1)])
+
+    R, x, y, z, u = ring("x,y,z,u", ZZ)
+    assert R.dmp_zz_factor(x**2*y**2*z**2*u**2 - 9) == \
+        (1, [(x*y*z*u - 3, 1),
+             (x*y*z*u + 3, 1)])
+
+    R, x, y, z = ring("x,y,z", ZZ)
+    assert R.dmp_zz_factor(f_1) == \
+        (1, [(x + y*z + 20, 1),
+             (x*y + z + 10, 1),
+             (x*z + y + 30, 1)])
+
+    assert R.dmp_zz_factor(f_2) == \
+        (1, [(x**2*y**2 + x**2*z**2 + y + 90, 1),
+             (x**3*y + x**3*z + z - 11, 1)])
+
+    assert R.dmp_zz_factor(f_3) == \
+        (1, [(x**2*y**2 + x*z**4 + x + z, 1),
+             (x**3 + x*y*z + y**2 + y*z**3, 1)])
+
+    assert R.dmp_zz_factor(f_4) == \
+        (-1, [(x*y**3 + z**2, 1),
+              (x**2*z + y**4*z**2 + 5, 1),
+              (x**3*y - z**2 - 3, 1),
+              (x**3*y**4 + z**2, 1)])
+
+    assert R.dmp_zz_factor(f_5) == \
+        (-1, [(x + y - z, 3)])
+
+    R, x, y, z, t = ring("x,y,z,t", ZZ)
+    assert R.dmp_zz_factor(f_6) == \
+        (1, [(47*x*y + z**3*t**2 - t**2, 1),
+             (45*x**3 - 9*y**3 - y**2 + 3*z**3 + 2*z*t, 1)])
+
+    R, x, y, z = ring("x,y,z", ZZ)
+    assert R.dmp_zz_factor(w_1) == \
+        (1, [(x**2*y**2 - x**2*z**2 + y - z**2, 1),
+             (x**2*y*z**2 + 3*x*z + 2*y, 1),
+             (4*x**2*y + 4*x**2*z + x*y*z - 1, 1)])
+
+    R, x, y = ring("x,y", ZZ)
+    f = -12*x**16*y + 240*x**12*y**3 - 768*x**10*y**4 + 1080*x**8*y**5 - 768*x**6*y**6 + 240*x**4*y**7 - 12*y**9
+
+    assert R.dmp_zz_factor(f) == \
+        (-12, [(y, 1),
+               (x**2 - y, 6),
+               (x**4 + 6*x**2*y + y**2, 1)])
+
+
+def test_dup_qq_i_factor():
+    R, x = ring("x", QQ_I)
+    i = QQ_I(0, 1)
+
+    assert R.dup_qq_i_factor(x**2 - 2) == (QQ_I(1, 0), [(x**2 - 2, 1)])
+
+    assert R.dup_qq_i_factor(x**2 - 1) == (QQ_I(1, 0), [(x - 1, 1), (x + 1, 1)])
+
+    assert R.dup_qq_i_factor(x**2 + 1) == (QQ_I(1, 0), [(x - i, 1), (x + i, 1)])
+
+    assert R.dup_qq_i_factor(x**2/4 + 1) == \
+            (QQ_I(QQ(1, 4), 0), [(x - 2*i, 1), (x + 2*i, 1)])
+
+    assert R.dup_qq_i_factor(x**2 + 4) == \
+            (QQ_I(1, 0), [(x - 2*i, 1), (x + 2*i, 1)])
+
+    assert R.dup_qq_i_factor(x**2 + 2*x + 1) == \
+            (QQ_I(1, 0), [(x + 1, 2)])
+
+    assert R.dup_qq_i_factor(x**2 + 2*i*x - 1) == \
+            (QQ_I(1, 0), [(x + i, 2)])
+
+    f = 8192*x**2 + x*(22656 + 175232*i) - 921416 + 242313*i
+
+    assert R.dup_qq_i_factor(f) == \
+            (QQ_I(8192, 0), [(x + QQ_I(QQ(177, 128), QQ(1369, 128)), 2)])
+
+
+def test_dmp_qq_i_factor():
+    R, x, y = ring("x, y", QQ_I)
+    i = QQ_I(0, 1)
+
+    assert R.dmp_qq_i_factor(x**2 + 2*y**2) == \
+            (QQ_I(1, 0), [(x**2 + 2*y**2, 1)])
+
+    assert R.dmp_qq_i_factor(x**2 + y**2) == \
+            (QQ_I(1, 0), [(x - i*y, 1), (x + i*y, 1)])
+
+    assert R.dmp_qq_i_factor(x**2 + y**2/4) == \
+            (QQ_I(1, 0), [(x - i*y/2, 1), (x + i*y/2, 1)])
+
+    assert R.dmp_qq_i_factor(4*x**2 + y**2) == \
+            (QQ_I(4, 0), [(x - i*y/2, 1), (x + i*y/2, 1)])
+
+
+def test_dup_zz_i_factor():
+    R, x = ring("x", ZZ_I)
+    i = ZZ_I(0, 1)
+
+    assert R.dup_zz_i_factor(x**2 - 2) == (ZZ_I(1, 0), [(x**2 - 2, 1)])
+
+    assert R.dup_zz_i_factor(x**2 - 1) == (ZZ_I(1, 0), [(x - 1, 1), (x + 1, 1)])
+
+    assert R.dup_zz_i_factor(x**2 + 1) == (ZZ_I(1, 0), [(x - i, 1), (x + i, 1)])
+
+    assert R.dup_zz_i_factor(x**2 + 4) == \
+            (ZZ_I(1, 0), [(x - 2*i, 1), (x + 2*i, 1)])
+
+    assert R.dup_zz_i_factor(x**2 + 2*x + 1) == \
+            (ZZ_I(1, 0), [(x + 1, 2)])
+
+    assert R.dup_zz_i_factor(x**2 + 2*i*x - 1) == \
+            (ZZ_I(1, 0), [(x + i, 2)])
+
+    f = 8192*x**2 + x*(22656 + 175232*i) - 921416 + 242313*i
+
+    assert R.dup_zz_i_factor(f) == \
+            (ZZ_I(0, 1), [((64 - 64*i)*x + (773 + 596*i), 2)])
+
+
+def test_dmp_zz_i_factor():
+    R, x, y = ring("x, y", ZZ_I)
+    i = ZZ_I(0, 1)
+
+    assert R.dmp_zz_i_factor(x**2 + 2*y**2) == \
+            (ZZ_I(1, 0), [(x**2 + 2*y**2, 1)])
+
+    assert R.dmp_zz_i_factor(x**2 + y**2) == \
+            (ZZ_I(1, 0), [(x - i*y, 1), (x + i*y, 1)])
+
+    assert R.dmp_zz_i_factor(4*x**2 + y**2) == \
+            (ZZ_I(1, 0), [(2*x - i*y, 1), (2*x + i*y, 1)])
+
+
+def test_dup_ext_factor():
+    R, x = ring("x", QQ.algebraic_field(I))
+    def anp(element):
+        return ANP(element, [QQ(1), QQ(0), QQ(1)], QQ)
+
+    assert R.dup_ext_factor(0) == (anp([]), [])
+
+    f = anp([QQ(1)])*x + anp([QQ(1)])
+
+    assert R.dup_ext_factor(f) == (anp([QQ(1)]), [(f, 1)])
+
+    g = anp([QQ(2)])*x + anp([QQ(2)])
+
+    assert R.dup_ext_factor(g) == (anp([QQ(2)]), [(f, 1)])
+
+    f = anp([QQ(7)])*x**4 + anp([QQ(1, 1)])
+    g = anp([QQ(1)])*x**4 + anp([QQ(1, 7)])
+
+    assert R.dup_ext_factor(f) == (anp([QQ(7)]), [(g, 1)])
+
+    f = anp([QQ(1)])*x**4 + anp([QQ(1)])
+
+    assert R.dup_ext_factor(f) == \
+        (anp([QQ(1, 1)]), [(anp([QQ(1)])*x**2 + anp([QQ(-1), QQ(0)]), 1),
+                           (anp([QQ(1)])*x**2 + anp([QQ( 1), QQ(0)]), 1)])
+
+    f = anp([QQ(4, 1)])*x**2 + anp([QQ(9, 1)])
+
+    assert R.dup_ext_factor(f) == \
+        (anp([QQ(4, 1)]), [(anp([QQ(1, 1)])*x + anp([-QQ(3, 2), QQ(0, 1)]), 1),
+                           (anp([QQ(1, 1)])*x + anp([ QQ(3, 2), QQ(0, 1)]), 1)])
+
+    f = anp([QQ(4, 1)])*x**4 + anp([QQ(8, 1)])*x**3 + anp([QQ(77, 1)])*x**2 + anp([QQ(18, 1)])*x + anp([QQ(153, 1)])
+
+    assert R.dup_ext_factor(f) == \
+        (anp([QQ(4, 1)]), [(anp([QQ(1, 1)])*x + anp([-QQ(4, 1), QQ(1, 1)]), 1),
+                           (anp([QQ(1, 1)])*x + anp([-QQ(3, 2), QQ(0, 1)]), 1),
+                           (anp([QQ(1, 1)])*x + anp([ QQ(3, 2), QQ(0, 1)]), 1),
+                           (anp([QQ(1, 1)])*x + anp([ QQ(4, 1), QQ(1, 1)]), 1)])
+
+    R, x = ring("x", QQ.algebraic_field(sqrt(2)))
+    def anp(element):
+        return ANP(element, [QQ(1), QQ(0), QQ(-2)], QQ)
+
+    f = anp([QQ(1)])*x**4 + anp([QQ(1, 1)])
+
+    assert R.dup_ext_factor(f) == \
+        (anp([QQ(1)]), [(anp([QQ(1)])*x**2 + anp([QQ(-1), QQ(0)])*x + anp([QQ(1)]), 1),
+                        (anp([QQ(1)])*x**2 + anp([QQ( 1), QQ(0)])*x + anp([QQ(1)]), 1)])
+
+    f = anp([QQ(1, 1)])*x**2 + anp([QQ(2), QQ(0)])*x + anp([QQ(2, 1)])
+
+    assert R.dup_ext_factor(f) == \
+        (anp([QQ(1, 1)]), [(anp([1])*x + anp([1, 0]), 2)])
+
+    assert R.dup_ext_factor(f**3) == \
+        (anp([QQ(1, 1)]), [(anp([1])*x + anp([1, 0]), 6)])
+
+    f *= anp([QQ(2, 1)])
+
+    assert R.dup_ext_factor(f) == \
+        (anp([QQ(2, 1)]), [(anp([1])*x + anp([1, 0]), 2)])
+
+    assert R.dup_ext_factor(f**3) == \
+        (anp([QQ(8, 1)]), [(anp([1])*x + anp([1, 0]), 6)])
+
+
+def test_dmp_ext_factor():
+    K = QQ.algebraic_field(sqrt(2))
+    R, x,y = ring("x,y", K)
+    sqrt2 = K.unit
+
+    def anp(x):
+        return ANP(x, [QQ(1), QQ(0), QQ(-2)], QQ)
+
+    assert R.dmp_ext_factor(0) == (anp([]), [])
+
+    f = anp([QQ(1)])*x + anp([QQ(1)])
+
+    assert R.dmp_ext_factor(f) == (anp([QQ(1)]), [(f, 1)])
+
+    g = anp([QQ(2)])*x + anp([QQ(2)])
+
+    assert R.dmp_ext_factor(g) == (anp([QQ(2)]), [(f, 1)])
+
+    f = anp([QQ(1)])*x**2 + anp([QQ(-2)])*y**2
+
+    assert R.dmp_ext_factor(f) == \
+        (anp([QQ(1)]), [(anp([QQ(1)])*x + anp([QQ(-1), QQ(0)])*y, 1),
+                        (anp([QQ(1)])*x + anp([QQ( 1), QQ(0)])*y, 1)])
+
+    f = anp([QQ(2)])*x**2 + anp([QQ(-4)])*y**2
+
+    assert R.dmp_ext_factor(f) == \
+        (anp([QQ(2)]), [(anp([QQ(1)])*x + anp([QQ(-1), QQ(0)])*y, 1),
+                        (anp([QQ(1)])*x + anp([QQ( 1), QQ(0)])*y, 1)])
+
+    f1 = y + 1
+    f2 = y + sqrt2
+    f3 = x**2 + x + 2 + 3*sqrt2
+    f = f1**2 * f2**2 * f3**2
+    assert R.dmp_ext_factor(f) == (K.one, [(f1, 2), (f2, 2), (f3, 2)])
+
+
+def test_dup_factor_list():
+    R, x = ring("x", ZZ)
+    assert R.dup_factor_list(0) == (0, [])
+    assert R.dup_factor_list(7) == (7, [])
+
+    R, x = ring("x", QQ)
+    assert R.dup_factor_list(0) == (0, [])
+    assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
+
+    R, x = ring("x", ZZ['t'])
+    assert R.dup_factor_list(0) == (0, [])
+    assert R.dup_factor_list(7) == (7, [])
+
+    R, x = ring("x", QQ['t'])
+    assert R.dup_factor_list(0) == (0, [])
+    assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
+
+    R, x = ring("x", ZZ)
+    assert R.dup_factor_list_include(0) == [(0, 1)]
+    assert R.dup_factor_list_include(7) == [(7, 1)]
+
+    assert R.dup_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
+    assert R.dup_factor_list_include(x**2 + 2*x + 1) == [(x + 1, 2)]
+    # issue 8037
+    assert R.dup_factor_list(6*x**2 - 5*x - 6) == (1, [(2*x - 3, 1), (3*x + 2, 1)])
+
+    R, x = ring("x", QQ)
+    assert R.dup_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1, 2), [(x + 1, 2)])
+
+    R, x = ring("x", FF(2))
+    assert R.dup_factor_list(x**2 + 1) == (1, [(x + 1, 2)])
+
+    R, x = ring("x", RR)
+    assert R.dup_factor_list(1.0*x**2 + 2.0*x + 1.0) == (1.0, [(1.0*x + 1.0, 2)])
+    assert R.dup_factor_list(2.0*x**2 + 4.0*x + 2.0) == (2.0, [(1.0*x + 1.0, 2)])
+
+    f = 6.7225336055071*x**2 - 10.6463972754741*x - 0.33469524022264
+    coeff, factors = R.dup_factor_list(f)
+    assert coeff == RR(10.6463972754741)
+    assert len(factors) == 1
+    assert factors[0][0].max_norm() == RR(1.0)
+    assert factors[0][1] == 1
+
+    Rt, t = ring("t", ZZ)
+    R, x = ring("x", Rt)
+
+    f = 4*t*x**2 + 4*t**2*x
+
+    assert R.dup_factor_list(f) == \
+        (4*t, [(x, 1),
+             (x + t, 1)])
+
+    Rt, t = ring("t", QQ)
+    R, x = ring("x", Rt)
+
+    f = QQ(1, 2)*t*x**2 + QQ(1, 2)*t**2*x
+
+    assert R.dup_factor_list(f) == \
+        (QQ(1, 2)*t, [(x, 1),
+                    (x + t, 1)])
+
+    R, x = ring("x", QQ.algebraic_field(I))
+    def anp(element):
+        return ANP(element, [QQ(1), QQ(0), QQ(1)], QQ)
+
+    f = anp([QQ(1, 1)])*x**4 + anp([QQ(2, 1)])*x**2
+
+    assert R.dup_factor_list(f) == \
+        (anp([QQ(1, 1)]), [(anp([QQ(1, 1)])*x, 2),
+                           (anp([QQ(1, 1)])*x**2 + anp([])*x + anp([QQ(2, 1)]), 1)])
+
+    R, x = ring("x", EX)
+    raises(DomainError, lambda: R.dup_factor_list(EX(sin(1))))
+
+
+def test_dmp_factor_list():
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_factor_list(0) == (ZZ(0), [])
+    assert R.dmp_factor_list(7) == (7, [])
+
+    R, x, y = ring("x,y", QQ)
+    assert R.dmp_factor_list(0) == (QQ(0), [])
+    assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
+
+    Rt, t = ring("t", ZZ)
+    R, x, y = ring("x,y", Rt)
+    assert R.dmp_factor_list(0) == (0, [])
+    assert R.dmp_factor_list(7) == (ZZ(7), [])
+
+    Rt, t = ring("t", QQ)
+    R, x, y = ring("x,y", Rt)
+    assert R.dmp_factor_list(0) == (0, [])
+    assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
+
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_factor_list_include(0) == [(0, 1)]
+    assert R.dmp_factor_list_include(7) == [(7, 1)]
+
+    R, X = xring("x:200", ZZ)
+
+    f, g = X[0]**2 + 2*X[0] + 1, X[0] + 1
+    assert R.dmp_factor_list(f) == (1, [(g, 2)])
+
+    f, g = X[-1]**2 + 2*X[-1] + 1, X[-1] + 1
+    assert R.dmp_factor_list(f) == (1, [(g, 2)])
+
+    R, x = ring("x", ZZ)
+    assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
+    R, x = ring("x", QQ)
+    assert R.dmp_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1,2), [(x + 1, 2)])
+
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
+    R, x, y = ring("x,y", QQ)
+    assert R.dmp_factor_list(QQ(1,2)*x**2 + x + QQ(1,2)) == (QQ(1,2), [(x + 1, 2)])
+
+    R, x, y = ring("x,y", ZZ)
+    f = 4*x**2*y + 4*x*y**2
+
+    assert R.dmp_factor_list(f) == \
+        (4, [(y, 1),
+             (x, 1),
+             (x + y, 1)])
+
+    assert R.dmp_factor_list_include(f) == \
+        [(4*y, 1),
+         (x, 1),
+         (x + y, 1)]
+
+    R, x, y = ring("x,y", QQ)
+    f = QQ(1,2)*x**2*y + QQ(1,2)*x*y**2
+
+    assert R.dmp_factor_list(f) == \
+        (QQ(1,2), [(y, 1),
+                   (x, 1),
+                   (x + y, 1)])
+
+    R, x, y = ring("x,y", RR)
+    f = 2.0*x**2 - 8.0*y**2
+
+    assert R.dmp_factor_list(f) == \
+        (RR(8.0), [(0.5*x - y, 1),
+                   (0.5*x + y, 1)])
+
+    f = 6.7225336055071*x**2*y**2 - 10.6463972754741*x*y - 0.33469524022264
+    coeff, factors = R.dmp_factor_list(f)
+    assert coeff == RR(10.6463972754741)
+    assert len(factors) == 1
+    assert factors[0][0].max_norm() == RR(1.0)
+    assert factors[0][1] == 1
+
+    Rt, t = ring("t", ZZ)
+    R, x, y = ring("x,y", Rt)
+    f = 4*t*x**2 + 4*t**2*x
+
+    assert R.dmp_factor_list(f) == \
+        (4*t, [(x, 1),
+             (x + t, 1)])
+
+    Rt, t = ring("t", QQ)
+    R, x, y = ring("x,y", Rt)
+    f = QQ(1, 2)*t*x**2 + QQ(1, 2)*t**2*x
+
+    assert R.dmp_factor_list(f) == \
+        (QQ(1, 2)*t, [(x, 1),
+                    (x + t, 1)])
+
+    R, x, y = ring("x,y", FF(2))
+    raises(NotImplementedError, lambda: R.dmp_factor_list(x**2 + y**2))
+
+    R, x, y = ring("x,y", EX)
+    raises(DomainError, lambda: R.dmp_factor_list(EX(sin(1))))
+
+
+def test_dup_irreducible_p():
+    R, x = ring("x", ZZ)
+    assert R.dup_irreducible_p(x**2 + x + 1) is True
+    assert R.dup_irreducible_p(x**2 + 2*x + 1) is False
+
+
+def test_dmp_irreducible_p():
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_irreducible_p(x**2 + x + 1) is True
+    assert R.dmp_irreducible_p(x**2 + 2*x + 1) is False

rl/Lib/site-packages/sympy/polys/tests/test_fields.py

@@ -0,0 +1,362 @@
+"""Test sparse rational functions. """
+
+from sympy.polys.fields import field, sfield, FracField, FracElement
+from sympy.polys.rings import ring
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.orderings import lex
+
+from sympy.testing.pytest import raises, XFAIL
+from sympy.core import symbols, E
+from sympy.core.numbers import Rational
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+
+def test_FracField___init__():
+    F1 = FracField("x,y", ZZ, lex)
+    F2 = FracField("x,y", ZZ, lex)
+    F3 = FracField("x,y,z", ZZ, lex)
+
+    assert F1.x == F1.gens[0]
+    assert F1.y == F1.gens[1]
+    assert F1.x == F2.x
+    assert F1.y == F2.y
+    assert F1.x != F3.x
+    assert F1.y != F3.y
+
+def test_FracField___hash__():
+    F, x, y, z = field("x,y,z", QQ)
+    assert hash(F)
+
+def test_FracField___eq__():
+    assert field("x,y,z", QQ)[0] == field("x,y,z", QQ)[0]
+    assert field("x,y,z", QQ)[0] is field("x,y,z", QQ)[0]
+
+    assert field("x,y,z", QQ)[0] != field("x,y,z", ZZ)[0]
+    assert field("x,y,z", QQ)[0] is not field("x,y,z", ZZ)[0]
+
+    assert field("x,y,z", ZZ)[0] != field("x,y,z", QQ)[0]
+    assert field("x,y,z", ZZ)[0] is not field("x,y,z", QQ)[0]
+
+    assert field("x,y,z", QQ)[0] != field("x,y", QQ)[0]
+    assert field("x,y,z", QQ)[0] is not field("x,y", QQ)[0]
+
+    assert field("x,y", QQ)[0] != field("x,y,z", QQ)[0]
+    assert field("x,y", QQ)[0] is not field("x,y,z", QQ)[0]
+
+def test_sfield():
+    x = symbols("x")
+
+    F = FracField((E, exp(exp(x)), exp(x)), ZZ, lex)
+    e, exex, ex = F.gens
+    assert sfield(exp(x)*exp(exp(x) + 1 + log(exp(x) + 3)/2)**2/(exp(x) + 3)) \
+        == (F, e**2*exex**2*ex)
+
+    F = FracField((x, exp(1/x), log(x), x**QQ(1, 3)), ZZ, lex)
+    _, ex, lg, x3 = F.gens
+    assert sfield(((x-3)*log(x)+4*x**2)*exp(1/x+log(x)/3)/x**2) == \
+        (F, (4*F.x**2*ex + F.x*ex*lg - 3*ex*lg)/x3**5)
+
+    F = FracField((x, log(x), sqrt(x + log(x))), ZZ, lex)
+    _, lg, srt = F.gens
+    assert sfield((x + 1) / (x * (x + log(x))**QQ(3, 2)) - 1/(x * log(x)**2)) \
+        == (F, (F.x*lg**2 - F.x*srt + lg**2 - lg*srt)/
+            (F.x**2*lg**2*srt + F.x*lg**3*srt))
+
+def test_FracElement___hash__():
+    F, x, y, z = field("x,y,z", QQ)
+    assert hash(x*y/z)
+
+def test_FracElement_copy():
+    F, x, y, z = field("x,y,z", ZZ)
+
+    f = x*y/3*z
+    g = f.copy()
+
+    assert f == g
+    g.numer[(1, 1, 1)] = 7
+    assert f != g
+
+def test_FracElement_as_expr():
+    F, x, y, z = field("x,y,z", ZZ)
+    f = (3*x**2*y - x*y*z)/(7*z**3 + 1)
+
+    X, Y, Z = F.symbols
+    g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1)
+
+    assert f != g
+    assert f.as_expr() == g
+
+    X, Y, Z = symbols("x,y,z")
+    g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1)
+
+    assert f != g
+    assert f.as_expr(X, Y, Z) == g
+
+    raises(ValueError, lambda: f.as_expr(X))
+
+def test_FracElement_from_expr():
+    x, y, z = symbols("x,y,z")
+    F, X, Y, Z = field((x, y, z), ZZ)
+
+    f = F.from_expr(1)
+    assert f == 1 and isinstance(f, F.dtype)
+
+    f = F.from_expr(Rational(3, 7))
+    assert f == F(3)/7 and isinstance(f, F.dtype)
+
+    f = F.from_expr(x)
+    assert f == X and isinstance(f, F.dtype)
+
+    f = F.from_expr(Rational(3,7)*x)
+    assert f == X*Rational(3, 7) and isinstance(f, F.dtype)
+
+    f = F.from_expr(1/x)
+    assert f == 1/X and isinstance(f, F.dtype)
+
+    f = F.from_expr(x*y*z)
+    assert f == X*Y*Z and isinstance(f, F.dtype)
+
+    f = F.from_expr(x*y/z)
+    assert f == X*Y/Z and isinstance(f, F.dtype)
+
+    f = F.from_expr(x*y*z + x*y + x)
+    assert f == X*Y*Z + X*Y + X and isinstance(f, F.dtype)
+
+    f = F.from_expr((x*y*z + x*y + x)/(x*y + 7))
+    assert f == (X*Y*Z + X*Y + X)/(X*Y + 7) and isinstance(f, F.dtype)
+
+    f = F.from_expr(x**3*y*z + x**2*y**7 + 1)
+    assert f == X**3*Y*Z + X**2*Y**7 + 1 and isinstance(f, F.dtype)
+
+    raises(ValueError, lambda: F.from_expr(2**x))
+    raises(ValueError, lambda: F.from_expr(7*x + sqrt(2)))
+
+    assert isinstance(ZZ[2**x].get_field().convert(2**(-x)),
+        FracElement)
+    assert isinstance(ZZ[x**2].get_field().convert(x**(-6)),
+        FracElement)
+    assert isinstance(ZZ[exp(Rational(1, 3))].get_field().convert(E),
+        FracElement)
+
+
+def test_FracField_nested():
+    a, b, x = symbols('a b x')
+    F1 = ZZ.frac_field(a, b)
+    F2 = F1.frac_field(x)
+    frac = F2(a + b)
+    assert frac.numer == F1.poly_ring(x)(a + b)
+    assert frac.numer.coeffs() == [F1(a + b)]
+    assert frac.denom == F1.poly_ring(x)(1)
+
+    F3 = ZZ.poly_ring(a, b)
+    F4 = F3.frac_field(x)
+    frac = F4(a + b)
+    assert frac.numer == F3.poly_ring(x)(a + b)
+    assert frac.numer.coeffs() == [F3(a + b)]
+    assert frac.denom == F3.poly_ring(x)(1)
+
+    frac = F2(F3(a + b))
+    assert frac.numer == F1.poly_ring(x)(a + b)
+    assert frac.numer.coeffs() == [F1(a + b)]
+    assert frac.denom == F1.poly_ring(x)(1)
+
+    frac = F4(F1(a + b))
+    assert frac.numer == F3.poly_ring(x)(a + b)
+    assert frac.numer.coeffs() == [F3(a + b)]
+    assert frac.denom == F3.poly_ring(x)(1)
+
+
+def test_FracElement__lt_le_gt_ge__():
+    F, x, y = field("x,y", ZZ)
+
+    assert F(1) < 1/x < 1/x**2 < 1/x**3
+    assert F(1) <= 1/x <= 1/x**2 <= 1/x**3
+
+    assert -7/x < 1/x < 3/x < y/x < 1/x**2
+    assert -7/x <= 1/x <= 3/x <= y/x <= 1/x**2
+
+    assert 1/x**3 > 1/x**2 > 1/x > F(1)
+    assert 1/x**3 >= 1/x**2 >= 1/x >= F(1)
+
+    assert 1/x**2 > y/x > 3/x > 1/x > -7/x
+    assert 1/x**2 >= y/x >= 3/x >= 1/x >= -7/x
+
+def test_FracElement___neg__():
+    F, x,y = field("x,y", QQ)
+
+    f = (7*x - 9)/y
+    g = (-7*x + 9)/y
+
+    assert -f == g
+    assert -g == f
+
+def test_FracElement___add__():
+    F, x,y = field("x,y", QQ)
+
+    f, g = 1/x, 1/y
+    assert f + g == g + f == (x + y)/(x*y)
+
+    assert x + F.ring.gens[0] == F.ring.gens[0] + x == 2*x
+
+    F, x,y = field("x,y", ZZ)
+    assert x + 3 == 3 + x
+    assert x + QQ(3,7) == QQ(3,7) + x == (7*x + 3)/7
+
+    Fuv, u,v = field("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
+
+    f = (u*v + x)/(y + u*v)
+    assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
+    assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
+
+    Ruv, u,v = ring("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
+
+    f = (u*v + x)/(y + u*v)
+    assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
+    assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
+
+def test_FracElement___sub__():
+    F, x,y = field("x,y", QQ)
+
+    f, g = 1/x, 1/y
+    assert f - g == (-x + y)/(x*y)
+
+    assert x - F.ring.gens[0] == F.ring.gens[0] - x == 0
+
+    F, x,y = field("x,y", ZZ)
+    assert x - 3 == -(3 - x)
+    assert x - QQ(3,7) == -(QQ(3,7) - x) == (7*x - 3)/7
+
+    Fuv, u,v = field("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
+
+    f = (u*v - x)/(y - u*v)
+    assert dict(f.numer) == {(1, 0, 0, 0):-1, (0, 0, 0, 0): u*v}
+    assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0):-u*v}
+
+    Ruv, u,v = ring("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
+
+    f = (u*v - x)/(y - u*v)
+    assert dict(f.numer) == {(1, 0, 0, 0):-1, (0, 0, 0, 0): u*v}
+    assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0):-u*v}
+
+def test_FracElement___mul__():
+    F, x,y = field("x,y", QQ)
+
+    f, g = 1/x, 1/y
+    assert f*g == g*f == 1/(x*y)
+
+    assert x*F.ring.gens[0] == F.ring.gens[0]*x == x**2
+
+    F, x,y = field("x,y", ZZ)
+    assert x*3 == 3*x
+    assert x*QQ(3,7) == QQ(3,7)*x == x*Rational(3, 7)
+
+    Fuv, u,v = field("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
+
+    f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
+    assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
+    assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
+
+    Ruv, u,v = ring("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
+
+    f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
+    assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
+    assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
+
+def test_FracElement___truediv__():
+    F, x,y = field("x,y", QQ)
+
+    f, g = 1/x, 1/y
+    assert f/g == y/x
+
+    assert x/F.ring.gens[0] == F.ring.gens[0]/x == 1
+
+    F, x,y = field("x,y", ZZ)
+    assert x*3 == 3*x
+    assert x/QQ(3,7) == (QQ(3,7)/x)**-1 == x*Rational(7, 3)
+
+    raises(ZeroDivisionError, lambda: x/0)
+    raises(ZeroDivisionError, lambda: 1/(x - x))
+    raises(ZeroDivisionError, lambda: x/(x - x))
+
+    Fuv, u,v = field("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
+
+    f = (u*v)/(x*y)
+    assert dict(f.numer) == {(0, 0, 0, 0): u*v}
+    assert dict(f.denom) == {(1, 1, 0, 0): 1}
+
+    g = (x*y)/(u*v)
+    assert dict(g.numer) == {(1, 1, 0, 0): 1}
+    assert dict(g.denom) == {(0, 0, 0, 0): u*v}
+
+    Ruv, u,v = ring("u,v", ZZ)
+    Fxyzt, x,y,z,t = field("x,y,z,t", Ruv)
+
+    f = (u*v)/(x*y)
+    assert dict(f.numer) == {(0, 0, 0, 0): u*v}
+    assert dict(f.denom) == {(1, 1, 0, 0): 1}
+
+    g = (x*y)/(u*v)
+    assert dict(g.numer) == {(1, 1, 0, 0): 1}
+    assert dict(g.denom) == {(0, 0, 0, 0): u*v}
+
+def test_FracElement___pow__():
+    F, x,y = field("x,y", QQ)
+
+    f, g = 1/x, 1/y
+
+    assert f**3 == 1/x**3
+    assert g**3 == 1/y**3
+
+    assert (f*g)**3 == 1/(x**3*y**3)
+    assert (f*g)**-3 == (x*y)**3
+
+    raises(ZeroDivisionError, lambda: (x - x)**-3)
+
+def test_FracElement_diff():
+    F, x,y,z = field("x,y,z", ZZ)
+
+    assert ((x**2 + y)/(z + 1)).diff(x) == 2*x/(z + 1)
+
+@XFAIL
+def test_FracElement___call__():
+    F, x,y,z = field("x,y,z", ZZ)
+    f = (x**2 + 3*y)/z
+
+    r = f(1, 1, 1)
+    assert r == 4 and not isinstance(r, FracElement)
+    raises(ZeroDivisionError, lambda: f(1, 1, 0))
+
+def test_FracElement_evaluate():
+    F, x,y,z = field("x,y,z", ZZ)
+    Fyz = field("y,z", ZZ)[0]
+    f = (x**2 + 3*y)/z
+
+    assert f.evaluate(x, 0) == 3*Fyz.y/Fyz.z
+    raises(ZeroDivisionError, lambda: f.evaluate(z, 0))
+
+def test_FracElement_subs():
+    F, x,y,z = field("x,y,z", ZZ)
+    f = (x**2 + 3*y)/z
+
+    assert f.subs(x, 0) == 3*y/z
+    raises(ZeroDivisionError, lambda: f.subs(z, 0))
+
+def test_FracElement_compose():
+    pass
+
+def test_FracField_index():
+    a = symbols("a")
+    F, x, y, z = field('x y z', QQ)
+    assert F.index(x) == 0
+    assert F.index(y) == 1
+
+    raises(ValueError, lambda: F.index(1))
+    raises(ValueError, lambda: F.index(a))
+    pass

rl/Lib/site-packages/sympy/polys/tests/test_galoistools.py

@@ -0,0 +1,875 @@
+from sympy.polys.galoistools import (
+    gf_crt, gf_crt1, gf_crt2, gf_int,
+    gf_degree, gf_strip, gf_trunc, gf_normal,
+    gf_from_dict, gf_to_dict,
+    gf_from_int_poly, gf_to_int_poly,
+    gf_neg, gf_add_ground, gf_sub_ground, gf_mul_ground,
+    gf_add, gf_sub, gf_add_mul, gf_sub_mul, gf_mul, gf_sqr,
+    gf_div, gf_rem, gf_quo, gf_exquo,
+    gf_lshift, gf_rshift, gf_expand,
+    gf_pow, gf_pow_mod,
+    gf_gcdex, gf_gcd, gf_lcm, gf_cofactors,
+    gf_LC, gf_TC, gf_monic,
+    gf_eval, gf_multi_eval,
+    gf_compose, gf_compose_mod,
+    gf_trace_map,
+    gf_diff,
+    gf_irreducible, gf_irreducible_p,
+    gf_irred_p_ben_or, gf_irred_p_rabin,
+    gf_sqf_list, gf_sqf_part, gf_sqf_p,
+    gf_Qmatrix, gf_Qbasis,
+    gf_ddf_zassenhaus, gf_ddf_shoup,
+    gf_edf_zassenhaus, gf_edf_shoup,
+    gf_berlekamp,
+    gf_factor_sqf, gf_factor,
+    gf_value, linear_congruence, _csolve_prime_las_vegas,
+    csolve_prime, gf_csolve, gf_frobenius_map, gf_frobenius_monomial_base
+)
+
+from sympy.polys.polyerrors import (
+    ExactQuotientFailed,
+)
+
+from sympy.polys import polyconfig as config
+
+from sympy.polys.domains import ZZ
+from sympy.core.numbers import pi
+from sympy.ntheory.generate import nextprime
+from sympy.testing.pytest import raises
+
+
+def test_gf_crt():
+    U = [49, 76, 65]
+    M = [99, 97, 95]
+
+    p = 912285
+    u = 639985
+
+    assert gf_crt(U, M, ZZ) == u
+
+    E = [9215, 9405, 9603]
+    S = [62, 24, 12]
+
+    assert gf_crt1(M, ZZ) == (p, E, S)
+    assert gf_crt2(U, M, p, E, S, ZZ) == u
+
+
+def test_gf_int():
+    assert gf_int(0, 5) == 0
+    assert gf_int(1, 5) == 1
+    assert gf_int(2, 5) == 2
+    assert gf_int(3, 5) == -2
+    assert gf_int(4, 5) == -1
+    assert gf_int(5, 5) == 0
+
+
+def test_gf_degree():
+    assert gf_degree([]) == -1
+    assert gf_degree([1]) == 0
+    assert gf_degree([1, 0]) == 1
+    assert gf_degree([1, 0, 0, 0, 1]) == 4
+
+
+def test_gf_strip():
+    assert gf_strip([]) == []
+    assert gf_strip([0]) == []
+    assert gf_strip([0, 0, 0]) == []
+
+    assert gf_strip([1]) == [1]
+    assert gf_strip([0, 1]) == [1]
+    assert gf_strip([0, 0, 0, 1]) == [1]
+
+    assert gf_strip([1, 2, 0]) == [1, 2, 0]
+    assert gf_strip([0, 1, 2, 0]) == [1, 2, 0]
+    assert gf_strip([0, 0, 0, 1, 2, 0]) == [1, 2, 0]
+
+
+def test_gf_trunc():
+    assert gf_trunc([], 11) == []
+    assert gf_trunc([1], 11) == [1]
+    assert gf_trunc([22], 11) == []
+    assert gf_trunc([12], 11) == [1]
+
+    assert gf_trunc([11, 22, 17, 1, 0], 11) == [6, 1, 0]
+    assert gf_trunc([12, 23, 17, 1, 0], 11) == [1, 1, 6, 1, 0]
+
+
+def test_gf_normal():
+    assert gf_normal([11, 22, 17, 1, 0], 11, ZZ) == [6, 1, 0]
+
+
+def test_gf_from_to_dict():
+    f = {11: 12, 6: 2, 0: 25}
+    F = {11: 1, 6: 2, 0: 3}
+    g = [1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3]
+
+    assert gf_from_dict(f, 11, ZZ) == g
+    assert gf_to_dict(g, 11) == F
+
+    f = {11: -5, 4: 0, 3: 1, 0: 12}
+    F = {11: -5, 3: 1, 0: 1}
+    g = [6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1]
+
+    assert gf_from_dict(f, 11, ZZ) == g
+    assert gf_to_dict(g, 11) == F
+
+    assert gf_to_dict([10], 11, symmetric=True) == {0: -1}
+    assert gf_to_dict([10], 11, symmetric=False) == {0: 10}
+
+
+def test_gf_from_to_int_poly():
+    assert gf_from_int_poly([1, 0, 7, 2, 20], 5) == [1, 0, 2, 2, 0]
+    assert gf_to_int_poly([1, 0, 4, 2, 3], 5) == [1, 0, -1, 2, -2]
+
+    assert gf_to_int_poly([10], 11, symmetric=True) == [-1]
+    assert gf_to_int_poly([10], 11, symmetric=False) == [10]
+
+
+def test_gf_LC():
+    assert gf_LC([], ZZ) == 0
+    assert gf_LC([1], ZZ) == 1
+    assert gf_LC([1, 2], ZZ) == 1
+
+
+def test_gf_TC():
+    assert gf_TC([], ZZ) == 0
+    assert gf_TC([1], ZZ) == 1
+    assert gf_TC([1, 2], ZZ) == 2
+
+
+def test_gf_monic():
+    assert gf_monic(ZZ.map([]), 11, ZZ) == (0, [])
+
+    assert gf_monic(ZZ.map([1]), 11, ZZ) == (1, [1])
+    assert gf_monic(ZZ.map([2]), 11, ZZ) == (2, [1])
+
+    assert gf_monic(ZZ.map([1, 2, 3, 4]), 11, ZZ) == (1, [1, 2, 3, 4])
+    assert gf_monic(ZZ.map([2, 3, 4, 5]), 11, ZZ) == (2, [1, 7, 2, 8])
+
+
+def test_gf_arith():
+    assert gf_neg([], 11, ZZ) == []
+    assert gf_neg([1], 11, ZZ) == [10]
+    assert gf_neg([1, 2, 3], 11, ZZ) == [10, 9, 8]
+
+    assert gf_add_ground([], 0, 11, ZZ) == []
+    assert gf_sub_ground([], 0, 11, ZZ) == []
+
+    assert gf_add_ground([], 3, 11, ZZ) == [3]
+    assert gf_sub_ground([], 3, 11, ZZ) == [8]
+
+    assert gf_add_ground([1], 3, 11, ZZ) == [4]
+    assert gf_sub_ground([1], 3, 11, ZZ) == [9]
+
+    assert gf_add_ground([8], 3, 11, ZZ) == []
+    assert gf_sub_ground([3], 3, 11, ZZ) == []
+
+    assert gf_add_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 6]
+    assert gf_sub_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 0]
+
+    assert gf_mul_ground([], 0, 11, ZZ) == []
+    assert gf_mul_ground([], 1, 11, ZZ) == []
+
+    assert gf_mul_ground([1], 0, 11, ZZ) == []
+    assert gf_mul_ground([1], 1, 11, ZZ) == [1]
+
+    assert gf_mul_ground([1, 2, 3], 0, 11, ZZ) == []
+    assert gf_mul_ground([1, 2, 3], 1, 11, ZZ) == [1, 2, 3]
+    assert gf_mul_ground([1, 2, 3], 7, 11, ZZ) == [7, 3, 10]
+
+    assert gf_add([], [], 11, ZZ) == []
+    assert gf_add([1], [], 11, ZZ) == [1]
+    assert gf_add([], [1], 11, ZZ) == [1]
+    assert gf_add([1], [1], 11, ZZ) == [2]
+    assert gf_add([1], [2], 11, ZZ) == [3]
+
+    assert gf_add([1, 2], [1], 11, ZZ) == [1, 3]
+    assert gf_add([1], [1, 2], 11, ZZ) == [1, 3]
+
+    assert gf_add([1, 2, 3], [8, 9, 10], 11, ZZ) == [9, 0, 2]
+
+    assert gf_sub([], [], 11, ZZ) == []
+    assert gf_sub([1], [], 11, ZZ) == [1]
+    assert gf_sub([], [1], 11, ZZ) == [10]
+    assert gf_sub([1], [1], 11, ZZ) == []
+    assert gf_sub([1], [2], 11, ZZ) == [10]
+
+    assert gf_sub([1, 2], [1], 11, ZZ) == [1, 1]
+    assert gf_sub([1], [1, 2], 11, ZZ) == [10, 10]
+
+    assert gf_sub([3, 2, 1], [8, 9, 10], 11, ZZ) == [6, 4, 2]
+
+    assert gf_add_mul(
+        [1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [1, 2, 10, 8, 9]
+    assert gf_sub_mul(
+        [1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [10, 9, 3, 2, 3]
+
+    assert gf_mul([], [], 11, ZZ) == []
+    assert gf_mul([], [1], 11, ZZ) == []
+    assert gf_mul([1], [], 11, ZZ) == []
+    assert gf_mul([1], [1], 11, ZZ) == [1]
+    assert gf_mul([5], [7], 11, ZZ) == [2]
+
+    assert gf_mul([3, 0, 0, 6, 1, 2], [4, 0, 1, 0], 11, ZZ) == [1, 0,
+                  3, 2, 4, 3, 1, 2, 0]
+    assert gf_mul([4, 0, 1, 0], [3, 0, 0, 6, 1, 2], 11, ZZ) == [1, 0,
+                  3, 2, 4, 3, 1, 2, 0]
+
+    assert gf_mul([2, 0, 0, 1, 7], [2, 0, 0, 1, 7], 11, ZZ) == [4, 0,
+                  0, 4, 6, 0, 1, 3, 5]
+
+    assert gf_sqr([], 11, ZZ) == []
+    assert gf_sqr([2], 11, ZZ) == [4]
+    assert gf_sqr([1, 2], 11, ZZ) == [1, 4, 4]
+
+    assert gf_sqr([2, 0, 0, 1, 7], 11, ZZ) == [4, 0, 0, 4, 6, 0, 1, 3, 5]
+
+
+def test_gf_division():
+    raises(ZeroDivisionError, lambda: gf_div([1, 2, 3], [], 11, ZZ))
+    raises(ZeroDivisionError, lambda: gf_rem([1, 2, 3], [], 11, ZZ))
+    raises(ZeroDivisionError, lambda: gf_quo([1, 2, 3], [], 11, ZZ))
+    raises(ZeroDivisionError, lambda: gf_quo([1, 2, 3], [], 11, ZZ))
+
+    assert gf_div([1], [1, 2, 3], 7, ZZ) == ([], [1])
+    assert gf_rem([1], [1, 2, 3], 7, ZZ) == [1]
+    assert gf_quo([1], [1, 2, 3], 7, ZZ) == []
+
+    f = ZZ.map([5, 4, 3, 2, 1, 0])
+    g = ZZ.map([1, 2, 3])
+    q = [5, 1, 0, 6]
+    r = [3, 3]
+
+    assert gf_div(f, g, 7, ZZ) == (q, r)
+    assert gf_rem(f, g, 7, ZZ) == r
+    assert gf_quo(f, g, 7, ZZ) == q
+
+    raises(ExactQuotientFailed, lambda: gf_exquo(f, g, 7, ZZ))
+
+    f = ZZ.map([5, 4, 3, 2, 1, 0])
+    g = ZZ.map([1, 2, 3, 0])
+    q = [5, 1, 0]
+    r = [6, 1, 0]
+
+    assert gf_div(f, g, 7, ZZ) == (q, r)
+    assert gf_rem(f, g, 7, ZZ) == r
+    assert gf_quo(f, g, 7, ZZ) == q
+
+    raises(ExactQuotientFailed, lambda: gf_exquo(f, g, 7, ZZ))
+
+    assert gf_quo(ZZ.map([1, 2, 1]), ZZ.map([1, 1]), 11, ZZ) == [1, 1]
+
+
+def test_gf_shift():
+    f = [1, 2, 3, 4, 5]
+
+    assert gf_lshift([], 5, ZZ) == []
+    assert gf_rshift([], 5, ZZ) == ([], [])
+
+    assert gf_lshift(f, 1, ZZ) == [1, 2, 3, 4, 5, 0]
+    assert gf_lshift(f, 2, ZZ) == [1, 2, 3, 4, 5, 0, 0]
+
+    assert gf_rshift(f, 0, ZZ) == (f, [])
+    assert gf_rshift(f, 1, ZZ) == ([1, 2, 3, 4], [5])
+    assert gf_rshift(f, 3, ZZ) == ([1, 2], [3, 4, 5])
+    assert gf_rshift(f, 5, ZZ) == ([], f)
+
+
+def test_gf_expand():
+    F = [([1, 1], 2), ([1, 2], 3)]
+
+    assert gf_expand(F, 11, ZZ) == [1, 8, 3, 5, 6, 8]
+    assert gf_expand((4, F), 11, ZZ) == [4, 10, 1, 9, 2, 10]
+
+
+def test_gf_powering():
+    assert gf_pow([1, 0, 0, 1, 8], 0, 11, ZZ) == [1]
+    assert gf_pow([1, 0, 0, 1, 8], 1, 11, ZZ) == [1, 0, 0, 1, 8]
+    assert gf_pow([1, 0, 0, 1, 8], 2, 11, ZZ) == [1, 0, 0, 2, 5, 0, 1, 5, 9]
+
+    assert gf_pow([1, 0, 0, 1, 8], 5, 11, ZZ) == \
+        [1, 0, 0, 5, 7, 0, 10, 6, 2, 10, 9, 6, 10, 6, 6, 0, 5, 2, 5, 9, 10]
+
+    assert gf_pow([1, 0, 0, 1, 8], 8, 11, ZZ) == \
+        [1, 0, 0, 8, 9, 0, 6, 8, 10, 1, 2, 5, 10, 7, 7, 9, 1, 2, 0, 0, 6, 2,
+         5, 2, 5, 7, 7, 9, 10, 10, 7, 5, 5]
+
+    assert gf_pow([1, 0, 0, 1, 8], 45, 11, ZZ) == \
+        [ 1, 0, 0,  1,  8, 0, 0, 0, 0, 0, 0,  0, 0, 0,  0,  0, 0, 0, 0, 0, 0, 0,
+          0, 0, 0,  0,  0, 0, 0, 0, 0, 0, 0,  4, 0, 0,  4, 10, 0, 0, 0, 0, 0, 0,
+         10, 0, 0, 10,  3, 0, 0, 0, 0, 0, 0,  0, 0, 0,  0,  0, 0, 0, 0, 0, 0, 0,
+          6, 0, 0,  6,  4, 0, 0, 0, 0, 0, 0,  8, 0, 0,  8,  9, 0, 0, 0, 0, 0, 0,
+         10, 0, 0, 10,  3, 0, 0, 0, 0, 0, 0,  4, 0, 0,  4, 10, 0, 0, 0, 0, 0, 0,
+          8, 0, 0,  8,  9, 0, 0, 0, 0, 0, 0,  9, 0, 0,  9,  6, 0, 0, 0, 0, 0, 0,
+          3, 0, 0,  3,  2, 0, 0, 0, 0, 0, 0, 10, 0, 0, 10,  3, 0, 0, 0, 0, 0, 0,
+         10, 0, 0, 10,  3, 0, 0, 0, 0, 0, 0,  2, 0, 0,  2,  5, 0, 0, 0, 0, 0, 0,
+          4, 0, 0,  4, 10]
+
+    assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 0, ZZ.map([2, 0, 7]), 11, ZZ) == [1]
+    assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 1, ZZ.map([2, 0, 7]), 11, ZZ) == [1, 1]
+    assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 2, ZZ.map([2, 0, 7]), 11, ZZ) == [2, 3]
+    assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 5, ZZ.map([2, 0, 7]), 11, ZZ) == [7, 8]
+    assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 8, ZZ.map([2, 0, 7]), 11, ZZ) == [1, 5]
+    assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 45, ZZ.map([2, 0, 7]), 11, ZZ) == [5, 4]
+
+
+def test_gf_gcdex():
+    assert gf_gcdex(ZZ.map([]), ZZ.map([]), 11, ZZ) == ([1], [], [])
+    assert gf_gcdex(ZZ.map([2]), ZZ.map([]), 11, ZZ) == ([6], [], [1])
+    assert gf_gcdex(ZZ.map([]), ZZ.map([2]), 11, ZZ) == ([], [6], [1])
+    assert gf_gcdex(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == ([], [6], [1])
+
+    assert gf_gcdex(ZZ.map([]), ZZ.map([3, 0]), 11, ZZ) == ([], [4], [1, 0])
+    assert gf_gcdex(ZZ.map([3, 0]), ZZ.map([]), 11, ZZ) == ([4], [], [1, 0])
+
+    assert gf_gcdex(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == ([], [4], [1, 0])
+
+    assert gf_gcdex(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == ([5, 6], [6], [1, 7])
+
+
+def test_gf_gcd():
+    assert gf_gcd(ZZ.map([]), ZZ.map([]), 11, ZZ) == []
+    assert gf_gcd(ZZ.map([2]), ZZ.map([]), 11, ZZ) == [1]
+    assert gf_gcd(ZZ.map([]), ZZ.map([2]), 11, ZZ) == [1]
+    assert gf_gcd(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == [1]
+
+    assert gf_gcd(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == [1, 0]
+    assert gf_gcd(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == [1, 0]
+
+    assert gf_gcd(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == [1, 0]
+    assert gf_gcd(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == [1, 7]
+
+
+def test_gf_lcm():
+    assert gf_lcm(ZZ.map([]), ZZ.map([]), 11, ZZ) == []
+    assert gf_lcm(ZZ.map([2]), ZZ.map([]), 11, ZZ) == []
+    assert gf_lcm(ZZ.map([]), ZZ.map([2]), 11, ZZ) == []
+    assert gf_lcm(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == [1]
+
+    assert gf_lcm(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == []
+    assert gf_lcm(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == []
+
+    assert gf_lcm(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == [1, 0]
+    assert gf_lcm(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == [1, 8, 8, 8, 7]
+
+
+def test_gf_cofactors():
+    assert gf_cofactors(ZZ.map([]), ZZ.map([]), 11, ZZ) == ([], [], [])
+    assert gf_cofactors(ZZ.map([2]), ZZ.map([]), 11, ZZ) == ([1], [2], [])
+    assert gf_cofactors(ZZ.map([]), ZZ.map([2]), 11, ZZ) == ([1], [], [2])
+    assert gf_cofactors(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == ([1], [2], [2])
+
+    assert gf_cofactors(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == ([1, 0], [], [1])
+    assert gf_cofactors(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == ([1, 0], [1], [])
+
+    assert gf_cofactors(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == (
+        [1, 0], [3], [3])
+    assert gf_cofactors(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == (
+        ([1, 7], [1, 1], [1, 0, 1]))
+
+
+def test_gf_diff():
+    assert gf_diff([], 11, ZZ) == []
+    assert gf_diff([7], 11, ZZ) == []
+
+    assert gf_diff([7, 3], 11, ZZ) == [7]
+    assert gf_diff([7, 3, 1], 11, ZZ) == [3, 3]
+
+    assert gf_diff([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 11, ZZ) == []
+
+
+def test_gf_eval():
+    assert gf_eval([], 4, 11, ZZ) == 0
+    assert gf_eval([], 27, 11, ZZ) == 0
+    assert gf_eval([7], 4, 11, ZZ) == 7
+    assert gf_eval([7], 27, 11, ZZ) == 7
+
+    assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 0, 11, ZZ) == 0
+    assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 4, 11, ZZ) == 9
+    assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 27, 11, ZZ) == 5
+
+    assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 0, 11, ZZ) == 5
+    assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 4, 11, ZZ) == 3
+    assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 27, 11, ZZ) == 9
+
+    assert gf_multi_eval([3, 2, 1], [0, 1, 2, 3], 11, ZZ) == [1, 6, 6, 1]
+
+
+def test_gf_compose():
+    assert gf_compose([], [1, 0], 11, ZZ) == []
+    assert gf_compose_mod([], [1, 0], [1, 0], 11, ZZ) == []
+
+    assert gf_compose([1], [], 11, ZZ) == [1]
+    assert gf_compose([1, 0], [], 11, ZZ) == []
+    assert gf_compose([1, 0], [1, 0], 11, ZZ) == [1, 0]
+
+    f = ZZ.map([1, 1, 4, 9, 1])
+    g = ZZ.map([1, 1, 1])
+    h = ZZ.map([1, 0, 0, 2])
+
+    assert gf_compose(g, h, 11, ZZ) == [1, 0, 0, 5, 0, 0, 7]
+    assert gf_compose_mod(g, h, f, 11, ZZ) == [3, 9, 6, 10]
+
+
+def test_gf_trace_map():
+    f = ZZ.map([1, 1, 4, 9, 1])
+    a = [1, 1, 1]
+    c = ZZ.map([1, 0])
+    b = gf_pow_mod(c, 11, f, 11, ZZ)
+
+    assert gf_trace_map(a, b, c, 0, f, 11, ZZ) == \
+        ([1, 1, 1], [1, 1, 1])
+    assert gf_trace_map(a, b, c, 1, f, 11, ZZ) == \
+        ([5, 2, 10, 3], [5, 3, 0, 4])
+    assert gf_trace_map(a, b, c, 2, f, 11, ZZ) == \
+        ([5, 9, 5, 3], [10, 1, 5, 7])
+    assert gf_trace_map(a, b, c, 3, f, 11, ZZ) == \
+        ([1, 10, 6, 0], [7])
+    assert gf_trace_map(a, b, c, 4, f, 11, ZZ) == \
+        ([1, 1, 1], [1, 1, 8])
+    assert gf_trace_map(a, b, c, 5, f, 11, ZZ) == \
+        ([5, 2, 10, 3], [5, 3, 0, 0])
+    assert gf_trace_map(a, b, c, 11, f, 11, ZZ) == \
+        ([1, 10, 6, 0], [10])
+
+
+def test_gf_irreducible():
+    assert gf_irreducible_p(gf_irreducible(1, 11, ZZ), 11, ZZ) is True
+    assert gf_irreducible_p(gf_irreducible(2, 11, ZZ), 11, ZZ) is True
+    assert gf_irreducible_p(gf_irreducible(3, 11, ZZ), 11, ZZ) is True
+    assert gf_irreducible_p(gf_irreducible(4, 11, ZZ), 11, ZZ) is True
+    assert gf_irreducible_p(gf_irreducible(5, 11, ZZ), 11, ZZ) is True
+    assert gf_irreducible_p(gf_irreducible(6, 11, ZZ), 11, ZZ) is True
+    assert gf_irreducible_p(gf_irreducible(7, 11, ZZ), 11, ZZ) is True
+
+
+def test_gf_irreducible_p():
+    assert gf_irred_p_ben_or(ZZ.map([7]), 11, ZZ) is True
+    assert gf_irred_p_ben_or(ZZ.map([7, 3]), 11, ZZ) is True
+    assert gf_irred_p_ben_or(ZZ.map([7, 3, 1]), 11, ZZ) is False
+
+    assert gf_irred_p_rabin(ZZ.map([7]), 11, ZZ) is True
+    assert gf_irred_p_rabin(ZZ.map([7, 3]), 11, ZZ) is True
+    assert gf_irred_p_rabin(ZZ.map([7, 3, 1]), 11, ZZ) is False
+
+    config.setup('GF_IRRED_METHOD', 'ben-or')
+
+    assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True
+    assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True
+    assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False
+
+    config.setup('GF_IRRED_METHOD', 'rabin')
+
+    assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True
+    assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True
+    assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False
+
+    config.setup('GF_IRRED_METHOD', 'other')
+    raises(KeyError, lambda: gf_irreducible_p([7], 11, ZZ))
+    config.setup('GF_IRRED_METHOD')
+
+    f = ZZ.map([1, 9, 9, 13, 16, 15, 6, 7, 7, 7, 10])
+    g = ZZ.map([1, 7, 16, 7, 15, 13, 13, 11, 16, 10, 9])
+
+    h = gf_mul(f, g, 17, ZZ)
+
+    assert gf_irred_p_ben_or(f, 17, ZZ) is True
+    assert gf_irred_p_ben_or(g, 17, ZZ) is True
+
+    assert gf_irred_p_ben_or(h, 17, ZZ) is False
+
+    assert gf_irred_p_rabin(f, 17, ZZ) is True
+    assert gf_irred_p_rabin(g, 17, ZZ) is True
+
+    assert gf_irred_p_rabin(h, 17, ZZ) is False
+
+
+def test_gf_squarefree():
+    assert gf_sqf_list([], 11, ZZ) == (0, [])
+    assert gf_sqf_list([1], 11, ZZ) == (1, [])
+    assert gf_sqf_list([1, 1], 11, ZZ) == (1, [([1, 1], 1)])
+
+    assert gf_sqf_p([], 11, ZZ) is True
+    assert gf_sqf_p([1], 11, ZZ) is True
+    assert gf_sqf_p([1, 1], 11, ZZ) is True
+
+    f = gf_from_dict({11: 1, 0: 1}, 11, ZZ)
+
+    assert gf_sqf_p(f, 11, ZZ) is False
+
+    assert gf_sqf_list(f, 11, ZZ) == \
+        (1, [([1, 1], 11)])
+
+    f = [1, 5, 8, 4]
+
+    assert gf_sqf_p(f, 11, ZZ) is False
+
+    assert gf_sqf_list(f, 11, ZZ) == \
+        (1, [([1, 1], 1),
+             ([1, 2], 2)])
+
+    assert gf_sqf_part(f, 11, ZZ) == [1, 3, 2]
+
+    f = [1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0]
+
+    assert gf_sqf_list(f, 3, ZZ) == \
+        (1, [([1, 0], 1),
+             ([1, 1], 3),
+             ([1, 2], 6)])
+
+def test_gf_frobenius_map():
+    f = ZZ.map([2, 0, 1, 0, 2, 2, 0, 2, 2, 2])
+    g = ZZ.map([1,1,0,2,0,1,0,2,0,1])
+    p = 3
+    b = gf_frobenius_monomial_base(g, p, ZZ)
+    h = gf_frobenius_map(f, g, b, p, ZZ)
+    h1 = gf_pow_mod(f, p, g, p, ZZ)
+    assert h == h1
+
+
+def test_gf_berlekamp():
+    f = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11)
+
+    Q = [[1, 0, 0, 0, 0, 0],
+         [3, 5, 8, 8, 6, 5],
+         [3, 6, 6, 1, 10, 0],
+         [9, 4, 10, 3, 7, 9],
+         [7, 8, 10, 0, 0, 8],
+         [8, 10, 7, 8, 10, 8]]
+
+    V = [[1, 0, 0, 0, 0, 0],
+         [0, 1, 1, 1, 1, 0],
+         [0, 0, 7, 9, 0, 1]]
+
+    assert gf_Qmatrix(f, 11, ZZ) == Q
+    assert gf_Qbasis(Q, 11, ZZ) == V
+
+    assert gf_berlekamp(f, 11, ZZ) == \
+        [[1, 1], [1, 5, 3], [1, 2, 3, 4]]
+
+    f = ZZ.map([1, 0, 1, 0, 10, 10, 8, 2, 8])
+
+    Q = ZZ.map([[1, 0, 0, 0, 0, 0, 0, 0],
+         [2, 1, 7, 11, 10, 12, 5, 11],
+         [3, 6, 4, 3, 0, 4, 7, 2],
+         [4, 3, 6, 5, 1, 6, 2, 3],
+         [2, 11, 8, 8, 3, 1, 3, 11],
+         [6, 11, 8, 6, 2, 7, 10, 9],
+         [5, 11, 7, 10, 0, 11, 7, 12],
+         [3, 3, 12, 5, 0, 11, 9, 12]])
+
+    V = [[1, 0, 0, 0, 0, 0, 0, 0],
+         [0, 5, 5, 0, 9, 5, 1, 0],
+         [0, 9, 11, 9, 10, 12, 0, 1]]
+
+    assert gf_Qmatrix(f, 13, ZZ) == Q
+    assert gf_Qbasis(Q, 13, ZZ) == V
+
+    assert gf_berlekamp(f, 13, ZZ) == \
+        [[1, 3], [1, 8, 4, 12], [1, 2, 3, 4, 6]]
+
+
+def test_gf_ddf():
+    f = gf_from_dict({15: ZZ(1), 0: ZZ(-1)}, 11, ZZ)
+    g = [([1, 0, 0, 0, 0, 10], 1),
+         ([1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], 2)]
+
+    assert gf_ddf_zassenhaus(f, 11, ZZ) == g
+    assert gf_ddf_shoup(f, 11, ZZ) == g
+
+    f = gf_from_dict({63: ZZ(1), 0: ZZ(1)}, 2, ZZ)
+    g = [([1, 1], 1),
+         ([1, 1, 1], 2),
+         ([1, 1, 1, 1, 1, 1, 1], 3),
+         ([1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0,
+           0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0,
+           0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1], 6)]
+
+    assert gf_ddf_zassenhaus(f, 2, ZZ) == g
+    assert gf_ddf_shoup(f, 2, ZZ) == g
+
+    f = gf_from_dict({6: ZZ(1), 5: ZZ(-1), 4: ZZ(1), 3: ZZ(1), 1: ZZ(-1)}, 3, ZZ)
+    g = [([1, 1, 0], 1),
+         ([1, 1, 0, 1, 2], 2)]
+
+    assert gf_ddf_zassenhaus(f, 3, ZZ) == g
+    assert gf_ddf_shoup(f, 3, ZZ) == g
+
+    f = ZZ.map([1, 2, 5, 26, 677, 436, 791, 325, 456, 24, 577])
+    g = [([1, 701], 1),
+         ([1, 110, 559, 532, 694, 151, 110, 70, 735, 122], 9)]
+
+    assert gf_ddf_zassenhaus(f, 809, ZZ) == g
+    assert gf_ddf_shoup(f, 809, ZZ) == g
+
+    p = ZZ(nextprime(int((2**15 * pi).evalf())))
+    f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
+    g = [([1, 22730, 68144], 2),
+         ([1, 64876, 83977, 10787, 12561, 68608, 52650, 88001, 84356], 4),
+         ([1, 15347, 95022, 84569, 94508, 92335], 5)]
+
+    assert gf_ddf_zassenhaus(f, p, ZZ) == g
+    assert gf_ddf_shoup(f, p, ZZ) == g
+
+
+def test_gf_edf():
+    f = ZZ.map([1, 1, 0, 1, 2])
+    g = ZZ.map([[1, 0, 1], [1, 1, 2]])
+
+    assert gf_edf_zassenhaus(f, 2, 3, ZZ) == g
+    assert gf_edf_shoup(f, 2, 3, ZZ) == g
+
+
+def test_issue_23174():
+    f = ZZ.map([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])
+    g = ZZ.map([[1, 0, 0, 1, 1, 1, 0, 0, 1], [1, 1, 1, 0, 1, 0, 1, 1, 1]])
+
+    assert gf_edf_zassenhaus(f, 8, 2, ZZ) == g
+
+
+def test_gf_factor():
+    assert gf_factor([], 11, ZZ) == (0, [])
+    assert gf_factor([1], 11, ZZ) == (1, [])
+    assert gf_factor([1, 1], 11, ZZ) == (1, [([1, 1], 1)])
+
+    assert gf_factor_sqf([], 11, ZZ) == (0, [])
+    assert gf_factor_sqf([1], 11, ZZ) == (1, [])
+    assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+
+    assert gf_factor_sqf([], 11, ZZ) == (0, [])
+    assert gf_factor_sqf([1], 11, ZZ) == (1, [])
+    assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+
+    assert gf_factor_sqf([], 11, ZZ) == (0, [])
+    assert gf_factor_sqf([1], 11, ZZ) == (1, [])
+    assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+
+    assert gf_factor_sqf(ZZ.map([]), 11, ZZ) == (0, [])
+    assert gf_factor_sqf(ZZ.map([1]), 11, ZZ) == (1, [])
+    assert gf_factor_sqf(ZZ.map([1, 1]), 11, ZZ) == (1, [[1, 1]])
+
+    f, p = ZZ.map([1, 0, 0, 1, 0]), 2
+
+    g = (1, [([1, 0], 1),
+             ([1, 1], 1),
+             ([1, 1, 1], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    g = (1, [[1, 0],
+             [1, 1],
+             [1, 1, 1]])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor_sqf(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor_sqf(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor_sqf(f, p, ZZ) == g
+
+    f, p = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11), 11
+
+    g = (1, [([1, 1], 1),
+             ([1, 5, 3], 1),
+             ([1, 2, 3, 4], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    f, p = [1, 5, 8, 4], 11
+
+    g = (1, [([1, 1], 1), ([1, 2], 2)])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11
+
+    g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11
+
+    g = (1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
+             ([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    f, p = gf_from_dict({32: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11
+
+    g = (8, [([1, 3], 1),
+             ([1, 8], 1),
+             ([1, 0, 9], 1),
+             ([1, 2, 2], 1),
+             ([1, 9, 2], 1),
+             ([1, 0, 5, 0, 7], 1),
+             ([1, 0, 6, 0, 7], 1),
+             ([1, 0, 0, 0, 1, 0, 0, 0, 6], 1),
+             ([1, 0, 0, 0, 10, 0, 0, 0, 6], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    f, p = gf_from_dict({63: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11
+
+    g = (8, [([1, 7], 1),
+             ([1, 4, 5], 1),
+             ([1, 6, 8, 2], 1),
+             ([1, 9, 9, 2], 1),
+             ([1, 0, 0, 9, 0, 0, 4], 1),
+             ([1, 2, 0, 8, 4, 6, 4], 1),
+             ([1, 2, 3, 8, 0, 6, 4], 1),
+             ([1, 2, 6, 0, 8, 4, 4], 1),
+             ([1, 3, 3, 1, 6, 8, 4], 1),
+             ([1, 5, 6, 0, 8, 6, 4], 1),
+             ([1, 6, 2, 7, 9, 8, 4], 1),
+             ([1, 10, 4, 7, 10, 7, 4], 1),
+             ([1, 10, 10, 1, 4, 9, 4], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'berlekamp')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    # Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
+
+    p = ZZ(nextprime(int((2**15 * pi).evalf())))
+    f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
+
+    assert gf_sqf_p(f, p, ZZ) is True
+
+    g = (1, [([1, 22730, 68144], 1),
+             ([1, 81553, 77449, 86810, 4724], 1),
+             ([1, 86276, 56779, 14859, 31575], 1),
+             ([1, 15347, 95022, 84569, 94508, 92335], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    g = (1, [[1, 22730, 68144],
+             [1, 81553, 77449, 86810, 4724],
+             [1, 86276, 56779, 14859, 31575],
+             [1, 15347, 95022, 84569, 94508, 92335]])
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor_sqf(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor_sqf(f, p, ZZ) == g
+
+    # Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
+    # (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
+
+    p = ZZ(nextprime(int((2**4 * pi).evalf())))
+    f = ZZ.map([1, 2, 5, 26, 41, 39, 38])
+
+    assert gf_sqf_p(f, p, ZZ) is True
+
+    g = (1, [([1, 44, 26], 1),
+             ([1, 11, 25, 18, 30], 1)])
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor(f, p, ZZ) == g
+
+    g = (1, [[1, 44, 26],
+             [1, 11, 25, 18, 30]])
+
+    config.setup('GF_FACTOR_METHOD', 'zassenhaus')
+    assert gf_factor_sqf(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'shoup')
+    assert gf_factor_sqf(f, p, ZZ) == g
+
+    config.setup('GF_FACTOR_METHOD', 'other')
+    raises(KeyError, lambda: gf_factor([1, 1], 11, ZZ))
+    config.setup('GF_FACTOR_METHOD')
+
+
+def test_gf_csolve():
+    assert gf_value([1, 7, 2, 4], 11) == 2204
+
+    assert linear_congruence(4, 3, 5) == [2]
+    assert linear_congruence(0, 3, 5) == []
+    assert linear_congruence(6, 1, 4) == []
+    assert linear_congruence(0, 5, 5) == [0, 1, 2, 3, 4]
+    assert linear_congruence(3, 12, 15) == [4, 9, 14]
+    assert linear_congruence(6, 0, 18) == [0, 3, 6, 9, 12, 15]
+    # _csolve_prime_las_vegas
+    assert _csolve_prime_las_vegas([2, 3, 1], 5) == [2, 4]
+    assert _csolve_prime_las_vegas([2, 0, 1], 5) == []
+    from sympy.ntheory import primerange
+    for p in primerange(2, 100):
+        # f = x**(p-1) - 1
+        f = gf_sub_ground(gf_pow([1, 0], p - 1, p, ZZ), 1, p, ZZ)
+        assert _csolve_prime_las_vegas(f, p) == list(range(1, p))
+    # with power = 1
+    assert csolve_prime([1, 3, 2, 17], 7) == [3]
+    assert csolve_prime([1, 3, 1, 5], 5) == [0, 1]
+    assert csolve_prime([3, 6, 9, 3], 3) == [0, 1, 2]
+    # with power > 1
+    assert csolve_prime(
+        [1, 1, 223], 3, 4) == [4, 13, 22, 31, 40, 49, 58, 67, 76]
+    assert csolve_prime([3, 5, 2, 25], 5, 3) == [16, 50, 99]
+    assert csolve_prime([3, 2, 2, 49], 7, 3) == [147, 190, 234]
+
+    assert gf_csolve([1, 1, 7], 189) == [13, 49, 76, 112, 139, 175]
+    assert gf_csolve([1, 3, 4, 1, 30], 60) == [10, 30]
+    assert gf_csolve([1, 1, 7], 15) == []

rl/Lib/site-packages/sympy/polys/tests/test_groebnertools.py

@@ -0,0 +1,533 @@
+"""Tests for Groebner bases. """
+
+from sympy.polys.groebnertools import (
+    groebner, sig, sig_key,
+    lbp, lbp_key, critical_pair,
+    cp_key, is_rewritable_or_comparable,
+    Sign, Polyn, Num, s_poly, f5_reduce,
+    groebner_lcm, groebner_gcd, is_groebner,
+    is_reduced
+)
+
+from sympy.polys.fglmtools import _representing_matrices
+from sympy.polys.orderings import lex, grlex
+
+from sympy.polys.rings import ring, xring
+from sympy.polys.domains import ZZ, QQ
+
+from sympy.testing.pytest import slow
+from sympy.polys import polyconfig as config
+
+def _do_test_groebner():
+    R, x,y = ring("x,y", QQ, lex)
+    f = x**2 + 2*x*y**2
+    g = x*y + 2*y**3 - 1
+
+    assert groebner([f, g], R) == [x, y**3 - QQ(1,2)]
+
+    R, y,x = ring("y,x", QQ, lex)
+    f = 2*x**2*y + y**2
+    g = 2*x**3 + x*y - 1
+
+    assert groebner([f, g], R) == [y, x**3 - QQ(1,2)]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = x - z**2
+    g = y - z**3
+
+    assert groebner([f, g], R) == [f, g]
+
+    R, x,y = ring("x,y", QQ, grlex)
+    f = x**3 - 2*x*y
+    g = x**2*y + x - 2*y**2
+
+    assert groebner([f, g], R) == [x**2, x*y, -QQ(1,2)*x + y**2]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = -x**2 + y
+    g = -x**3 + z
+
+    assert groebner([f, g], R) == [x**2 - y, x*y - z, x*z - y**2, y**3 - z**2]
+
+    R, x,y,z = ring("x,y,z", QQ, grlex)
+    f = -x**2 + y
+    g = -x**3 + z
+
+    assert groebner([f, g], R) == [y**3 - z**2, x**2 - y, x*y - z, x*z - y**2]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = -x**2 + z
+    g = -x**3 + y
+
+    assert groebner([f, g], R) == [x**2 - z, x*y - z**2, x*z - y, y**2 - z**3]
+
+    R, x,y,z = ring("x,y,z", QQ, grlex)
+    f = -x**2 + z
+    g = -x**3 + y
+
+    assert groebner([f, g], R) == [-y**2 + z**3, x**2 - z, x*y - z**2, x*z - y]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = x - y**2
+    g = -y**3 + z
+
+    assert groebner([f, g], R) == [x - y**2, y**3 - z]
+
+    R, x,y,z = ring("x,y,z", QQ, grlex)
+    f = x - y**2
+    g = -y**3 + z
+
+    assert groebner([f, g], R) == [x**2 - y*z, x*y - z, -x + y**2]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = x - z**2
+    g = y - z**3
+
+    assert groebner([f, g], R) == [x - z**2, y - z**3]
+
+    R, x,y,z = ring("x,y,z", QQ, grlex)
+    f = x - z**2
+    g = y - z**3
+
+    assert groebner([f, g], R) == [x**2 - y*z, x*z - y, -x + z**2]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = -y**2 + z
+    g = x - y**3
+
+    assert groebner([f, g], R) == [x - y*z, y**2 - z]
+
+    R, x,y,z = ring("x,y,z", QQ, grlex)
+    f = -y**2 + z
+    g = x - y**3
+
+    assert groebner([f, g], R) == [-x**2 + z**3, x*y - z**2, y**2 - z, -x + y*z]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = y - z**2
+    g = x - z**3
+
+    assert groebner([f, g], R) == [x - z**3, y - z**2]
+
+    R, x,y,z = ring("x,y,z", QQ, grlex)
+    f = y - z**2
+    g = x - z**3
+
+    assert groebner([f, g], R) == [-x**2 + y**3, x*z - y**2, -x + y*z, -y + z**2]
+
+    R, x,y,z = ring("x,y,z", QQ, lex)
+    f = 4*x**2*y**2 + 4*x*y + 1
+    g = x**2 + y**2 - 1
+
+    assert groebner([f, g], R) == [
+        x - 4*y**7 + 8*y**5 - 7*y**3 + 3*y,
+        y**8 - 2*y**6 + QQ(3,2)*y**4 - QQ(1,2)*y**2 + QQ(1,16),
+    ]
+
+def test_groebner_buchberger():
+    with config.using(groebner='buchberger'):
+        _do_test_groebner()
+
+def test_groebner_f5b():
+    with config.using(groebner='f5b'):
+        _do_test_groebner()
+
+def _do_test_benchmark_minpoly():
+    R, x,y,z = ring("x,y,z", QQ, lex)
+
+    F = [x**3 + x + 1, y**2 + y + 1, (x + y) * z - (x**2 + y)]
+    G = [x + QQ(155,2067)*z**5 - QQ(355,689)*z**4 + QQ(6062,2067)*z**3 - QQ(3687,689)*z**2 + QQ(6878,2067)*z - QQ(25,53),
+         y + QQ(4,53)*z**5 - QQ(91,159)*z**4 + QQ(523,159)*z**3 - QQ(387,53)*z**2 + QQ(1043,159)*z - QQ(308,159),
+         z**6 - 7*z**5 + 41*z**4 - 82*z**3 + 89*z**2 - 46*z + 13]
+
+    assert groebner(F, R) == G
+
+def test_benchmark_minpoly_buchberger():
+    with config.using(groebner='buchberger'):
+        _do_test_benchmark_minpoly()
+
+def test_benchmark_minpoly_f5b():
+    with config.using(groebner='f5b'):
+        _do_test_benchmark_minpoly()
+
+
+def test_benchmark_coloring():
+    V = range(1, 12 + 1)
+    E = [(1, 2), (2, 3), (1, 4), (1, 6), (1, 12), (2, 5), (2, 7), (3, 8), (3, 10),
+         (4, 11), (4, 9), (5, 6), (6, 7), (7, 8), (8, 9), (9, 10), (10, 11),
+         (11, 12), (5, 12), (5, 9), (6, 10), (7, 11), (8, 12), (3, 4)]
+
+    R, V = xring([ "x%d" % v for v in V ], QQ, lex)
+    E = [(V[i - 1], V[j - 1]) for i, j in E]
+
+    x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = V
+
+    I3 = [x**3 - 1 for x in V]
+    Ig = [x**2 + x*y + y**2 for x, y in E]
+
+    I = I3 + Ig
+
+    assert groebner(I[:-1], R) == [
+        x1 + x11 + x12,
+        x2 - x11,
+        x3 - x12,
+        x4 - x12,
+        x5 + x11 + x12,
+        x6 - x11,
+        x7 - x12,
+        x8 + x11 + x12,
+        x9 - x11,
+        x10 + x11 + x12,
+        x11**2 + x11*x12 + x12**2,
+        x12**3 - 1,
+    ]
+
+    assert groebner(I, R) == [1]
+
+
+def _do_test_benchmark_katsura_3():
+    R, x0,x1,x2 = ring("x:3", ZZ, lex)
+    I = [x0 + 2*x1 + 2*x2 - 1,
+         x0**2 + 2*x1**2 + 2*x2**2 - x0,
+         2*x0*x1 + 2*x1*x2 - x1]
+
+    assert groebner(I, R) == [
+        -7 + 7*x0 + 8*x2 + 158*x2**2 - 420*x2**3,
+        7*x1 + 3*x2 - 79*x2**2 + 210*x2**3,
+        x2 + x2**2 - 40*x2**3 + 84*x2**4,
+    ]
+
+    R, x0,x1,x2 = ring("x:3", ZZ, grlex)
+    I = [ i.set_ring(R) for i in I ]
+
+    assert groebner(I, R) == [
+        7*x1 + 3*x2 - 79*x2**2 + 210*x2**3,
+        -x1 + x2 - 3*x2**2 + 5*x1**2,
+        -x1 - 4*x2 + 10*x1*x2 + 12*x2**2,
+        -1 + x0 + 2*x1 + 2*x2,
+    ]
+
+def test_benchmark_katsura3_buchberger():
+    with config.using(groebner='buchberger'):
+        _do_test_benchmark_katsura_3()
+
+def test_benchmark_katsura3_f5b():
+    with config.using(groebner='f5b'):
+        _do_test_benchmark_katsura_3()
+
+def _do_test_benchmark_katsura_4():
+    R, x0,x1,x2,x3 = ring("x:4", ZZ, lex)
+    I = [x0 + 2*x1 + 2*x2 + 2*x3 - 1,
+         x0**2 + 2*x1**2 + 2*x2**2 + 2*x3**2 - x0,
+         2*x0*x1 + 2*x1*x2 + 2*x2*x3 - x1,
+         x1**2 + 2*x0*x2 + 2*x1*x3 - x2]
+
+    assert groebner(I, R) == [
+        5913075*x0 - 159690237696*x3**7 + 31246269696*x3**6 + 27439610544*x3**5 - 6475723368*x3**4 - 838935856*x3**3 + 275119624*x3**2 + 4884038*x3 - 5913075,
+        1971025*x1 - 97197721632*x3**7 + 73975630752*x3**6 - 12121915032*x3**5 - 2760941496*x3**4 + 814792828*x3**3 - 1678512*x3**2 - 9158924*x3,
+        5913075*x2 + 371438283744*x3**7 - 237550027104*x3**6 + 22645939824*x3**5 + 11520686172*x3**4 - 2024910556*x3**3 - 132524276*x3**2 + 30947828*x3,
+        128304*x3**8 - 93312*x3**7 + 15552*x3**6 + 3144*x3**5 -
+        1120*x3**4 + 36*x3**3 + 15*x3**2 - x3,
+    ]
+
+    R, x0,x1,x2,x3 = ring("x:4", ZZ, grlex)
+    I = [ i.set_ring(R) for i in I ]
+
+    assert groebner(I, R) == [
+        393*x1 - 4662*x2**2 + 4462*x2*x3 - 59*x2 + 224532*x3**4 - 91224*x3**3 - 678*x3**2 + 2046*x3,
+        -x1 + 196*x2**3 - 21*x2**2 + 60*x2*x3 - 18*x2 - 168*x3**3 + 83*x3**2 - 9*x3,
+        -6*x1 + 1134*x2**2*x3 - 189*x2**2 - 466*x2*x3 + 32*x2 - 630*x3**3 + 57*x3**2 + 51*x3,
+        33*x1 + 63*x2**2 + 2268*x2*x3**2 - 188*x2*x3 + 34*x2 + 2520*x3**3 - 849*x3**2 + 3*x3,
+        7*x1**2 - x1 - 7*x2**2 - 24*x2*x3 + 3*x2 - 15*x3**2 + 5*x3,
+        14*x1*x2 - x1 + 14*x2**2 + 18*x2*x3 - 4*x2 + 6*x3**2 - 2*x3,
+        14*x1*x3 - x1 + 7*x2**2 + 32*x2*x3 - 4*x2 + 27*x3**2 - 9*x3,
+        x0 + 2*x1 + 2*x2 + 2*x3 - 1,
+    ]
+
+def test_benchmark_kastura_4_buchberger():
+    with config.using(groebner='buchberger'):
+        _do_test_benchmark_katsura_4()
+
+def test_benchmark_kastura_4_f5b():
+    with config.using(groebner='f5b'):
+        _do_test_benchmark_katsura_4()
+
+def _do_test_benchmark_czichowski():
+    R, x,t = ring("x,t", ZZ, lex)
+    I = [9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9,
+         (-72 - 72*t)*x**7 + (-256 - 252*t)*x**6 + (192 + 192*t)*x**5 + (1280 + 1260*t)*x**4 + (312 + 312*t)*x**3 + (-404*t)*x**2 + (-576 - 576*t)*x + 96 + 108*t]
+
+    assert groebner(I, R) == [
+        3725588592068034903797967297424801242396746870413359539263038139343329273586196480000*x -
+        160420835591776763325581422211936558925462474417709511019228211783493866564923546661604487873*t**7 -
+        1406108495478033395547109582678806497509499966197028487131115097902188374051595011248311352864*t**6 -
+        5241326875850889518164640374668786338033653548841427557880599579174438246266263602956254030352*t**5 -
+        10758917262823299139373269714910672770004760114329943852726887632013485035262879510837043892416*t**4 -
+        13119383576444715672578819534846747735372132018341964647712009275306635391456880068261130581248*t**3 -
+        9491412317016197146080450036267011389660653495578680036574753839055748080962214787557853941760*t**2 -
+        3767520915562795326943800040277726397326609797172964377014046018280260848046603967211258368000*t -
+        632314652371226552085897259159210286886724229880266931574701654721512325555116066073245696000,
+        610733380717522355121*t**8 +
+        6243748742141230639968*t**7 +
+        27761407182086143225024*t**6 +
+        70066148869420956398592*t**5 +
+        109701225644313784229376*t**4 +
+        109009005495588442152960*t**3 +
+        67072101084384786432000*t**2 +
+        23339979742629593088000*t +
+        3513592776846090240000,
+    ]
+
+    R, x,t = ring("x,t", ZZ, grlex)
+    I = [ i.set_ring(R) for i in I ]
+
+    assert groebner(I, R) == [
+        16996618586000601590732959134095643086442*t**3*x -
+        32936701459297092865176560282688198064839*t**3 +
+        78592411049800639484139414821529525782364*t**2*x -
+        120753953358671750165454009478961405619916*t**2 +
+        120988399875140799712152158915653654637280*t*x -
+        144576390266626470824138354942076045758736*t +
+        60017634054270480831259316163620768960*x**2 +
+        61976058033571109604821862786675242894400*x -
+        56266268491293858791834120380427754600960,
+        576689018321912327136790519059646508441672750656050290242749*t**4 +
+        2326673103677477425562248201573604572527893938459296513327336*t**3 +
+        110743790416688497407826310048520299245819959064297990236000*t**2*x +
+        3308669114229100853338245486174247752683277925010505284338016*t**2 +
+        323150205645687941261103426627818874426097912639158572428800*t*x +
+        1914335199925152083917206349978534224695445819017286960055680*t +
+        861662882561803377986838989464278045397192862768588480000*x**2 +
+        235296483281783440197069672204341465480107019878814196672000*x +
+        361850798943225141738895123621685122544503614946436727532800,
+       -117584925286448670474763406733005510014188341867*t**3 +
+        68566565876066068463853874568722190223721653044*t**2*x -
+        435970731348366266878180788833437896139920683940*t**2 +
+        196297602447033751918195568051376792491869233408*t*x -
+        525011527660010557871349062870980202067479780112*t +
+        517905853447200553360289634770487684447317120*x**3 +
+        569119014870778921949288951688799397569321920*x**2 +
+        138877356748142786670127389526667463202210102080*x -
+        205109210539096046121625447192779783475018619520,
+       -3725142681462373002731339445216700112264527*t**3 +
+        583711207282060457652784180668273817487940*t**2*x -
+        12381382393074485225164741437227437062814908*t**2 +
+        151081054097783125250959636747516827435040*t*x**2 +
+        1814103857455163948531448580501928933873280*t*x -
+        13353115629395094645843682074271212731433648*t +
+        236415091385250007660606958022544983766080*x**2 +
+        1390443278862804663728298060085399578417600*x -
+        4716885828494075789338754454248931750698880,
+    ]
+
+# NOTE: This is very slow (> 2 minutes on 3.4 GHz) without GMPY
+@slow
+def test_benchmark_czichowski_buchberger():
+    with config.using(groebner='buchberger'):
+        _do_test_benchmark_czichowski()
+
+def test_benchmark_czichowski_f5b():
+    with config.using(groebner='f5b'):
+        _do_test_benchmark_czichowski()
+
+def _do_test_benchmark_cyclic_4():
+    R, a,b,c,d = ring("a,b,c,d", ZZ, lex)
+
+    I = [a + b + c + d,
+         a*b + a*d + b*c + b*d,
+         a*b*c + a*b*d + a*c*d + b*c*d,
+         a*b*c*d - 1]
+
+    assert groebner(I, R) == [
+        4*a + 3*d**9 - 4*d**5 - 3*d,
+        4*b + 4*c - 3*d**9 + 4*d**5 + 7*d,
+        4*c**2 + 3*d**10 - 4*d**6 - 3*d**2,
+        4*c*d**4 + 4*c - d**9 + 4*d**5 + 5*d, d**12 - d**8 - d**4 + 1
+    ]
+
+    R, a,b,c,d = ring("a,b,c,d", ZZ, grlex)
+    I = [ i.set_ring(R) for i in I ]
+
+    assert groebner(I, R) == [
+        3*b*c - c**2 + d**6 - 3*d**2,
+        -b + 3*c**2*d**3 - c - d**5 - 4*d,
+        -b + 3*c*d**4 + 2*c + 2*d**5 + 2*d,
+        c**4 + 2*c**2*d**2 - d**4 - 2,
+        c**3*d + c*d**3 + d**4 + 1,
+        b*c**2 - c**3 - c**2*d - 2*c*d**2 - d**3,
+        b**2 - c**2, b*d + c**2 + c*d + d**2,
+        a + b + c + d
+    ]
+
+def test_benchmark_cyclic_4_buchberger():
+    with config.using(groebner='buchberger'):
+        _do_test_benchmark_cyclic_4()
+
+def test_benchmark_cyclic_4_f5b():
+    with config.using(groebner='f5b'):
+        _do_test_benchmark_cyclic_4()
+
+def test_sig_key():
+    s1 = sig((0,) * 3, 2)
+    s2 = sig((1,) * 3, 4)
+    s3 = sig((2,) * 3, 2)
+
+    assert sig_key(s1, lex) > sig_key(s2, lex)
+    assert sig_key(s2, lex) < sig_key(s3, lex)
+
+
+def test_lbp_key():
+    R, x,y,z,t = ring("x,y,z,t", ZZ, lex)
+
+    p1 = lbp(sig((0,) * 4, 3), R.zero, 12)
+    p2 = lbp(sig((0,) * 4, 4), R.zero, 13)
+    p3 = lbp(sig((0,) * 4, 4), R.zero, 12)
+
+    assert lbp_key(p1) > lbp_key(p2)
+    assert lbp_key(p2) < lbp_key(p3)
+
+
+def test_critical_pair():
+    # from cyclic4 with grlex
+    R, x,y,z,t = ring("x,y,z,t", QQ, grlex)
+
+    p1 = (((0, 0, 0, 0), 4), y*z*t**2 + z**2*t**2 - t**4 - 1, 4)
+    q1 = (((0, 0, 0, 0), 2), -y**2 - y*t - z*t - t**2, 2)
+
+    p2 = (((0, 0, 0, 2), 3), z**3*t**2 + z**2*t**3 - z - t, 5)
+    q2 = (((0, 0, 2, 2), 2), y*z + z*t**5 + z*t + t**6, 13)
+
+    assert critical_pair(p1, q1, R) == (
+        ((0, 0, 1, 2), 2), ((0, 0, 1, 2), QQ(-1, 1)), (((0, 0, 0, 0), 2), -y**2 - y*t - z*t - t**2, 2),
+        ((0, 1, 0, 0), 4), ((0, 1, 0, 0), QQ(1, 1)), (((0, 0, 0, 0), 4), y*z*t**2 + z**2*t**2 - t**4 - 1, 4)
+    )
+    assert critical_pair(p2, q2, R) == (
+        ((0, 0, 4, 2), 2), ((0, 0, 2, 0), QQ(1, 1)), (((0, 0, 2, 2), 2), y*z + z*t**5 + z*t + t**6, 13),
+        ((0, 0, 0, 5), 3), ((0, 0, 0, 3), QQ(1, 1)), (((0, 0, 0, 2), 3), z**3*t**2 + z**2*t**3 - z - t, 5)
+    )
+
+def test_cp_key():
+    # from cyclic4 with grlex
+    R, x,y,z,t = ring("x,y,z,t", QQ, grlex)
+
+    p1 = (((0, 0, 0, 0), 4), y*z*t**2 + z**2*t**2 - t**4 - 1, 4)
+    q1 = (((0, 0, 0, 0), 2), -y**2 - y*t - z*t - t**2, 2)
+
+    p2 = (((0, 0, 0, 2), 3), z**3*t**2 + z**2*t**3 - z - t, 5)
+    q2 = (((0, 0, 2, 2), 2), y*z + z*t**5 + z*t + t**6, 13)
+
+    cp1 = critical_pair(p1, q1, R)
+    cp2 = critical_pair(p2, q2, R)
+
+    assert cp_key(cp1, R) < cp_key(cp2, R)
+
+    cp1 = critical_pair(p1, p2, R)
+    cp2 = critical_pair(q1, q2, R)
+
+    assert cp_key(cp1, R) < cp_key(cp2, R)
+
+
+def test_is_rewritable_or_comparable():
+    # from katsura4 with grlex
+    R, x,y,z,t = ring("x,y,z,t", QQ, grlex)
+
+    p = lbp(sig((0, 0, 2, 1), 2), R.zero, 2)
+    B = [lbp(sig((0, 0, 0, 1), 2), QQ(2,45)*y**2 + QQ(1,5)*y*z + QQ(5,63)*y*t + z**2*t + QQ(4,45)*z**2 + QQ(76,35)*z*t**2 - QQ(32,105)*z*t + QQ(13,7)*t**3 - QQ(13,21)*t**2, 6)]
+
+    # rewritable:
+    assert is_rewritable_or_comparable(Sign(p), Num(p), B) is True
+
+    p = lbp(sig((0, 1, 1, 0), 2), R.zero, 7)
+    B = [lbp(sig((0, 0, 0, 0), 3), QQ(10,3)*y*z + QQ(4,3)*y*t - QQ(1,3)*y + 4*z**2 + QQ(22,3)*z*t - QQ(4,3)*z + 4*t**2 - QQ(4,3)*t, 3)]
+
+    # comparable:
+    assert is_rewritable_or_comparable(Sign(p), Num(p), B) is True
+
+
+def test_f5_reduce():
+    # katsura3 with lex
+    R, x,y,z = ring("x,y,z", QQ, lex)
+
+    F = [(((0, 0, 0), 1), x + 2*y + 2*z - 1, 1),
+         (((0, 0, 0), 2), 6*y**2 + 8*y*z - 2*y + 6*z**2 - 2*z, 2),
+         (((0, 0, 0), 3), QQ(10,3)*y*z - QQ(1,3)*y + 4*z**2 - QQ(4,3)*z, 3),
+         (((0, 0, 1), 2), y + 30*z**3 - QQ(79,7)*z**2 + QQ(3,7)*z, 4),
+         (((0, 0, 2), 2), z**4 - QQ(10,21)*z**3 + QQ(1,84)*z**2 + QQ(1,84)*z, 5)]
+
+    cp = critical_pair(F[0], F[1], R)
+    s = s_poly(cp)
+
+    assert f5_reduce(s, F) == (((0, 2, 0), 1), R.zero, 1)
+
+    s = lbp(sig(Sign(s)[0], 100), Polyn(s), Num(s))
+    assert f5_reduce(s, F) == s
+
+
+def test_representing_matrices():
+    R, x,y = ring("x,y", QQ, grlex)
+
+    basis = [(0, 0), (0, 1), (1, 0), (1, 1)]
+    F = [x**2 - x - 3*y + 1, -2*x + y**2 + y - 1]
+
+    assert _representing_matrices(basis, F, R) == [
+        [[QQ(0, 1), QQ(0, 1),-QQ(1, 1), QQ(3, 1)],
+         [QQ(0, 1), QQ(0, 1), QQ(3, 1),-QQ(4, 1)],
+         [QQ(1, 1), QQ(0, 1), QQ(1, 1), QQ(6, 1)],
+         [QQ(0, 1), QQ(1, 1), QQ(0, 1), QQ(1, 1)]],
+        [[QQ(0, 1), QQ(1, 1), QQ(0, 1),-QQ(2, 1)],
+         [QQ(1, 1),-QQ(1, 1), QQ(0, 1), QQ(6, 1)],
+         [QQ(0, 1), QQ(2, 1), QQ(0, 1), QQ(3, 1)],
+         [QQ(0, 1), QQ(0, 1), QQ(1, 1),-QQ(1, 1)]]]
+
+def test_groebner_lcm():
+    R, x,y,z = ring("x,y,z", ZZ)
+
+    assert groebner_lcm(x**2 - y**2, x - y) == x**2 - y**2
+    assert groebner_lcm(2*x**2 - 2*y**2, 2*x - 2*y) == 2*x**2 - 2*y**2
+
+    R, x,y,z = ring("x,y,z", QQ)
+
+    assert groebner_lcm(x**2 - y**2, x - y) == x**2 - y**2
+    assert groebner_lcm(2*x**2 - 2*y**2, 2*x - 2*y) == 2*x**2 - 2*y**2
+
+    R, x,y = ring("x,y", ZZ)
+
+    assert groebner_lcm(x**2*y, x*y**2) == x**2*y**2
+
+    f = 2*x*y**5 - 3*x*y**4 - 2*x*y**3 + 3*x*y**2
+    g = y**5 - 2*y**3 + y
+    h = 2*x*y**7 - 3*x*y**6 - 4*x*y**5 + 6*x*y**4 + 2*x*y**3 - 3*x*y**2
+
+    assert groebner_lcm(f, g) == h
+
+    f = x**3 - 3*x**2*y - 9*x*y**2 - 5*y**3
+    g = x**4 + 6*x**3*y + 12*x**2*y**2 + 10*x*y**3 + 3*y**4
+    h = x**5 + x**4*y - 18*x**3*y**2 - 50*x**2*y**3 - 47*x*y**4 - 15*y**5
+
+    assert groebner_lcm(f, g) == h
+
+def test_groebner_gcd():
+    R, x,y,z = ring("x,y,z", ZZ)
+
+    assert groebner_gcd(x**2 - y**2, x - y) == x - y
+    assert groebner_gcd(2*x**2 - 2*y**2, 2*x - 2*y) == 2*x - 2*y
+
+    R, x,y,z = ring("x,y,z", QQ)
+
+    assert groebner_gcd(x**2 - y**2, x - y) == x - y
+    assert groebner_gcd(2*x**2 - 2*y**2, 2*x - 2*y) == x - y
+
+def test_is_groebner():
+    R, x,y = ring("x,y", QQ, grlex)
+    valid_groebner = [x**2, x*y, -QQ(1,2)*x + y**2]
+    invalid_groebner = [x**3, x*y, -QQ(1,2)*x + y**2]
+    assert is_groebner(valid_groebner, R) is True
+    assert is_groebner(invalid_groebner, R) is False
+
+def test_is_reduced():
+    R, x, y = ring("x,y", QQ, lex)
+    f = x**2 + 2*x*y**2
+    g = x*y + 2*y**3 - 1
+    assert is_reduced([f, g], R) ==  False
+    G = groebner([f, g], R)
+    assert is_reduced(G, R) == True

rl/Lib/site-packages/sympy/polys/tests/test_heuristicgcd.py

@@ -0,0 +1,152 @@
+from sympy.polys.rings import ring
+from sympy.polys.domains import ZZ
+from sympy.polys.heuristicgcd import heugcd
+
+
+def test_heugcd_univariate_integers():
+    R, x = ring("x", ZZ)
+
+    f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
+    g = x**3 + 6*x**2 + 11*x + 6
+
+    h = x**2 + 3*x + 2
+
+    cff = x**2 + 5*x + 4
+    cfg = x + 3
+
+    assert heugcd(f, g) == (h, cff, cfg)
+
+    f = x**4 - 4
+    g = x**4 + 4*x**2 + 4
+
+    h = x**2 + 2
+
+    cff = x**2 - 2
+    cfg = x**2 + 2
+
+    assert heugcd(f, g) == (h, cff, cfg)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    h = 1
+
+    cff = f
+    cfg = g
+
+    assert heugcd(f, g) == (h, cff, cfg)
+
+    f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+        + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+        + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+        + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
+        - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+        + 289127344604779611146960547954288113529690984687482920704*x**14 \
+        + 19007977035740498977629742919480623972236450681*x**7 \
+        + 311973482284542371301330321821976049
+
+    g =   365431878023781158602430064717380211405897160759702125019136*x**21 \
+        + 197599133478719444145775798221171663643171734081650688*x**14 \
+        - 9504116979659010018253915765478924103928886144*x**7 \
+        - 311973482284542371301330321821976049
+
+    # TODO: assert heugcd(f, f.diff(x))[0] == g
+
+    f = 1317378933230047068160*x + 2945748836994210856960
+    g = 120352542776360960*x + 269116466014453760
+
+    h = 120352542776360960*x + 269116466014453760
+    cff = 10946
+    cfg = 1
+
+    assert heugcd(f, g) == (h, cff, cfg)
+
+def test_heugcd_multivariate_integers():
+    R, x, y = ring("x,y", ZZ)
+
+    f, g = 2*x**2 + 4*x + 2, x + 1
+    assert heugcd(f, g) == (x + 1, 2*x + 2, 1)
+
+    f, g = x + 1, 2*x**2 + 4*x + 2
+    assert heugcd(f, g) == (x + 1, 1, 2*x + 2)
+
+    R, x, y, z, u = ring("x,y,z,u", ZZ)
+
+    f, g = u**2 + 2*u + 1, 2*u + 2
+    assert heugcd(f, g) == (u + 1, u + 1, 2)
+
+    f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1
+    h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1
+
+    assert heugcd(f, g) == (h, cff, cfg)
+    assert heugcd(g, f) == (h, cfg, cff)
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g, h = R.fateman_poly_F_2()
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    f, g, h = R.fateman_poly_F_3()
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, t = ring("x,y,z,t", ZZ)
+
+    f, g, h = R.fateman_poly_F_3()
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+
+def test_issue_10996():
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f = 12*x**6*y**7*z**3 - 3*x**4*y**9*z**3 + 12*x**3*y**5*z**4
+    g = -48*x**7*y**8*z**3 + 12*x**5*y**10*z**3 - 48*x**5*y**7*z**2 + \
+    36*x**4*y**7*z - 48*x**4*y**6*z**4 + 12*x**3*y**9*z**2 - 48*x**3*y**4 \
+    - 9*x**2*y**9*z - 48*x**2*y**5*z**3 + 12*x*y**6 + 36*x*y**5*z**2 - 48*y**2*z
+
+    H, cff, cfg = heugcd(f, g)
+
+    assert H == 12*x**3*y**4 - 3*x*y**6 + 12*y**2*z
+    assert H*cff == f and H*cfg == g
+
+
+def test_issue_25793():
+    R, x = ring("x", ZZ)
+    f = x - 4851  # failure starts for values more than 4850
+    g = f*(2*x + 1)
+    H, cff, cfg = R.dup_zz_heu_gcd(f, g)
+    assert H == f
+    # needs a test for dmp, too, that fails in master before this change

rl/Lib/site-packages/sympy/polys/tests/test_hypothesis.py

@@ -0,0 +1,36 @@
+from hypothesis import given
+from hypothesis import strategies as st
+from sympy.abc import x
+from sympy.polys.polytools import Poly
+
+
+def polys(*, nonzero=False, domain="ZZ"):
+    # This is a simple strategy, but sufficient the tests below
+    elems = {"ZZ": st.integers(), "QQ": st.fractions()}
+    coeff_st = st.lists(elems[domain])
+    if nonzero:
+        coeff_st = coeff_st.filter(any)
+    return st.builds(Poly, coeff_st, st.just(x), domain=st.just(domain))
+
+
+@given(f=polys(), g=polys(), r=polys())
+def test_gcd_hypothesis(f, g, r):
+    gcd_1 = f.gcd(g)
+    gcd_2 = g.gcd(f)
+    assert gcd_1 == gcd_2
+
+    # multiply by r
+    gcd_3 = g.gcd(f + r * g)
+    assert gcd_1 == gcd_3
+
+
+@given(f_z=polys(), g_z=polys(nonzero=True))
+def test_poly_hypothesis_integers(f_z, g_z):
+    remainder_z = f_z.rem(g_z)
+    assert g_z.degree() >= remainder_z.degree() or remainder_z.degree() == 0
+
+
+@given(f_q=polys(domain="QQ"), g_q=polys(nonzero=True, domain="QQ"))
+def test_poly_hypothesis_rationals(f_q, g_q):
+    remainder_q = f_q.rem(g_q)
+    assert g_q.degree() >= remainder_q.degree() or remainder_q.degree() == 0

rl/Lib/site-packages/sympy/polys/tests/test_injections.py

@@ -0,0 +1,39 @@
+"""Tests for functions that inject symbols into the global namespace. """
+
+from sympy.polys.rings import vring
+from sympy.polys.fields import vfield
+from sympy.polys.domains import QQ
+
+def test_vring():
+    ns = {'vring':vring, 'QQ':QQ}
+    exec('R = vring("r", QQ)', ns)
+    exec('assert r == R.gens[0]', ns)
+
+    exec('R = vring("rb rbb rcc rzz _rx", QQ)', ns)
+    exec('assert rb == R.gens[0]', ns)
+    exec('assert rbb == R.gens[1]', ns)
+    exec('assert rcc == R.gens[2]', ns)
+    exec('assert rzz == R.gens[3]', ns)
+    exec('assert _rx == R.gens[4]', ns)
+
+    exec('R = vring(["rd", "re", "rfg"], QQ)', ns)
+    exec('assert rd == R.gens[0]', ns)
+    exec('assert re == R.gens[1]', ns)
+    exec('assert rfg == R.gens[2]', ns)
+
+def test_vfield():
+    ns = {'vfield':vfield, 'QQ':QQ}
+    exec('F = vfield("f", QQ)', ns)
+    exec('assert f == F.gens[0]', ns)
+
+    exec('F = vfield("fb fbb fcc fzz _fx", QQ)', ns)
+    exec('assert fb == F.gens[0]', ns)
+    exec('assert fbb == F.gens[1]', ns)
+    exec('assert fcc == F.gens[2]', ns)
+    exec('assert fzz == F.gens[3]', ns)
+    exec('assert _fx == F.gens[4]', ns)
+
+    exec('F = vfield(["fd", "fe", "ffg"], QQ)', ns)
+    exec('assert fd == F.gens[0]', ns)
+    exec('assert fe == F.gens[1]', ns)
+    exec('assert ffg == F.gens[2]', ns)

rl/Lib/site-packages/sympy/polys/tests/test_modulargcd.py

@@ -0,0 +1,325 @@
+from sympy.polys.rings import ring
+from sympy.polys.domains import ZZ, QQ, AlgebraicField
+from sympy.polys.modulargcd import (
+    modgcd_univariate,
+    modgcd_bivariate,
+    _chinese_remainder_reconstruction_multivariate,
+    modgcd_multivariate,
+    _to_ZZ_poly,
+    _to_ANP_poly,
+    func_field_modgcd,
+    _func_field_modgcd_m)
+from sympy.functions.elementary.miscellaneous import sqrt
+
+
+def test_modgcd_univariate_integers():
+    R, x = ring("x", ZZ)
+
+    f, g = R.zero, R.zero
+    assert modgcd_univariate(f, g) == (0, 0, 0)
+
+    f, g = R.zero, x
+    assert modgcd_univariate(f, g) == (x, 0, 1)
+    assert modgcd_univariate(g, f) == (x, 1, 0)
+
+    f, g = R.zero, -x
+    assert modgcd_univariate(f, g) == (x, 0, -1)
+    assert modgcd_univariate(g, f) == (x, -1, 0)
+
+    f, g = 2*x, R(2)
+    assert modgcd_univariate(f, g) == (2, x, 1)
+
+    f, g = 2*x + 2, 6*x**2 - 6
+    assert modgcd_univariate(f, g) == (2*x + 2, 1, 3*x - 3)
+
+    f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
+    g = x**3 + 6*x**2 + 11*x + 6
+
+    h = x**2 + 3*x + 2
+
+    cff = x**2 + 5*x + 4
+    cfg = x + 3
+
+    assert modgcd_univariate(f, g) == (h, cff, cfg)
+
+    f = x**4 - 4
+    g = x**4 + 4*x**2 + 4
+
+    h = x**2 + 2
+
+    cff = x**2 - 2
+    cfg = x**2 + 2
+
+    assert modgcd_univariate(f, g) == (h, cff, cfg)
+
+    f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+
+    h = 1
+
+    cff = f
+    cfg = g
+
+    assert modgcd_univariate(f, g) == (h, cff, cfg)
+
+    f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+        + 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+        + 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+        + 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
+        - 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+        + 289127344604779611146960547954288113529690984687482920704*x**14 \
+        + 19007977035740498977629742919480623972236450681*x**7 \
+        + 311973482284542371301330321821976049
+
+    g =   365431878023781158602430064717380211405897160759702125019136*x**21 \
+        + 197599133478719444145775798221171663643171734081650688*x**14 \
+        - 9504116979659010018253915765478924103928886144*x**7 \
+        - 311973482284542371301330321821976049
+
+    assert modgcd_univariate(f, f.diff(x))[0] == g
+
+    f = 1317378933230047068160*x + 2945748836994210856960
+    g = 120352542776360960*x + 269116466014453760
+
+    h = 120352542776360960*x + 269116466014453760
+    cff = 10946
+    cfg = 1
+
+    assert modgcd_univariate(f, g) == (h, cff, cfg)
+
+
+def test_modgcd_bivariate_integers():
+    R, x, y = ring("x,y", ZZ)
+
+    f, g = R.zero, R.zero
+    assert modgcd_bivariate(f, g) == (0, 0, 0)
+
+    f, g = 2*x, R(2)
+    assert modgcd_bivariate(f, g) == (2, x, 1)
+
+    f, g = x + 2*y, x + y
+    assert modgcd_bivariate(f, g) == (1, f, g)
+
+    f, g = x**2 + 2*x*y + y**2, x**3 + y**3
+    assert modgcd_bivariate(f, g) == (x + y, x + y, x**2 - x*y + y**2)
+
+    f, g = x*y**2 + 2*x*y + x, x*y**3 + x
+    assert modgcd_bivariate(f, g) == (x*y + x, y + 1, y**2 - y + 1)
+
+    f, g = x**2*y**2 + x**2*y + 1, x*y**2 + x*y + 1
+    assert modgcd_bivariate(f, g) == (1, f, g)
+
+    f = 2*x*y**2 + 4*x*y + 2*x + y**2 + 2*y + 1
+    g = 2*x*y**3 + 2*x + y**3 + 1
+    assert modgcd_bivariate(f, g) == (2*x*y + 2*x + y + 1, y + 1, y**2 - y + 1)
+
+    f, g = 2*x**2 + 4*x + 2, x + 1
+    assert modgcd_bivariate(f, g) == (x + 1, 2*x + 2, 1)
+
+    f, g = x + 1, 2*x**2 + 4*x + 2
+    assert modgcd_bivariate(f, g) == (x + 1, 1, 2*x + 2)
+
+    f = 2*x**2 + 4*x*y - 2*x - 4*y
+    g = x**2 + x - 2
+    assert modgcd_bivariate(f, g) == (x - 1, 2*x + 4*y, x + 2)
+
+    f = 2*x**2 + 2*x*y - 3*x - 3*y
+    g = 4*x*y - 2*x + 4*y**2 - 2*y
+    assert modgcd_bivariate(f, g) == (x + y, 2*x - 3, 4*y - 2)
+
+
+def test_chinese_remainder():
+    R, x, y = ring("x, y", ZZ)
+    p, q = 3, 5
+
+    hp = x**3*y - x**2 - 1
+    hq = -x**3*y - 2*x*y**2 + 2
+
+    hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q)
+
+    assert hpq.trunc_ground(p) == hp
+    assert hpq.trunc_ground(q) == hq
+
+    T, z = ring("z", R)
+    p, q = 3, 7
+
+    hp = (x*y + 1)*z**2 + x
+    hq = (x**2 - 3*y)*z + 2
+
+    hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q)
+
+    assert hpq.trunc_ground(p) == hp
+    assert hpq.trunc_ground(q) == hq
+
+
+def test_modgcd_multivariate_integers():
+    R, x, y = ring("x,y", ZZ)
+
+    f, g = R.zero, R.zero
+    assert modgcd_multivariate(f, g) == (0, 0, 0)
+
+    f, g = 2*x**2 + 4*x + 2, x + 1
+    assert modgcd_multivariate(f, g) == (x + 1, 2*x + 2, 1)
+
+    f, g = x + 1, 2*x**2 + 4*x + 2
+    assert modgcd_multivariate(f, g) == (x + 1, 1, 2*x + 2)
+
+    f = 2*x**2 + 2*x*y - 3*x - 3*y
+    g = 4*x*y - 2*x + 4*y**2 - 2*y
+    assert modgcd_multivariate(f, g) == (x + y, 2*x - 3, 4*y - 2)
+
+    f, g = x*y**2 + 2*x*y + x, x*y**3 + x
+    assert modgcd_multivariate(f, g) == (x*y + x, y + 1, y**2 - y + 1)
+
+    f, g = x**2*y**2 + x**2*y + 1, x*y**2 + x*y + 1
+    assert modgcd_multivariate(f, g) == (1, f, g)
+
+    f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
+    g = x**3 + 6*x**2 + 11*x + 6
+
+    h = x**2 + 3*x + 2
+
+    cff = x**2 + 5*x + 4
+    cfg = x + 3
+
+    assert modgcd_multivariate(f, g) == (h, cff, cfg)
+
+    R, x, y, z, u = ring("x,y,z,u", ZZ)
+
+    f, g = x + y + z, -x - y - z - u
+    assert modgcd_multivariate(f, g) == (1, f, g)
+
+    f, g = u**2 + 2*u + 1, 2*u + 2
+    assert modgcd_multivariate(f, g) == (u + 1, u + 1, 2)
+
+    f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1
+    h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1
+
+    assert modgcd_multivariate(f, g) == (h, cff, cfg)
+    assert modgcd_multivariate(g, f) == (h, cfg, cff)
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g = x - y*z, x - y*z
+    assert modgcd_multivariate(f, g) == (x - y*z, 1, 1)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = modgcd_multivariate(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = modgcd_multivariate(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = modgcd_multivariate(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ)
+
+    f, g, h = R.fateman_poly_F_1()
+    H, cff, cfg = modgcd_multivariate(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z = ring("x,y,z", ZZ)
+
+    f, g, h = R.fateman_poly_F_2()
+    H, cff, cfg = modgcd_multivariate(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    f, g, h = R.fateman_poly_F_3()
+    H, cff, cfg = modgcd_multivariate(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+    R, x, y, z, t = ring("x,y,z,t", ZZ)
+
+    f, g, h = R.fateman_poly_F_3()
+    H, cff, cfg = modgcd_multivariate(f, g)
+
+    assert H == h and H*cff == f and H*cfg == g
+
+
+def test_to_ZZ_ANP_poly():
+    A = AlgebraicField(QQ, sqrt(2))
+    R, x = ring("x", A)
+    f = x*(sqrt(2) + 1)
+
+    T, x_, z_ = ring("x_, z_", ZZ)
+    f_ = x_*z_ + x_
+
+    assert _to_ZZ_poly(f, T) == f_
+    assert _to_ANP_poly(f_, R) == f
+
+    R, x, t, s = ring("x, t, s", A)
+    f = x*t**2 + x*s + sqrt(2)
+
+    D, t_, s_ = ring("t_, s_", ZZ)
+    T, x_, z_ = ring("x_, z_", D)
+    f_ = (t_**2 + s_)*x_ + z_
+
+    assert _to_ZZ_poly(f, T) == f_
+    assert _to_ANP_poly(f_, R) == f
+
+
+def test_modgcd_algebraic_field():
+    A = AlgebraicField(QQ, sqrt(2))
+    R, x = ring("x", A)
+    one = A.one
+
+    f, g = 2*x, R(2)
+    assert func_field_modgcd(f, g) == (one, f, g)
+
+    f, g = 2*x, R(sqrt(2))
+    assert func_field_modgcd(f, g) == (one, f, g)
+
+    f, g = 2*x + 2, 6*x**2 - 6
+    assert func_field_modgcd(f, g) == (x + 1, R(2), 6*x - 6)
+
+    R, x, y = ring("x, y", A)
+
+    f, g = x + sqrt(2)*y, x + y
+    assert func_field_modgcd(f, g) == (one, f, g)
+
+    f, g = x*y + sqrt(2)*y**2, R(sqrt(2))*y
+    assert func_field_modgcd(f, g) == (y, x + sqrt(2)*y, R(sqrt(2)))
+
+    f, g = x**2 + 2*sqrt(2)*x*y + 2*y**2, x + sqrt(2)*y
+    assert func_field_modgcd(f, g) == (g, g, one)
+
+    A = AlgebraicField(QQ, sqrt(2), sqrt(3))
+    R, x, y, z = ring("x, y, z", A)
+
+    h = x**2*y**7 + sqrt(6)/21*z
+    f, g = h*(27*y**3 + 1), h*(y + x)
+    assert func_field_modgcd(f, g) == (h, 27*y**3+1, y+x)
+
+    h = x**13*y**3 + 1/2*x**10 + 1/sqrt(2)
+    f, g = h*(x + 1), h*sqrt(2)/sqrt(3)
+    assert func_field_modgcd(f, g) == (h, x + 1, R(sqrt(2)/sqrt(3)))
+
+    A = AlgebraicField(QQ, sqrt(2)**(-1)*sqrt(3))
+    R, x = ring("x", A)
+
+    f, g = x + 1, x - 1
+    assert func_field_modgcd(f, g) == (A.one, f, g)
+
+
+# when func_field_modgcd suppors function fields, this test can be changed
+def test_modgcd_func_field():
+    D, t = ring("t", ZZ)
+    R, x, z = ring("x, z", D)
+
+    minpoly = (z**2*t**2 + z**2*t - 1).drop(0)
+    f, g = x + 1, x - 1
+
+    assert _func_field_modgcd_m(f, g, minpoly) == R.one

rl/Lib/site-packages/sympy/polys/tests/test_monomials.py

@@ -0,0 +1,269 @@
+"""Tests for tools and arithmetics for monomials of distributed polynomials. """
+
+from sympy.polys.monomials import (
+    itermonomials, monomial_count,
+    monomial_mul, monomial_div,
+    monomial_gcd, monomial_lcm,
+    monomial_max, monomial_min,
+    monomial_divides, monomial_pow,
+    Monomial,
+)
+
+from sympy.polys.polyerrors import ExactQuotientFailed
+
+from sympy.abc import a, b, c, x, y, z
+from sympy.core import S, symbols
+from sympy.testing.pytest import raises
+
+def test_monomials():
+
+    # total_degree tests
+    assert set(itermonomials([], 0)) == {S.One}
+    assert set(itermonomials([], 1)) == {S.One}
+    assert set(itermonomials([], 2)) == {S.One}
+
+    assert set(itermonomials([], 0, 0)) == {S.One}
+    assert set(itermonomials([], 1, 0)) == {S.One}
+    assert set(itermonomials([], 2, 0)) == {S.One}
+
+    raises(StopIteration, lambda: next(itermonomials([], 0, 1)))
+    raises(StopIteration, lambda: next(itermonomials([], 0, 2)))
+    raises(StopIteration, lambda: next(itermonomials([], 0, 3)))
+
+    assert set(itermonomials([], 0, 1)) == set()
+    assert set(itermonomials([], 0, 2)) == set()
+    assert set(itermonomials([], 0, 3)) == set()
+
+    raises(ValueError, lambda: set(itermonomials([], -1)))
+    raises(ValueError, lambda: set(itermonomials([x], -1)))
+    raises(ValueError, lambda: set(itermonomials([x, y], -1)))
+
+    assert set(itermonomials([x], 0)) == {S.One}
+    assert set(itermonomials([x], 1)) == {S.One, x}
+    assert set(itermonomials([x], 2)) == {S.One, x, x**2}
+    assert set(itermonomials([x], 3)) == {S.One, x, x**2, x**3}
+
+    assert set(itermonomials([x, y], 0)) == {S.One}
+    assert set(itermonomials([x, y], 1)) == {S.One, x, y}
+    assert set(itermonomials([x, y], 2)) == {S.One, x, y, x**2, y**2, x*y}
+    assert set(itermonomials([x, y], 3)) == \
+            {S.One, x, y, x**2, x**3, y**2, y**3, x*y, x*y**2, y*x**2}
+
+    i, j, k = symbols('i j k', commutative=False)
+    assert set(itermonomials([i, j, k], 0)) == {S.One}
+    assert set(itermonomials([i, j, k], 1)) == {S.One, i, j, k}
+    assert set(itermonomials([i, j, k], 2)) == \
+           {S.One, i, j, k, i**2, j**2, k**2, i*j, i*k, j*i, j*k, k*i, k*j}
+
+    assert set(itermonomials([i, j, k], 3)) == \
+            {S.One, i, j, k, i**2, j**2, k**2, i*j, i*k, j*i, j*k, k*i, k*j,
+                    i**3, j**3, k**3,
+                    i**2 * j, i**2 * k, j * i**2, k * i**2,
+                    j**2 * i, j**2 * k, i * j**2, k * j**2,
+                    k**2 * i, k**2 * j, i * k**2, j * k**2,
+                    i*j*i, i*k*i, j*i*j, j*k*j, k*i*k, k*j*k,
+                    i*j*k, i*k*j, j*i*k, j*k*i, k*i*j, k*j*i,
+            }
+
+    assert set(itermonomials([x, i, j], 0)) == {S.One}
+    assert set(itermonomials([x, i, j], 1)) == {S.One, x, i, j}
+    assert set(itermonomials([x, i, j], 2)) == {S.One, x, i, j, x*i, x*j, i*j, j*i, x**2, i**2, j**2}
+    assert set(itermonomials([x, i, j], 3)) == \
+            {S.One, x, i, j, x*i, x*j, i*j, j*i, x**2, i**2, j**2,
+                            x**3, i**3, j**3,
+                            x**2 * i, x**2 * j,
+                            x * i**2, j * i**2, i**2 * j, i*j*i,
+                            x * j**2, i * j**2, j**2 * i, j*i*j,
+                            x * i * j, x * j * i
+            }
+
+    # degree_list tests
+    assert set(itermonomials([], [])) == {S.One}
+
+    raises(ValueError, lambda: set(itermonomials([], [0])))
+    raises(ValueError, lambda: set(itermonomials([], [1])))
+    raises(ValueError, lambda: set(itermonomials([], [2])))
+
+    raises(ValueError, lambda: set(itermonomials([x], [1], [])))
+    raises(ValueError, lambda: set(itermonomials([x], [1, 2], [])))
+    raises(ValueError, lambda: set(itermonomials([x], [1, 2, 3], [])))
+
+    raises(ValueError, lambda: set(itermonomials([x], [], [1])))
+    raises(ValueError, lambda: set(itermonomials([x], [], [1, 2])))
+    raises(ValueError, lambda: set(itermonomials([x], [], [1, 2, 3])))
+
+    raises(ValueError, lambda: set(itermonomials([x, y], [1, 2], [1, 2, 3])))
+    raises(ValueError, lambda: set(itermonomials([x, y, z], [1, 2, 3], [0, 1])))
+
+    raises(ValueError, lambda: set(itermonomials([x], [1], [-1])))
+    raises(ValueError, lambda: set(itermonomials([x, y], [1, 2], [1, -1])))
+
+    raises(ValueError, lambda: set(itermonomials([], [], 1)))
+    raises(ValueError, lambda: set(itermonomials([], [], 2)))
+    raises(ValueError, lambda: set(itermonomials([], [], 3)))
+
+    raises(ValueError, lambda: set(itermonomials([x, y], [0, 1], [1, 2])))
+    raises(ValueError, lambda: set(itermonomials([x, y, z], [0, 0, 3], [0, 1, 2])))
+
+    assert set(itermonomials([x], [0])) == {S.One}
+    assert set(itermonomials([x], [1])) == {S.One, x}
+    assert set(itermonomials([x], [2])) == {S.One, x, x**2}
+    assert set(itermonomials([x], [3])) == {S.One, x, x**2, x**3}
+
+    assert set(itermonomials([x], [3], [1])) == {x, x**3, x**2}
+    assert set(itermonomials([x], [3], [2])) == {x**3, x**2}
+
+    assert set(itermonomials([x, y], 3, 3)) == {x**3, x**2*y, x*y**2, y**3}
+    assert set(itermonomials([x, y], 3, 2)) == {x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3}
+
+    assert set(itermonomials([x, y], [0, 0])) == {S.One}
+    assert set(itermonomials([x, y], [0, 1])) == {S.One, y}
+    assert set(itermonomials([x, y], [0, 2])) == {S.One, y, y**2}
+    assert set(itermonomials([x, y], [0, 2], [0, 1])) == {y, y**2}
+    assert set(itermonomials([x, y], [0, 2], [0, 2])) == {y**2}
+
+    assert set(itermonomials([x, y], [1, 0])) == {S.One, x}
+    assert set(itermonomials([x, y], [1, 1])) == {S.One, x, y, x*y}
+    assert set(itermonomials([x, y], [1, 2])) == {S.One, x, y, x*y, y**2, x*y**2}
+    assert set(itermonomials([x, y], [1, 2], [1, 1])) == {x*y, x*y**2}
+    assert set(itermonomials([x, y], [1, 2], [1, 2])) == {x*y**2}
+
+    assert set(itermonomials([x, y], [2, 0])) == {S.One, x, x**2}
+    assert set(itermonomials([x, y], [2, 1])) == {S.One, x, y, x*y, x**2, x**2*y}
+    assert set(itermonomials([x, y], [2, 2])) == \
+            {S.One, y**2, x*y**2, x, x*y, x**2, x**2*y**2, y, x**2*y}
+
+    i, j, k = symbols('i j k', commutative=False)
+    assert set(itermonomials([i, j, k], 2, 2)) == \
+            {k*i, i**2, i*j, j*k, j*i, k**2, j**2, k*j, i*k}
+    assert set(itermonomials([i, j, k], 3, 2)) == \
+            {j*k**2, i*k**2, k*i*j, k*i**2, k**2, j*k*j, k*j**2, i*k*i, i*j,
+                    j**2*k, i**2*j, j*i*k, j**3, i**3, k*j*i, j*k*i, j*i,
+                    k**2*j, j*i**2, k*j, k*j*k, i*j*i, j*i*j, i*j**2, j**2,
+                    k*i*k, i**2, j*k, i*k, i*k*j, k**3, i**2*k, j**2*i, k**2*i,
+                    i*j*k, k*i
+            }
+    assert set(itermonomials([i, j, k], [0, 0, 0])) == {S.One}
+    assert set(itermonomials([i, j, k], [0, 0, 1])) == {1, k}
+    assert set(itermonomials([i, j, k], [0, 1, 0])) == {1, j}
+    assert set(itermonomials([i, j, k], [1, 0, 0])) == {i, 1}
+    assert set(itermonomials([i, j, k], [0, 0, 2])) == {k**2, 1, k}
+    assert set(itermonomials([i, j, k], [0, 2, 0])) == {1, j, j**2}
+    assert set(itermonomials([i, j, k], [2, 0, 0])) == {i, 1, i**2}
+    assert set(itermonomials([i, j, k], [1, 1, 1])) == {1, k, j, j*k, i*k, i, i*j, i*j*k}
+    assert set(itermonomials([i, j, k], [2, 2, 2])) == \
+            {1, k, i**2*k**2, j*k, j**2, i, i*k, j*k**2, i*j**2*k**2,
+                    i**2*j, i**2*j**2, k**2, j**2*k, i*j**2*k,
+                    j**2*k**2, i*j, i**2*k, i**2*j**2*k, j, i**2*j*k,
+                    i*j**2, i*k**2, i*j*k, i**2*j**2*k**2, i*j*k**2, i**2, i**2*j*k**2
+            }
+
+    assert set(itermonomials([x, j, k], [0, 0, 0])) == {S.One}
+    assert set(itermonomials([x, j, k], [0, 0, 1])) == {1, k}
+    assert set(itermonomials([x, j, k], [0, 1, 0])) == {1, j}
+    assert set(itermonomials([x, j, k], [1, 0, 0])) == {x, 1}
+    assert set(itermonomials([x, j, k], [0, 0, 2])) == {k**2, 1, k}
+    assert set(itermonomials([x, j, k], [0, 2, 0])) == {1, j, j**2}
+    assert set(itermonomials([x, j, k], [2, 0, 0])) == {x, 1, x**2}
+    assert set(itermonomials([x, j, k], [1, 1, 1])) == {1, k, j, j*k, x*k, x, x*j, x*j*k}
+    assert set(itermonomials([x, j, k], [2, 2, 2])) == \
+            {1, k, x**2*k**2, j*k, j**2, x, x*k, j*k**2, x*j**2*k**2,
+                    x**2*j, x**2*j**2, k**2, j**2*k, x*j**2*k,
+                    j**2*k**2, x*j, x**2*k, x**2*j**2*k, j, x**2*j*k,
+                    x*j**2, x*k**2, x*j*k, x**2*j**2*k**2, x*j*k**2, x**2, x**2*j*k**2
+            }
+
+def test_monomial_count():
+    assert monomial_count(2, 2) == 6
+    assert monomial_count(2, 3) == 10
+
+def test_monomial_mul():
+    assert monomial_mul((3, 4, 1), (1, 2, 0)) == (4, 6, 1)
+
+def test_monomial_div():
+    assert monomial_div((3, 4, 1), (1, 2, 0)) == (2, 2, 1)
+
+def test_monomial_gcd():
+    assert monomial_gcd((3, 4, 1), (1, 2, 0)) == (1, 2, 0)
+
+def test_monomial_lcm():
+    assert monomial_lcm((3, 4, 1), (1, 2, 0)) == (3, 4, 1)
+
+def test_monomial_max():
+    assert monomial_max((3, 4, 5), (0, 5, 1), (6, 3, 9)) == (6, 5, 9)
+
+def test_monomial_pow():
+    assert monomial_pow((1, 2, 3), 3) == (3, 6, 9)
+
+def test_monomial_min():
+    assert monomial_min((3, 4, 5), (0, 5, 1), (6, 3, 9)) == (0, 3, 1)
+
+def test_monomial_divides():
+    assert monomial_divides((1, 2, 3), (4, 5, 6)) is True
+    assert monomial_divides((1, 2, 3), (0, 5, 6)) is False
+
+def test_Monomial():
+    m = Monomial((3, 4, 1), (x, y, z))
+    n = Monomial((1, 2, 0), (x, y, z))
+
+    assert m.as_expr() == x**3*y**4*z
+    assert n.as_expr() == x**1*y**2
+
+    assert m.as_expr(a, b, c) == a**3*b**4*c
+    assert n.as_expr(a, b, c) == a**1*b**2
+
+    assert m.exponents == (3, 4, 1)
+    assert m.gens == (x, y, z)
+
+    assert n.exponents == (1, 2, 0)
+    assert n.gens == (x, y, z)
+
+    assert m == (3, 4, 1)
+    assert n != (3, 4, 1)
+    assert m != (1, 2, 0)
+    assert n == (1, 2, 0)
+    assert (m == 1) is False
+
+    assert m[0] == m[-3] == 3
+    assert m[1] == m[-2] == 4
+    assert m[2] == m[-1] == 1
+
+    assert n[0] == n[-3] == 1
+    assert n[1] == n[-2] == 2
+    assert n[2] == n[-1] == 0
+
+    assert m[:2] == (3, 4)
+    assert n[:2] == (1, 2)
+
+    assert m*n == Monomial((4, 6, 1))
+    assert m/n == Monomial((2, 2, 1))
+
+    assert m*(1, 2, 0) == Monomial((4, 6, 1))
+    assert m/(1, 2, 0) == Monomial((2, 2, 1))
+
+    assert m.gcd(n) == Monomial((1, 2, 0))
+    assert m.lcm(n) == Monomial((3, 4, 1))
+
+    assert m.gcd((1, 2, 0)) == Monomial((1, 2, 0))
+    assert m.lcm((1, 2, 0)) == Monomial((3, 4, 1))
+
+    assert m**0 == Monomial((0, 0, 0))
+    assert m**1 == m
+    assert m**2 == Monomial((6, 8, 2))
+    assert m**3 == Monomial((9, 12, 3))
+    _a = Monomial((0, 0, 0))
+    for n in range(10):
+        assert _a == m**n
+        _a *= m
+
+    raises(ExactQuotientFailed, lambda: m/Monomial((5, 2, 0)))
+
+    mm = Monomial((1, 2, 3))
+    raises(ValueError, lambda: mm.as_expr())
+    assert str(mm) == 'Monomial((1, 2, 3))'
+    assert str(m) == 'x**3*y**4*z**1'
+    raises(NotImplementedError, lambda: m*1)
+    raises(NotImplementedError, lambda: m/1)
+    raises(ValueError, lambda: m**-1)
+    raises(TypeError, lambda: m.gcd(3))
+    raises(TypeError, lambda: m.lcm(3))

rl/Lib/site-packages/sympy/polys/tests/test_multivariate_resultants.py

@@ -0,0 +1,294 @@
+"""Tests for Dixon's and Macaulay's classes. """
+
+from sympy.matrices.dense import Matrix
+from sympy.polys.polytools import factor
+from sympy.core import symbols
+from sympy.tensor.indexed import IndexedBase
+
+from sympy.polys.multivariate_resultants import (DixonResultant,
+                                                 MacaulayResultant)
+
+c, d = symbols("a, b")
+x, y = symbols("x, y")
+
+p =  c * x + y
+q =  x + d * y
+
+dixon = DixonResultant(polynomials=[p, q], variables=[x, y])
+macaulay = MacaulayResultant(polynomials=[p, q], variables=[x, y])
+
+def test_dixon_resultant_init():
+    """Test init method of DixonResultant."""
+    a = IndexedBase("alpha")
+
+    assert dixon.polynomials == [p, q]
+    assert dixon.variables == [x, y]
+    assert dixon.n == 2
+    assert dixon.m == 2
+    assert dixon.dummy_variables == [a[0], a[1]]
+
+def test_get_dixon_polynomial_numerical():
+    """Test Dixon's polynomial for a numerical example."""
+    a = IndexedBase("alpha")
+
+    p = x + y
+    q = x ** 2 + y **3
+    h = x ** 2 + y
+
+    dixon = DixonResultant([p, q, h], [x, y])
+    polynomial = -x * y ** 2 * a[0] - x * y ** 2 * a[1] - x * y * a[0] \
+    * a[1] - x * y * a[1] ** 2 - x * a[0] * a[1] ** 2 + x * a[0] - \
+    y ** 2 * a[0] * a[1] + y ** 2 * a[1] - y * a[0] * a[1] ** 2 + y * \
+    a[1] ** 2
+
+    assert dixon.get_dixon_polynomial().as_expr().expand() == polynomial
+
+def test_get_max_degrees():
+    """Tests max degrees function."""
+
+    p = x + y
+    q = x ** 2 + y **3
+    h = x ** 2 + y
+
+    dixon = DixonResultant(polynomials=[p, q, h], variables=[x, y])
+    dixon_polynomial = dixon.get_dixon_polynomial()
+
+    assert dixon.get_max_degrees(dixon_polynomial) == [1, 2]
+
+def test_get_dixon_matrix():
+    """Test Dixon's resultant for a numerical example."""
+
+    x, y = symbols('x, y')
+
+    p = x + y
+    q = x ** 2 + y ** 3
+    h = x ** 2 + y
+
+    dixon = DixonResultant([p, q, h], [x, y])
+    polynomial = dixon.get_dixon_polynomial()
+
+    assert dixon.get_dixon_matrix(polynomial).det() == 0
+
+def test_get_dixon_matrix_example_two():
+    """Test Dixon's matrix for example from [Palancz08]_."""
+    x, y, z = symbols('x, y, z')
+
+    f = x ** 2 + y ** 2 - 1 + z * 0
+    g = x ** 2 + z ** 2 - 1 + y * 0
+    h = y ** 2 + z ** 2 - 1
+
+    example_two = DixonResultant([f, g, h], [y, z])
+    poly = example_two.get_dixon_polynomial()
+    matrix = example_two.get_dixon_matrix(poly)
+
+    expr = 1 - 8 * x ** 2 + 24 * x ** 4 - 32 * x ** 6 + 16 * x ** 8
+    assert (matrix.det() - expr).expand() == 0
+
+def test_KSY_precondition():
+    """Tests precondition for KSY Resultant."""
+    A, B, C = symbols('A, B, C')
+
+    m1 = Matrix([[1, 2, 3],
+                 [4, 5, 12],
+                 [6, 7, 18]])
+
+    m2 = Matrix([[0, C**2],
+                 [-2 * C, -C ** 2]])
+
+    m3 = Matrix([[1, 0],
+                 [0, 1]])
+
+    m4 = Matrix([[A**2, 0, 1],
+                 [A, 1, 1 / A]])
+
+    m5 = Matrix([[5, 1],
+                 [2, B],
+                 [0, 1],
+                 [0, 0]])
+
+    assert dixon.KSY_precondition(m1) == False
+    assert dixon.KSY_precondition(m2) == True
+    assert dixon.KSY_precondition(m3) == True
+    assert dixon.KSY_precondition(m4) == False
+    assert dixon.KSY_precondition(m5) == True
+
+def test_delete_zero_rows_and_columns():
+    """Tests method for deleting rows and columns containing only zeros."""
+    A, B, C = symbols('A, B, C')
+
+    m1 = Matrix([[0, 0],
+                 [0, 0],
+                 [1, 2]])
+
+    m2 = Matrix([[0, 1, 2],
+                 [0, 3, 4],
+                 [0, 5, 6]])
+
+    m3 = Matrix([[0, 0, 0, 0],
+                 [0, 1, 2, 0],
+                 [0, 3, 4, 0],
+                 [0, 0, 0, 0]])
+
+    m4 = Matrix([[1, 0, 2],
+                 [0, 0, 0],
+                 [3, 0, 4]])
+
+    m5 = Matrix([[0, 0, 0, 1],
+                 [0, 0, 0, 2],
+                 [0, 0, 0, 3],
+                 [0, 0, 0, 4]])
+
+    m6 = Matrix([[0, 0, A],
+                 [B, 0, 0],
+                 [0, 0, C]])
+
+    assert dixon.delete_zero_rows_and_columns(m1) == Matrix([[1, 2]])
+
+    assert dixon.delete_zero_rows_and_columns(m2) == Matrix([[1, 2],
+                                                             [3, 4],
+                                                             [5, 6]])
+
+    assert dixon.delete_zero_rows_and_columns(m3) == Matrix([[1, 2],
+                                                             [3, 4]])
+
+    assert dixon.delete_zero_rows_and_columns(m4) == Matrix([[1, 2],
+                                                             [3, 4]])
+
+    assert dixon.delete_zero_rows_and_columns(m5) == Matrix([[1],
+                                                             [2],
+                                                             [3],
+                                                             [4]])
+
+    assert dixon.delete_zero_rows_and_columns(m6) == Matrix([[0, A],
+                                                             [B, 0],
+                                                             [0, C]])
+
+def test_product_leading_entries():
+    """Tests product of leading entries method."""
+    A, B = symbols('A, B')
+
+    m1 = Matrix([[1, 2, 3],
+                 [0, 4, 5],
+                 [0, 0, 6]])
+
+    m2 = Matrix([[0, 0, 1],
+                 [2, 0, 3]])
+
+    m3 = Matrix([[0, 0, 0],
+                 [1, 2, 3],
+                 [0, 0, 0]])
+
+    m4 = Matrix([[0, 0, A],
+                 [1, 2, 3],
+                 [B, 0, 0]])
+
+    assert dixon.product_leading_entries(m1) == 24
+    assert dixon.product_leading_entries(m2) == 2
+    assert dixon.product_leading_entries(m3) == 1
+    assert dixon.product_leading_entries(m4) == A * B
+
+def test_get_KSY_Dixon_resultant_example_one():
+    """Tests the KSY Dixon resultant for example one"""
+    x, y, z = symbols('x, y, z')
+
+    p = x * y * z
+    q = x**2 - z**2
+    h = x + y + z
+    dixon = DixonResultant([p, q, h], [x, y])
+    dixon_poly = dixon.get_dixon_polynomial()
+    dixon_matrix = dixon.get_dixon_matrix(dixon_poly)
+    D = dixon.get_KSY_Dixon_resultant(dixon_matrix)
+
+    assert D == -z**3
+
+def test_get_KSY_Dixon_resultant_example_two():
+    """Tests the KSY Dixon resultant for example two"""
+    x, y, A = symbols('x, y, A')
+
+    p = x * y + x * A + x - A**2 - A + y**2 + y
+    q = x**2 + x * A - x + x * y + y * A - y
+    h = x**2 + x * y + 2 * x - x * A - y * A - 2 * A
+
+    dixon = DixonResultant([p, q, h], [x, y])
+    dixon_poly = dixon.get_dixon_polynomial()
+    dixon_matrix = dixon.get_dixon_matrix(dixon_poly)
+    D = factor(dixon.get_KSY_Dixon_resultant(dixon_matrix))
+
+    assert D == -8*A*(A - 1)*(A + 2)*(2*A - 1)**2
+
+def test_macaulay_resultant_init():
+    """Test init method of MacaulayResultant."""
+
+    assert macaulay.polynomials == [p, q]
+    assert macaulay.variables == [x, y]
+    assert macaulay.n == 2
+    assert macaulay.degrees == [1, 1]
+    assert macaulay.degree_m == 1
+    assert macaulay.monomials_size == 2
+
+def test_get_degree_m():
+    assert macaulay._get_degree_m() == 1
+
+def test_get_size():
+    assert macaulay.get_size() == 2
+
+def test_macaulay_example_one():
+    """Tests the Macaulay for example from [Bruce97]_"""
+
+    x, y, z = symbols('x, y, z')
+    a_1_1, a_1_2, a_1_3 = symbols('a_1_1, a_1_2, a_1_3')
+    a_2_2, a_2_3, a_3_3 = symbols('a_2_2, a_2_3, a_3_3')
+    b_1_1, b_1_2, b_1_3 = symbols('b_1_1, b_1_2, b_1_3')
+    b_2_2, b_2_3, b_3_3 = symbols('b_2_2, b_2_3, b_3_3')
+    c_1, c_2, c_3 = symbols('c_1, c_2, c_3')
+
+    f_1 = a_1_1 * x ** 2 + a_1_2 * x * y + a_1_3 * x * z + \
+          a_2_2 * y ** 2 + a_2_3 * y * z + a_3_3 * z ** 2
+    f_2 = b_1_1 * x ** 2 + b_1_2 * x * y + b_1_3 * x * z + \
+          b_2_2 * y ** 2 + b_2_3 * y * z + b_3_3 * z ** 2
+    f_3 = c_1 * x + c_2 * y + c_3 * z
+
+    mac = MacaulayResultant([f_1, f_2, f_3], [x, y, z])
+
+    assert mac.degrees == [2, 2, 1]
+    assert mac.degree_m == 3
+
+    assert mac.monomial_set == [x ** 3, x ** 2 * y, x ** 2 * z,
+                                x * y ** 2,
+                                x * y * z, x * z ** 2, y ** 3,
+                                y ** 2 *z, y * z ** 2, z ** 3]
+    assert mac.monomials_size == 10
+    assert mac.get_row_coefficients() == [[x, y, z], [x, y, z],
+                                          [x * y, x * z, y * z, z ** 2]]
+
+    matrix = mac.get_matrix()
+    assert matrix.shape == (mac.monomials_size, mac.monomials_size)
+    assert mac.get_submatrix(matrix) == Matrix([[a_1_1, a_2_2],
+                                                [b_1_1, b_2_2]])
+
+def test_macaulay_example_two():
+    """Tests the Macaulay formulation for example from [Stiller96]_."""
+
+    x, y, z = symbols('x, y, z')
+    a_0, a_1, a_2 = symbols('a_0, a_1, a_2')
+    b_0, b_1, b_2 = symbols('b_0, b_1, b_2')
+    c_0, c_1, c_2, c_3, c_4 = symbols('c_0, c_1, c_2, c_3, c_4')
+
+    f = a_0 * y -  a_1 * x + a_2 * z
+    g = b_1 * x ** 2 + b_0 * y ** 2 - b_2 * z ** 2
+    h = c_0 * y - c_1 * x ** 3 + c_2 * x ** 2 * z - c_3 * x * z ** 2 + \
+        c_4 * z ** 3
+
+    mac = MacaulayResultant([f, g, h], [x, y, z])
+
+    assert mac.degrees == [1, 2, 3]
+    assert mac.degree_m == 4
+    assert mac.monomials_size == 15
+    assert len(mac.get_row_coefficients()) == mac.n
+
+    matrix = mac.get_matrix()
+    assert matrix.shape == (mac.monomials_size, mac.monomials_size)
+    assert mac.get_submatrix(matrix) == Matrix([[-a_1, a_0, a_2, 0],
+                                                [0, -a_1, 0, 0],
+                                                [0, 0, -a_1, 0],
+                                                [0, 0, 0, -a_1]])

rl/Lib/site-packages/sympy/polys/tests/test_orderings.py

@@ -0,0 +1,124 @@
+"""Tests of monomial orderings. """
+
+from sympy.polys.orderings import (
+    monomial_key, lex, grlex, grevlex, ilex, igrlex,
+    LexOrder, InverseOrder, ProductOrder, build_product_order,
+)
+
+from sympy.abc import x, y, z, t
+from sympy.core import S
+from sympy.testing.pytest import raises
+
+def test_lex_order():
+    assert lex((1, 2, 3)) == (1, 2, 3)
+    assert str(lex) == 'lex'
+
+    assert lex((1, 2, 3)) == lex((1, 2, 3))
+
+    assert lex((2, 2, 3)) > lex((1, 2, 3))
+    assert lex((1, 3, 3)) > lex((1, 2, 3))
+    assert lex((1, 2, 4)) > lex((1, 2, 3))
+
+    assert lex((0, 2, 3)) < lex((1, 2, 3))
+    assert lex((1, 1, 3)) < lex((1, 2, 3))
+    assert lex((1, 2, 2)) < lex((1, 2, 3))
+
+    assert lex.is_global is True
+    assert lex == LexOrder()
+    assert lex != grlex
+
+def test_grlex_order():
+    assert grlex((1, 2, 3)) == (6, (1, 2, 3))
+    assert str(grlex) == 'grlex'
+
+    assert grlex((1, 2, 3)) == grlex((1, 2, 3))
+
+    assert grlex((2, 2, 3)) > grlex((1, 2, 3))
+    assert grlex((1, 3, 3)) > grlex((1, 2, 3))
+    assert grlex((1, 2, 4)) > grlex((1, 2, 3))
+
+    assert grlex((0, 2, 3)) < grlex((1, 2, 3))
+    assert grlex((1, 1, 3)) < grlex((1, 2, 3))
+    assert grlex((1, 2, 2)) < grlex((1, 2, 3))
+
+    assert grlex((2, 2, 3)) > grlex((1, 2, 4))
+    assert grlex((1, 3, 3)) > grlex((1, 2, 4))
+
+    assert grlex((0, 2, 3)) < grlex((1, 2, 2))
+    assert grlex((1, 1, 3)) < grlex((1, 2, 2))
+
+    assert grlex((0, 1, 1)) > grlex((0, 0, 2))
+    assert grlex((0, 3, 1)) < grlex((2, 2, 1))
+
+    assert grlex.is_global is True
+
+def test_grevlex_order():
+    assert grevlex((1, 2, 3)) == (6, (-3, -2, -1))
+    assert str(grevlex) == 'grevlex'
+
+    assert grevlex((1, 2, 3)) == grevlex((1, 2, 3))
+
+    assert grevlex((2, 2, 3)) > grevlex((1, 2, 3))
+    assert grevlex((1, 3, 3)) > grevlex((1, 2, 3))
+    assert grevlex((1, 2, 4)) > grevlex((1, 2, 3))
+
+    assert grevlex((0, 2, 3)) < grevlex((1, 2, 3))
+    assert grevlex((1, 1, 3)) < grevlex((1, 2, 3))
+    assert grevlex((1, 2, 2)) < grevlex((1, 2, 3))
+
+    assert grevlex((2, 2, 3)) > grevlex((1, 2, 4))
+    assert grevlex((1, 3, 3)) > grevlex((1, 2, 4))
+
+    assert grevlex((0, 2, 3)) < grevlex((1, 2, 2))
+    assert grevlex((1, 1, 3)) < grevlex((1, 2, 2))
+
+    assert grevlex((0, 1, 1)) > grevlex((0, 0, 2))
+    assert grevlex((0, 3, 1)) < grevlex((2, 2, 1))
+
+    assert grevlex.is_global is True
+
+def test_InverseOrder():
+    ilex = InverseOrder(lex)
+    igrlex = InverseOrder(grlex)
+
+    assert ilex((1, 2, 3)) > ilex((2, 0, 3))
+    assert igrlex((1, 2, 3)) < igrlex((0, 2, 3))
+    assert str(ilex) == "ilex"
+    assert str(igrlex) == "igrlex"
+    assert ilex.is_global is False
+    assert igrlex.is_global is False
+    assert ilex != igrlex
+    assert ilex == InverseOrder(LexOrder())
+
+def test_ProductOrder():
+    P = ProductOrder((grlex, lambda m: m[:2]), (grlex, lambda m: m[2:]))
+    assert P((1, 3, 3, 4, 5)) > P((2, 1, 5, 5, 5))
+    assert str(P) == "ProductOrder(grlex, grlex)"
+    assert P.is_global is True
+    assert ProductOrder((grlex, None), (ilex, None)).is_global is None
+    assert ProductOrder((igrlex, None), (ilex, None)).is_global is False
+
+def test_monomial_key():
+    assert monomial_key() == lex
+
+    assert monomial_key('lex') == lex
+    assert monomial_key('grlex') == grlex
+    assert monomial_key('grevlex') == grevlex
+
+    raises(ValueError, lambda: monomial_key('foo'))
+    raises(ValueError, lambda: monomial_key(1))
+
+    M = [x, x**2*z**2, x*y, x**2, S.One, y**2, x**3, y, z, x*y**2*z, x**2*y**2]
+    assert sorted(M, key=monomial_key('lex', [z, y, x])) == \
+        [S.One, x, x**2, x**3, y, x*y, y**2, x**2*y**2, z, x*y**2*z, x**2*z**2]
+    assert sorted(M, key=monomial_key('grlex', [z, y, x])) == \
+        [S.One, x, y, z, x**2, x*y, y**2, x**3, x**2*y**2, x*y**2*z, x**2*z**2]
+    assert sorted(M, key=monomial_key('grevlex', [z, y, x])) == \
+        [S.One, x, y, z, x**2, x*y, y**2, x**3, x**2*y**2, x**2*z**2, x*y**2*z]
+
+def test_build_product_order():
+    assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t])((4, 5, 6, 7)) == \
+        ((9, (4, 5)), (13, (6, 7)))
+
+    assert build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t]) == \
+        build_product_order((("grlex", x, y), ("grlex", z, t)), [x, y, z, t])

rl/Lib/site-packages/sympy/polys/tests/test_orthopolys.py

@@ -0,0 +1,175 @@
+"""Tests for efficient functions for generating orthogonal polynomials. """
+
+from sympy.core.numbers import Rational as Q
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.polys.polytools import Poly
+from sympy.testing.pytest import raises
+
+from sympy.polys.orthopolys import (
+    jacobi_poly,
+    gegenbauer_poly,
+    chebyshevt_poly,
+    chebyshevu_poly,
+    hermite_poly,
+    hermite_prob_poly,
+    legendre_poly,
+    laguerre_poly,
+    spherical_bessel_fn,
+)
+
+from sympy.abc import x, a, b
+
+
+def test_jacobi_poly():
+    raises(ValueError, lambda: jacobi_poly(-1, a, b, x))
+
+    assert jacobi_poly(1, a, b, x, polys=True) == Poly(
+        (a/2 + b/2 + 1)*x + a/2 - b/2, x, domain='ZZ(a,b)')
+
+    assert jacobi_poly(0, a, b, x) == 1
+    assert jacobi_poly(1, a, b, x) == a/2 - b/2 + x*(a/2 + b/2 + 1)
+    assert jacobi_poly(2, a, b, x) == (a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 +
+                                       x**2*(a**2/8 + a*b/4 + a*Q(7, 8) + b**2/8 +
+                                             b*Q(7, 8) + Q(3, 2)) + x*(a**2/4 +
+                                            a*Q(3, 4) - b**2/4 - b*Q(3, 4)) - S.Half)
+
+    assert jacobi_poly(1, a, b, polys=True) == Poly(
+        (a/2 + b/2 + 1)*x + a/2 - b/2, x, domain='ZZ(a,b)')
+
+
+def test_gegenbauer_poly():
+    raises(ValueError, lambda: gegenbauer_poly(-1, a, x))
+
+    assert gegenbauer_poly(
+        1, a, x, polys=True) == Poly(2*a*x, x, domain='ZZ(a)')
+
+    assert gegenbauer_poly(0, a, x) == 1
+    assert gegenbauer_poly(1, a, x) == 2*a*x
+    assert gegenbauer_poly(2, a, x) == -a + x**2*(2*a**2 + 2*a)
+    assert gegenbauer_poly(
+        3, a, x) == x**3*(4*a**3/3 + 4*a**2 + a*Q(8, 3)) + x*(-2*a**2 - 2*a)
+
+    assert gegenbauer_poly(1, S.Half).dummy_eq(x)
+    assert gegenbauer_poly(1, a, polys=True) == Poly(2*a*x, x, domain='ZZ(a)')
+
+
+def test_chebyshevt_poly():
+    raises(ValueError, lambda: chebyshevt_poly(-1, x))
+
+    assert chebyshevt_poly(1, x, polys=True) == Poly(x)
+
+    assert chebyshevt_poly(0, x) == 1
+    assert chebyshevt_poly(1, x) == x
+    assert chebyshevt_poly(2, x) == 2*x**2 - 1
+    assert chebyshevt_poly(3, x) == 4*x**3 - 3*x
+    assert chebyshevt_poly(4, x) == 8*x**4 - 8*x**2 + 1
+    assert chebyshevt_poly(5, x) == 16*x**5 - 20*x**3 + 5*x
+    assert chebyshevt_poly(6, x) == 32*x**6 - 48*x**4 + 18*x**2 - 1
+    assert chebyshevt_poly(75, x) == (2*chebyshevt_poly(37, x)*chebyshevt_poly(38, x) - x).expand()
+    assert chebyshevt_poly(100, x) == (2*chebyshevt_poly(50, x)**2 - 1).expand()
+
+    assert chebyshevt_poly(1).dummy_eq(x)
+    assert chebyshevt_poly(1, polys=True) == Poly(x)
+
+
+def test_chebyshevu_poly():
+    raises(ValueError, lambda: chebyshevu_poly(-1, x))
+
+    assert chebyshevu_poly(1, x, polys=True) == Poly(2*x)
+
+    assert chebyshevu_poly(0, x) == 1
+    assert chebyshevu_poly(1, x) == 2*x
+    assert chebyshevu_poly(2, x) == 4*x**2 - 1
+    assert chebyshevu_poly(3, x) == 8*x**3 - 4*x
+    assert chebyshevu_poly(4, x) == 16*x**4 - 12*x**2 + 1
+    assert chebyshevu_poly(5, x) == 32*x**5 - 32*x**3 + 6*x
+    assert chebyshevu_poly(6, x) == 64*x**6 - 80*x**4 + 24*x**2 - 1
+
+    assert chebyshevu_poly(1).dummy_eq(2*x)
+    assert chebyshevu_poly(1, polys=True) == Poly(2*x)
+
+
+def test_hermite_poly():
+    raises(ValueError, lambda: hermite_poly(-1, x))
+
+    assert hermite_poly(1, x, polys=True) == Poly(2*x)
+
+    assert hermite_poly(0, x) == 1
+    assert hermite_poly(1, x) == 2*x
+    assert hermite_poly(2, x) == 4*x**2 - 2
+    assert hermite_poly(3, x) == 8*x**3 - 12*x
+    assert hermite_poly(4, x) == 16*x**4 - 48*x**2 + 12
+    assert hermite_poly(5, x) == 32*x**5 - 160*x**3 + 120*x
+    assert hermite_poly(6, x) == 64*x**6 - 480*x**4 + 720*x**2 - 120
+
+    assert hermite_poly(1).dummy_eq(2*x)
+    assert hermite_poly(1, polys=True) == Poly(2*x)
+
+
+def test_hermite_prob_poly():
+    raises(ValueError, lambda: hermite_prob_poly(-1, x))
+
+    assert hermite_prob_poly(1, x, polys=True) == Poly(x)
+
+    assert hermite_prob_poly(0, x) == 1
+    assert hermite_prob_poly(1, x) == x
+    assert hermite_prob_poly(2, x) == x**2 - 1
+    assert hermite_prob_poly(3, x) == x**3 - 3*x
+    assert hermite_prob_poly(4, x) == x**4 - 6*x**2 + 3
+    assert hermite_prob_poly(5, x) == x**5 - 10*x**3 + 15*x
+    assert hermite_prob_poly(6, x) == x**6 - 15*x**4 + 45*x**2 - 15
+
+    assert hermite_prob_poly(1).dummy_eq(x)
+    assert hermite_prob_poly(1, polys=True) == Poly(x)
+
+
+def test_legendre_poly():
+    raises(ValueError, lambda: legendre_poly(-1, x))
+
+    assert legendre_poly(1, x, polys=True) == Poly(x, domain='QQ')
+
+    assert legendre_poly(0, x) == 1
+    assert legendre_poly(1, x) == x
+    assert legendre_poly(2, x) == Q(3, 2)*x**2 - Q(1, 2)
+    assert legendre_poly(3, x) == Q(5, 2)*x**3 - Q(3, 2)*x
+    assert legendre_poly(4, x) == Q(35, 8)*x**4 - Q(30, 8)*x**2 + Q(3, 8)
+    assert legendre_poly(5, x) == Q(63, 8)*x**5 - Q(70, 8)*x**3 + Q(15, 8)*x
+    assert legendre_poly(6, x) == Q(
+        231, 16)*x**6 - Q(315, 16)*x**4 + Q(105, 16)*x**2 - Q(5, 16)
+
+    assert legendre_poly(1).dummy_eq(x)
+    assert legendre_poly(1, polys=True) == Poly(x)
+
+
+def test_laguerre_poly():
+    raises(ValueError, lambda: laguerre_poly(-1, x))
+
+    assert laguerre_poly(1, x, polys=True) == Poly(-x + 1, domain='QQ')
+
+    assert laguerre_poly(0, x) == 1
+    assert laguerre_poly(1, x) == -x + 1
+    assert laguerre_poly(2, x) == Q(1, 2)*x**2 - Q(4, 2)*x + 1
+    assert laguerre_poly(3, x) == -Q(1, 6)*x**3 + Q(9, 6)*x**2 - Q(18, 6)*x + 1
+    assert laguerre_poly(4, x) == Q(
+        1, 24)*x**4 - Q(16, 24)*x**3 + Q(72, 24)*x**2 - Q(96, 24)*x + 1
+    assert laguerre_poly(5, x) == -Q(1, 120)*x**5 + Q(25, 120)*x**4 - Q(
+        200, 120)*x**3 + Q(600, 120)*x**2 - Q(600, 120)*x + 1
+    assert laguerre_poly(6, x) == Q(1, 720)*x**6 - Q(36, 720)*x**5 + Q(450, 720)*x**4 - Q(2400, 720)*x**3 + Q(5400, 720)*x**2 - Q(4320, 720)*x + 1
+
+    assert laguerre_poly(0, x, a) == 1
+    assert laguerre_poly(1, x, a) == -x + a + 1
+    assert laguerre_poly(2, x, a) == x**2/2 + (-a - 2)*x + a**2/2 + a*Q(3, 2) + 1
+    assert laguerre_poly(3, x, a) == -x**3/6 + (a/2 + Q(
+        3)/2)*x**2 + (-a**2/2 - a*Q(5, 2) - 3)*x + a**3/6 + a**2 + a*Q(11, 6) + 1
+
+    assert laguerre_poly(1).dummy_eq(-x + 1)
+    assert laguerre_poly(1, polys=True) == Poly(-x + 1)
+
+
+def test_spherical_bessel_fn():
+    x, z = symbols("x z")
+    assert spherical_bessel_fn(1, z) == 1/z**2
+    assert spherical_bessel_fn(2, z) == -1/z + 3/z**3
+    assert spherical_bessel_fn(3, z) == -6/z**2 + 15/z**4
+    assert spherical_bessel_fn(4, z) == 1/z - 45/z**3 + 105/z**5

rl/Lib/site-packages/sympy/polys/tests/test_partfrac.py

@@ -0,0 +1,249 @@
+"""Tests for algorithms for partial fraction decomposition of rational
+functions. """
+
+from sympy.polys.partfrac import (
+    apart_undetermined_coeffs,
+    apart,
+    apart_list, assemble_partfrac_list
+)
+
+from sympy.core.expr import Expr
+from sympy.core.function import Lambda
+from sympy.core.numbers import (E, I, Rational, pi, all_close)
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.matrices.dense import Matrix
+from sympy.polys.polytools import (Poly, factor)
+from sympy.polys.rationaltools import together
+from sympy.polys.rootoftools import RootSum
+from sympy.testing.pytest import raises, XFAIL
+from sympy.abc import x, y, a, b, c
+
+
+def test_apart():
+    assert apart(1) == 1
+    assert apart(1, x) == 1
+
+    f, g = (x**2 + 1)/(x + 1), 2/(x + 1) + x - 1
+
+    assert apart(f, full=False) == g
+    assert apart(f, full=True) == g
+
+    f, g = 1/(x + 2)/(x + 1), 1/(1 + x) - 1/(2 + x)
+
+    assert apart(f, full=False) == g
+    assert apart(f, full=True) == g
+
+    f, g = 1/(x + 1)/(x + 5), -1/(5 + x)/4 + 1/(1 + x)/4
+
+    assert apart(f, full=False) == g
+    assert apart(f, full=True) == g
+
+    assert apart((E*x + 2)/(x - pi)*(x - 1), x) == \
+        2 - E + E*pi + E*x + (E*pi + 2)*(pi - 1)/(x - pi)
+
+    assert apart(Eq((x**2 + 1)/(x + 1), x), x) == Eq(x - 1 + 2/(x + 1), x)
+
+    assert apart(x/2, y) == x/2
+
+    f, g = (x+y)/(2*x - y), Rational(3, 2)*y/(2*x - y) + S.Half
+
+    assert apart(f, x, full=False) == g
+    assert apart(f, x, full=True) == g
+
+    f, g = (x+y)/(2*x - y), 3*x/(2*x - y) - 1
+
+    assert apart(f, y, full=False) == g
+    assert apart(f, y, full=True) == g
+
+    raises(NotImplementedError, lambda: apart(1/(x + 1)/(y + 2)))
+
+
+def test_apart_matrix():
+    M = Matrix(2, 2, lambda i, j: 1/(x + i + 1)/(x + j))
+
+    assert apart(M) == Matrix([
+        [1/x - 1/(x + 1), (x + 1)**(-2)],
+        [1/(2*x) - (S.Half)/(x + 2), 1/(x + 1) - 1/(x + 2)],
+    ])
+
+
+def test_apart_symbolic():
+    f = a*x**4 + (2*b + 2*a*c)*x**3 + (4*b*c - a**2 + a*c**2)*x**2 + \
+        (-2*a*b + 2*b*c**2)*x - b**2
+    g = a**2*x**4 + (2*a*b + 2*c*a**2)*x**3 + (4*a*b*c + b**2 +
+        a**2*c**2)*x**2 + (2*c*b**2 + 2*a*b*c**2)*x + b**2*c**2
+
+    assert apart(f/g, x) == 1/a - 1/(x + c)**2 - b**2/(a*(a*x + b)**2)
+
+    assert apart(1/((x + a)*(x + b)*(x + c)), x) == \
+        1/((a - c)*(b - c)*(c + x)) - 1/((a - b)*(b - c)*(b + x)) + \
+        1/((a - b)*(a - c)*(a + x))
+
+
+def _make_extension_example():
+    # https://github.com/sympy/sympy/issues/18531
+    from sympy.core import Mul
+    def mul2(expr):
+        # 2-arg mul hack...
+        return Mul(2, expr, evaluate=False)
+
+    f = ((x**2 + 1)**3/((x - 1)**2*(x + 1)**2*(-x**2 + 2*x + 1)*(x**2 + 2*x - 1)))
+    g = (1/mul2(x - sqrt(2) + 1)
+       - 1/mul2(x - sqrt(2) - 1)
+       + 1/mul2(x + 1 + sqrt(2))
+       - 1/mul2(x - 1 + sqrt(2))
+       + 1/mul2((x + 1)**2)
+       + 1/mul2((x - 1)**2))
+    return f, g
+
+
+def test_apart_extension():
+    f = 2/(x**2 + 1)
+    g = I/(x + I) - I/(x - I)
+
+    assert apart(f, extension=I) == g
+    assert apart(f, gaussian=True) == g
+
+    f = x/((x - 2)*(x + I))
+
+    assert factor(together(apart(f)).expand()) == f
+
+    f, g = _make_extension_example()
+
+    # XXX: Only works with dotprodsimp. See test_apart_extension_xfail below
+    from sympy.matrices import dotprodsimp
+    with dotprodsimp(True):
+        assert apart(f, x, extension={sqrt(2)}) == g
+
+
+def test_apart_extension_xfail():
+    f, g = _make_extension_example()
+    assert apart(f, x, extension={sqrt(2)}) == g
+
+
+def test_apart_full():
+    f = 1/(x**2 + 1)
+
+    assert apart(f, full=False) == f
+    assert apart(f, full=True).dummy_eq(
+        -RootSum(x**2 + 1, Lambda(a, a/(x - a)), auto=False)/2)
+
+    f = 1/(x**3 + x + 1)
+
+    assert apart(f, full=False) == f
+    assert apart(f, full=True).dummy_eq(
+        RootSum(x**3 + x + 1,
+        Lambda(a, (a**2*Rational(6, 31) - a*Rational(9, 31) + Rational(4, 31))/(x - a)), auto=False))
+
+    f = 1/(x**5 + 1)
+
+    assert apart(f, full=False) == \
+        (Rational(-1, 5))*((x**3 - 2*x**2 + 3*x - 4)/(x**4 - x**3 + x**2 -
+         x + 1)) + (Rational(1, 5))/(x + 1)
+    assert apart(f, full=True).dummy_eq(
+        -RootSum(x**4 - x**3 + x**2 - x + 1,
+        Lambda(a, a/(x - a)), auto=False)/5 + (Rational(1, 5))/(x + 1))
+
+
+def test_apart_full_floats():
+    # https://github.com/sympy/sympy/issues/26648
+    f = (
+        6.43369157032015e-9*x**3 + 1.35203404799555e-5*x**2
+        + 0.00357538393743079*x + 0.085
+        )/(
+        4.74334912634438e-11*x**4 + 4.09576274286244e-6*x**3
+        + 0.00334241812250921*x**2 + 0.15406018058983*x + 1.0
+    )
+
+    expected = (
+        133.599202650992/(x + 85524.0054884464)
+        + 1.07757928431867/(x + 774.88576677949)
+        + 0.395006955518971/(x + 40.7977016133126)
+        + 0.564264854137341/(x + 7.79746609204661)
+    )
+
+    f_apart = apart(f, full=True).evalf()
+
+    # There is a significant floating point error in this operation.
+    assert all_close(f_apart, expected, rtol=1e-3, atol=1e-5)
+
+
+def test_apart_undetermined_coeffs():
+    p = Poly(2*x - 3)
+    q = Poly(x**9 - x**8 - x**6 + x**5 - 2*x**2 + 3*x - 1)
+    r = (-x**7 - x**6 - x**5 + 4)/(x**8 - x**5 - 2*x + 1) + 1/(x - 1)
+
+    assert apart_undetermined_coeffs(p, q) == r
+
+    p = Poly(1, x, domain='ZZ[a,b]')
+    q = Poly((x + a)*(x + b), x, domain='ZZ[a,b]')
+    r = 1/((a - b)*(b + x)) - 1/((a - b)*(a + x))
+
+    assert apart_undetermined_coeffs(p, q) == r
+
+
+def test_apart_list():
+    from sympy.utilities.iterables import numbered_symbols
+    def dummy_eq(i, j):
+        if type(i) in (list, tuple):
+            return all(dummy_eq(i, j) for i, j in zip(i, j))
+        return i == j or i.dummy_eq(j)
+
+    w0, w1, w2 = Symbol("w0"), Symbol("w1"), Symbol("w2")
+    _a = Dummy("a")
+
+    f = (-2*x - 2*x**2) / (3*x**2 - 6*x)
+    got = apart_list(f, x, dummies=numbered_symbols("w"))
+    ans = (-1, Poly(Rational(2, 3), x, domain='QQ'),
+        [(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 2), Lambda(_a, -_a + x), 1)])
+    assert dummy_eq(got, ans)
+
+    got = apart_list(2/(x**2-2), x, dummies=numbered_symbols("w"))
+    ans = (1, Poly(0, x, domain='ZZ'), [(Poly(w0**2 - 2, w0, domain='ZZ'),
+        Lambda(_a, _a/2),
+        Lambda(_a, -_a + x), 1)])
+    assert dummy_eq(got, ans)
+
+    f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
+    got = apart_list(f, x, dummies=numbered_symbols("w"))
+    ans = (1, Poly(0, x, domain='ZZ'),
+        [(Poly(w0 - 2, w0, domain='ZZ'), Lambda(_a, 4), Lambda(_a, -_a + x), 1),
+        (Poly(w1**2 - 1, w1, domain='ZZ'), Lambda(_a, -3*_a - 6), Lambda(_a, -_a + x), 2),
+        (Poly(w2 + 1, w2, domain='ZZ'), Lambda(_a, -4), Lambda(_a, -_a + x), 1)])
+    assert dummy_eq(got, ans)
+
+
+def test_assemble_partfrac_list():
+    f = 36 / (x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2)
+    pfd = apart_list(f)
+    assert assemble_partfrac_list(pfd) == -4/(x + 1) - 3/(x + 1)**2 - 9/(x - 1)**2 + 4/(x - 2)
+
+    a = Dummy("a")
+    pfd = (1, Poly(0, x, domain='ZZ'), [([sqrt(2),-sqrt(2)], Lambda(a, a/2), Lambda(a, -a + x), 1)])
+    assert assemble_partfrac_list(pfd) == -1/(sqrt(2)*(x + sqrt(2))) + 1/(sqrt(2)*(x - sqrt(2)))
+
+
+@XFAIL
+def test_noncommutative_pseudomultivariate():
+    # apart doesn't go inside noncommutative expressions
+    class foo(Expr):
+        is_commutative=False
+    e = x/(x + x*y)
+    c = 1/(1 + y)
+    assert apart(e + foo(e)) == c + foo(c)
+    assert apart(e*foo(e)) == c*foo(c)
+
+def test_noncommutative():
+    class foo(Expr):
+        is_commutative=False
+    e = x/(x + x*y)
+    c = 1/(1 + y)
+    assert apart(e + foo()) == c + foo()
+
+def test_issue_5798():
+    assert apart(
+        2*x/(x**2 + 1) - (x - 1)/(2*(x**2 + 1)) + 1/(2*(x + 1)) - 2/x) == \
+        (3*x + 1)/(x**2 + 1)/2 + 1/(x + 1)/2 - 2/x

rl/Lib/site-packages/sympy/polys/tests/test_polyclasses.py

@@ -0,0 +1,588 @@
+"""Tests for OO layer of several polynomial representations. """
+
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.polyclasses import DMP, DMF, ANP
+from sympy.polys.polyerrors import (CoercionFailed, ExactQuotientFailed,
+                                    NotInvertible)
+from sympy.polys.specialpolys import f_polys
+from sympy.testing.pytest import raises, warns_deprecated_sympy
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
+
+def test_DMP___init__():
+    f = DMP([[ZZ(0)], [], [ZZ(0), ZZ(1), ZZ(2)], [ZZ(3)]], ZZ)
+
+    assert f._rep == [[1, 2], [3]]
+    assert f.dom == ZZ
+    assert f.lev == 1
+
+    f = DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ, 1)
+
+    assert f._rep == [[1, 2], [3]]
+    assert f.dom == ZZ
+    assert f.lev == 1
+
+    f = DMP.from_dict({(1, 1): ZZ(1), (0, 0): ZZ(2)}, 1, ZZ)
+
+    assert f._rep == [[1, 0], [2]]
+    assert f.dom == ZZ
+    assert f.lev == 1
+
+
+def test_DMP_rep_deprecation():
+    f = DMP([1, 2, 3], ZZ)
+
+    with warns_deprecated_sympy():
+        assert f.rep == [1, 2, 3]
+
+
+def test_DMP___eq__():
+    assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) == \
+        DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ)
+
+    assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) == \
+        DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ)
+    assert DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ) == \
+        DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ)
+
+    assert DMP([[[ZZ(1)]]], ZZ) != DMP([[ZZ(1)]], ZZ)
+    assert DMP([[ZZ(1)]], ZZ) != DMP([[[ZZ(1)]]], ZZ)
+
+
+def test_DMP___bool__():
+    assert bool(DMP([[]], ZZ)) is False
+    assert bool(DMP([[ZZ(1)]], ZZ)) is True
+
+
+def test_DMP_to_dict():
+    f = DMP([[ZZ(3)], [], [ZZ(2)], [], [ZZ(8)]], ZZ)
+
+    assert f.to_dict() == \
+        {(4, 0): 3, (2, 0): 2, (0, 0): 8}
+    assert f.to_sympy_dict() == \
+        {(4, 0): ZZ.to_sympy(3), (2, 0): ZZ.to_sympy(2), (0, 0):
+         ZZ.to_sympy(8)}
+
+
+def test_DMP_properties():
+    assert DMP([[]], ZZ).is_zero is True
+    assert DMP([[ZZ(1)]], ZZ).is_zero is False
+
+    assert DMP([[ZZ(1)]], ZZ).is_one is True
+    assert DMP([[ZZ(2)]], ZZ).is_one is False
+
+    assert DMP([[ZZ(1)]], ZZ).is_ground is True
+    assert DMP([[ZZ(1)], [ZZ(2)], [ZZ(1)]], ZZ).is_ground is False
+
+    assert DMP([[ZZ(1)], [ZZ(2), ZZ(0)], [ZZ(1), ZZ(0)]], ZZ).is_sqf is True
+    assert DMP([[ZZ(1)], [ZZ(2), ZZ(0)], [ZZ(1), ZZ(0), ZZ(0)]], ZZ).is_sqf is False
+
+    assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ).is_monic is True
+    assert DMP([[ZZ(2), ZZ(2)], [ZZ(3)]], ZZ).is_monic is False
+
+    assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ).is_primitive is True
+    assert DMP([[ZZ(2), ZZ(4)], [ZZ(6)]], ZZ).is_primitive is False
+
+
+def test_DMP_arithmetics():
+    f = DMP([[ZZ(2)], [ZZ(2), ZZ(0)]], ZZ)
+
+    assert f.mul_ground(2) == DMP([[ZZ(4)], [ZZ(4), ZZ(0)]], ZZ)
+    assert f.quo_ground(2) == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
+
+    raises(ExactQuotientFailed, lambda: f.exquo_ground(3))
+
+    f = DMP([[ZZ(-5)]], ZZ)
+    g = DMP([[ZZ(5)]], ZZ)
+
+    assert f.abs() == g
+    assert abs(f) == g
+
+    assert g.neg() == f
+    assert -g == f
+
+    h = DMP([[]], ZZ)
+
+    assert f.add(g) == h
+    assert f + g == h
+    assert g + f == h
+    assert f + 5 == h
+    assert 5 + f == h
+
+    h = DMP([[ZZ(-10)]], ZZ)
+
+    assert f.sub(g) == h
+    assert f - g == h
+    assert g - f == -h
+    assert f - 5 == h
+    assert 5 - f == -h
+
+    h = DMP([[ZZ(-25)]], ZZ)
+
+    assert f.mul(g) == h
+    assert f * g == h
+    assert g * f == h
+    assert f * 5 == h
+    assert 5 * f == h
+
+    h = DMP([[ZZ(25)]], ZZ)
+
+    assert f.sqr() == h
+    assert f.pow(2) == h
+    assert f**2 == h
+
+    raises(TypeError, lambda: f.pow('x'))
+
+    f = DMP([[ZZ(1)], [], [ZZ(1), ZZ(0), ZZ(0)]], ZZ)
+    g = DMP([[ZZ(2)], [ZZ(-2), ZZ(0)]], ZZ)
+
+    q = DMP([[ZZ(2)], [ZZ(2), ZZ(0)]], ZZ)
+    r = DMP([[ZZ(8), ZZ(0), ZZ(0)]], ZZ)
+
+    assert f.pdiv(g) == (q, r)
+    assert f.pquo(g) == q
+    assert f.prem(g) == r
+
+    raises(ExactQuotientFailed, lambda: f.pexquo(g))
+
+    f = DMP([[ZZ(1)], [], [ZZ(1), ZZ(0), ZZ(0)]], ZZ)
+    g = DMP([[ZZ(1)], [ZZ(-1), ZZ(0)]], ZZ)
+
+    q = DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
+    r = DMP([[ZZ(2), ZZ(0), ZZ(0)]], ZZ)
+
+    assert f.div(g) == (q, r)
+    assert f.quo(g) == q
+    assert f.rem(g) == r
+
+    assert divmod(f, g) == (q, r)
+    assert f // g == q
+    assert f % g == r
+
+    raises(ExactQuotientFailed, lambda: f.exquo(g))
+
+    f = DMP([ZZ(1), ZZ(0), ZZ(-1)], ZZ)
+    g = DMP([ZZ(2), ZZ(-2)], ZZ)
+
+    q = DMP([], ZZ)
+    r = f
+
+    pq = DMP([ZZ(2), ZZ(2)], ZZ)
+    pr = DMP([], ZZ)
+
+    assert f.div(g) == (q, r)
+    assert f.quo(g) == q
+    assert f.rem(g) == r
+
+    assert divmod(f, g) == (q, r)
+    assert f // g == q
+    assert f % g == r
+
+    raises(ExactQuotientFailed, lambda: f.exquo(g))
+
+    assert f.pdiv(g) == (pq, pr)
+    assert f.pquo(g) == pq
+    assert f.prem(g) == pr
+    assert f.pexquo(g) == pq
+
+
+def test_DMP_functionality():
+    f = DMP([[ZZ(1)], [ZZ(2), ZZ(0)], [ZZ(1), ZZ(0), ZZ(0)]], ZZ)
+    g = DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
+    h = DMP([[ZZ(1)]], ZZ)
+
+    assert f.degree() == 2
+    assert f.degree_list() == (2, 2)
+    assert f.total_degree() == 2
+
+    assert f.LC() == ZZ(1)
+    assert f.TC() == ZZ(0)
+    assert f.nth(1, 1) == ZZ(2)
+
+    raises(TypeError, lambda: f.nth(0, 'x'))
+
+    assert f.max_norm() == 2
+    assert f.l1_norm() == 4
+
+    u = DMP([[ZZ(2)], [ZZ(2), ZZ(0)]], ZZ)
+
+    assert f.diff(m=1, j=0) == u
+    assert f.diff(m=1, j=1) == u
+
+    raises(TypeError, lambda: f.diff(m='x', j=0))
+
+    u = DMP([ZZ(1), ZZ(2), ZZ(1)], ZZ)
+    v = DMP([ZZ(1), ZZ(2), ZZ(1)], ZZ)
+
+    assert f.eval(a=1, j=0) == u
+    assert f.eval(a=1, j=1) == v
+
+    assert f.eval(1).eval(1) == ZZ(4)
+
+    assert f.cofactors(g) == (g, g, h)
+    assert f.gcd(g) == g
+    assert f.lcm(g) == f
+
+    u = DMP([[QQ(45), QQ(30), QQ(5)]], QQ)
+    v = DMP([[QQ(1), QQ(2, 3), QQ(1, 9)]], QQ)
+
+    assert u.monic() == v
+
+    assert (4*f).content() == ZZ(4)
+    assert (4*f).primitive() == (ZZ(4), f)
+
+    f = DMP([QQ(1,3), QQ(1)], QQ)
+    g = DMP([QQ(1,7), QQ(1)], QQ)
+
+    assert f.cancel(g) == f.cancel(g, include=True) == (
+        DMP([QQ(7), QQ(21)], QQ),
+        DMP([QQ(3), QQ(21)], QQ)
+    )
+    assert f.cancel(g, include=False) == (
+        QQ(7),
+        QQ(3),
+        DMP([QQ(1), QQ(3)], QQ),
+        DMP([QQ(1), QQ(7)], QQ)
+    )
+
+    f = DMP([[ZZ(1)], [ZZ(2)], [ZZ(3)], [ZZ(4)], [ZZ(5)], [ZZ(6)]], ZZ)
+
+    assert f.trunc(3) == DMP([[ZZ(1)], [ZZ(-1)], [], [ZZ(1)], [ZZ(-1)], []], ZZ)
+
+    f = DMP(f_4, ZZ)
+
+    assert f.sqf_part() == -f
+    assert f.sqf_list() == (ZZ(-1), [(-f, 1)])
+
+    f = DMP([[ZZ(-1)], [], [], [ZZ(5)]], ZZ)
+    g = DMP([[ZZ(3), ZZ(1)], [], []], ZZ)
+    h = DMP([[ZZ(45), ZZ(30), ZZ(5)]], ZZ)
+
+    r = DMP([ZZ(675), ZZ(675), ZZ(225), ZZ(25)], ZZ)
+
+    assert f.subresultants(g) == [f, g, h]
+    assert f.resultant(g) == r
+
+    f = DMP([ZZ(1), ZZ(3), ZZ(9), ZZ(-13)], ZZ)
+
+    assert f.discriminant() == -11664
+
+    f = DMP([QQ(2), QQ(0)], QQ)
+    g = DMP([QQ(1), QQ(0), QQ(-16)], QQ)
+
+    s = DMP([QQ(1, 32), QQ(0)], QQ)
+    t = DMP([QQ(-1, 16)], QQ)
+    h = DMP([QQ(1)], QQ)
+
+    assert f.half_gcdex(g) == (s, h)
+    assert f.gcdex(g) == (s, t, h)
+
+    assert f.invert(g) == s
+
+    f = DMP([[QQ(1)], [QQ(2)], [QQ(3)]], QQ)
+
+    raises(ValueError, lambda: f.half_gcdex(f))
+    raises(ValueError, lambda: f.gcdex(f))
+
+    raises(ValueError, lambda: f.invert(f))
+
+    f = DMP(ZZ.map([1, 0, 20, 0, 150, 0, 500, 0, 625, -2, 0, -10, 9]), ZZ)
+    g = DMP([ZZ(1), ZZ(0), ZZ(0), ZZ(-2), ZZ(9)], ZZ)
+    h = DMP([ZZ(1), ZZ(0), ZZ(5), ZZ(0)], ZZ)
+
+    assert g.compose(h) == f
+    assert f.decompose() == [g, h]
+
+    f = DMP([[QQ(1)], [QQ(2)], [QQ(3)]], QQ)
+
+    raises(ValueError, lambda: f.decompose())
+    raises(ValueError, lambda: f.sturm())
+
+
+def test_DMP_exclude():
+    f = [[[[[[[[[[[[[[[[[[[[[[[[[[ZZ(1)]], [[]]]]]]]]]]]]]]]]]]]]]]]]]]
+    J = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
+        18, 19, 20, 21, 22, 24, 25]
+
+    assert DMP(f, ZZ).exclude() == (J, DMP([ZZ(1), ZZ(0)], ZZ))
+    assert DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ).exclude() ==\
+            ([], DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ))
+
+
+def test_DMF__init__():
+    f = DMF(([[0], [], [0, 1, 2], [3]], [[1, 2, 3]]), ZZ)
+
+    assert f.num == [[1, 2], [3]]
+    assert f.den == [[1, 2, 3]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(([[1, 2], [3]], [[1, 2, 3]]), ZZ, 1)
+
+    assert f.num == [[1, 2], [3]]
+    assert f.den == [[1, 2, 3]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(([[-1], [-2]], [[3], [-4]]), ZZ)
+
+    assert f.num == [[-1], [-2]]
+    assert f.den == [[3], [-4]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(([[1], [2]], [[-3], [4]]), ZZ)
+
+    assert f.num == [[-1], [-2]]
+    assert f.den == [[3], [-4]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(([[1], [2]], [[-3], [4]]), ZZ)
+
+    assert f.num == [[-1], [-2]]
+    assert f.den == [[3], [-4]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(([[]], [[-3], [4]]), ZZ)
+
+    assert f.num == [[]]
+    assert f.den == [[1]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(17, ZZ, 1)
+
+    assert f.num == [[17]]
+    assert f.den == [[1]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(([[1], [2]]), ZZ)
+
+    assert f.num == [[1], [2]]
+    assert f.den == [[1]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF([[0], [], [0, 1, 2], [3]], ZZ)
+
+    assert f.num == [[1, 2], [3]]
+    assert f.den == [[1]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF({(1, 1): 1, (0, 0): 2}, ZZ, 1)
+
+    assert f.num == [[1, 0], [2]]
+    assert f.den == [[1]]
+    assert f.lev == 1
+    assert f.dom == ZZ
+
+    f = DMF(([[QQ(1)], [QQ(2)]], [[-QQ(3)], [QQ(4)]]), QQ)
+
+    assert f.num == [[-QQ(1)], [-QQ(2)]]
+    assert f.den == [[QQ(3)], [-QQ(4)]]
+    assert f.lev == 1
+    assert f.dom == QQ
+
+    f = DMF(([[QQ(1, 5)], [QQ(2, 5)]], [[-QQ(3, 7)], [QQ(4, 7)]]), QQ)
+
+    assert f.num == [[-QQ(7)], [-QQ(14)]]
+    assert f.den == [[QQ(15)], [-QQ(20)]]
+    assert f.lev == 1
+    assert f.dom == QQ
+
+    raises(ValueError, lambda: DMF(([1], [[1]]), ZZ))
+    raises(ZeroDivisionError, lambda: DMF(([1], []), ZZ))
+
+
+def test_DMF__bool__():
+    assert bool(DMF([[]], ZZ)) is False
+    assert bool(DMF([[1]], ZZ)) is True
+
+
+def test_DMF_properties():
+    assert DMF([[]], ZZ).is_zero is True
+    assert DMF([[]], ZZ).is_one is False
+
+    assert DMF([[1]], ZZ).is_zero is False
+    assert DMF([[1]], ZZ).is_one is True
+
+    assert DMF(([[1]], [[2]]), ZZ).is_one is False
+
+
+def test_DMF_arithmetics():
+    f = DMF([[7], [-9]], ZZ)
+    g = DMF([[-7], [9]], ZZ)
+
+    assert f.neg() == -f == g
+
+    f = DMF(([[1]], [[1], []]), ZZ)
+    g = DMF(([[1]], [[1, 0]]), ZZ)
+
+    h = DMF(([[1], [1, 0]], [[1, 0], []]), ZZ)
+
+    assert f.add(g) == f + g == h
+    assert g.add(f) == g + f == h
+
+    h = DMF(([[-1], [1, 0]], [[1, 0], []]), ZZ)
+
+    assert f.sub(g) == f - g == h
+
+    h = DMF(([[1]], [[1, 0], []]), ZZ)
+
+    assert f.mul(g) == f*g == h
+    assert g.mul(f) == g*f == h
+
+    h = DMF(([[1, 0]], [[1], []]), ZZ)
+
+    assert f.quo(g) == f/g == h
+
+    h = DMF(([[1]], [[1], [], [], []]), ZZ)
+
+    assert f.pow(3) == f**3 == h
+
+    h = DMF(([[1]], [[1, 0, 0, 0]]), ZZ)
+
+    assert g.pow(3) == g**3 == h
+
+    h = DMF(([[1, 0]], [[1]]), ZZ)
+
+    assert g.pow(-1) == g**-1 == h
+
+
+def test_ANP___init__():
+    rep = [QQ(1), QQ(1)]
+    mod = [QQ(1), QQ(0), QQ(1)]
+
+    f = ANP(rep, mod, QQ)
+
+    assert f.to_list() == [QQ(1), QQ(1)]
+    assert f.mod_to_list() == [QQ(1), QQ(0), QQ(1)]
+    assert f.dom == QQ
+
+    rep = {1: QQ(1), 0: QQ(1)}
+    mod = {2: QQ(1), 0: QQ(1)}
+
+    f = ANP(rep, mod, QQ)
+
+    assert f.to_list() == [QQ(1), QQ(1)]
+    assert f.mod_to_list() == [QQ(1), QQ(0), QQ(1)]
+    assert f.dom == QQ
+
+    f = ANP(1, mod, QQ)
+
+    assert f.to_list() == [QQ(1)]
+    assert f.mod_to_list() == [QQ(1), QQ(0), QQ(1)]
+    assert f.dom == QQ
+
+    f = ANP([1, 0.5], mod, QQ)
+
+    assert all(QQ.of_type(a) for a in f.to_list())
+
+    raises(CoercionFailed, lambda: ANP([sqrt(2)], mod, QQ))
+
+
+def test_ANP___eq__():
+    a = ANP([QQ(1), QQ(1)], [QQ(1), QQ(0), QQ(1)], QQ)
+    b = ANP([QQ(1), QQ(1)], [QQ(1), QQ(0), QQ(2)], QQ)
+
+    assert (a == a) is True
+    assert (a != a) is False
+
+    assert (a == b) is False
+    assert (a != b) is True
+
+    b = ANP([QQ(1), QQ(2)], [QQ(1), QQ(0), QQ(1)], QQ)
+
+    assert (a == b) is False
+    assert (a != b) is True
+
+
+def test_ANP___bool__():
+    assert bool(ANP([], [QQ(1), QQ(0), QQ(1)], QQ)) is False
+    assert bool(ANP([QQ(1)], [QQ(1), QQ(0), QQ(1)], QQ)) is True
+
+
+def test_ANP_properties():
+    mod = [QQ(1), QQ(0), QQ(1)]
+
+    assert ANP([QQ(0)], mod, QQ).is_zero is True
+    assert ANP([QQ(1)], mod, QQ).is_zero is False
+
+    assert ANP([QQ(1)], mod, QQ).is_one is True
+    assert ANP([QQ(2)], mod, QQ).is_one is False
+
+
+def test_ANP_arithmetics():
+    mod = [QQ(1), QQ(0), QQ(0), QQ(-2)]
+
+    a = ANP([QQ(2), QQ(-1), QQ(1)], mod, QQ)
+    b = ANP([QQ(1), QQ(2)], mod, QQ)
+
+    c = ANP([QQ(-2), QQ(1), QQ(-1)], mod, QQ)
+
+    assert a.neg() == -a == c
+
+    c = ANP([QQ(2), QQ(0), QQ(3)], mod, QQ)
+
+    assert a.add(b) == a + b == c
+    assert b.add(a) == b + a == c
+
+    c = ANP([QQ(2), QQ(-2), QQ(-1)], mod, QQ)
+
+    assert a.sub(b) == a - b == c
+
+    c = ANP([QQ(-2), QQ(2), QQ(1)], mod, QQ)
+
+    assert b.sub(a) == b - a == c
+
+    c = ANP([QQ(3), QQ(-1), QQ(6)], mod, QQ)
+
+    assert a.mul(b) == a*b == c
+    assert b.mul(a) == b*a == c
+
+    c = ANP([QQ(-1, 43), QQ(9, 43), QQ(5, 43)], mod, QQ)
+
+    assert a.pow(0) == a**(0) == ANP(1, mod, QQ)
+    assert a.pow(1) == a**(1) == a
+
+    assert a.pow(-1) == a**(-1) == c
+
+    assert a.quo(a) == a.mul(a.pow(-1)) == a*a**(-1) == ANP(1, mod, QQ)
+
+    c = ANP([], [1, 0, 0, -2], QQ)
+    r1 = a.rem(b)
+
+    (q, r2) = a.div(b)
+
+    assert r1 == r2 == c == a % b
+
+    raises(NotInvertible, lambda: a.div(c))
+    raises(NotInvertible, lambda: a.rem(c))
+
+    # Comparison with "hard-coded" value fails despite looking identical
+    # from sympy import Rational
+    # c = ANP([Rational(11, 10), Rational(-1, 5), Rational(-3, 5)], [1, 0, 0, -2], QQ)
+
+    assert q == a/b # == c
+
+def test_ANP_unify():
+    mod_z = [ZZ(1), ZZ(0), ZZ(-2)]
+    mod_q = [QQ(1), QQ(0), QQ(-2)]
+
+    a = ANP([QQ(1)], mod_q, QQ)
+    b = ANP([ZZ(1)], mod_z, ZZ)
+
+    assert a.unify(b)[0] == QQ
+    assert b.unify(a)[0] == QQ
+    assert a.unify(a)[0] == QQ
+    assert b.unify(b)[0] == ZZ
+
+    assert a.unify_ANP(b)[-1] == QQ
+    assert b.unify_ANP(a)[-1] == QQ
+    assert a.unify_ANP(a)[-1] == QQ
+    assert b.unify_ANP(b)[-1] == ZZ

rl/Lib/site-packages/sympy/polys/tests/test_polyfuncs.py

@@ -0,0 +1,126 @@
+"""Tests for high-level polynomials manipulation functions. """
+
+from sympy.polys.polyfuncs import (
+    symmetrize, horner, interpolate, rational_interpolate, viete,
+)
+
+from sympy.polys.polyerrors import (
+    MultivariatePolynomialError,
+)
+
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.testing.pytest import raises
+
+from sympy.abc import a, b, c, d, e, x, y, z
+
+
+def test_symmetrize():
+    assert symmetrize(0, x, y, z) == (0, 0)
+    assert symmetrize(1, x, y, z) == (1, 0)
+
+    s1 = x + y + z
+    s2 = x*y + x*z + y*z
+
+    assert symmetrize(1) == (1, 0)
+    assert symmetrize(1, formal=True) == (1, 0, [])
+
+    assert symmetrize(x) == (x, 0)
+    assert symmetrize(x + 1) == (x + 1, 0)
+
+    assert symmetrize(x, x, y) == (x + y, -y)
+    assert symmetrize(x + 1, x, y) == (x + y + 1, -y)
+
+    assert symmetrize(x, x, y, z) == (s1, -y - z)
+    assert symmetrize(x + 1, x, y, z) == (s1 + 1, -y - z)
+
+    assert symmetrize(x**2, x, y, z) == (s1**2 - 2*s2, -y**2 - z**2)
+
+    assert symmetrize(x**2 + y**2) == (-2*x*y + (x + y)**2, 0)
+    assert symmetrize(x**2 - y**2) == (-2*x*y + (x + y)**2, -2*y**2)
+
+    assert symmetrize(x**3 + y**2 + a*x**2 + b*y**3, x, y) == \
+        (-3*x*y*(x + y) - 2*a*x*y + a*(x + y)**2 + (x + y)**3,
+         y**2*(1 - a) + y**3*(b - 1))
+
+    U = [u0, u1, u2] = symbols('u:3')
+
+    assert symmetrize(x + 1, x, y, z, formal=True, symbols=U) == \
+        (u0 + 1, -y - z, [(u0, x + y + z), (u1, x*y + x*z + y*z), (u2, x*y*z)])
+
+    assert symmetrize([1, 2, 3]) == [(1, 0), (2, 0), (3, 0)]
+    assert symmetrize([1, 2, 3], formal=True) == ([(1, 0), (2, 0), (3, 0)], [])
+
+    assert symmetrize([x + y, x - y]) == [(x + y, 0), (x + y, -2*y)]
+
+
+def test_horner():
+    assert horner(0) == 0
+    assert horner(1) == 1
+    assert horner(x) == x
+
+    assert horner(x + 1) == x + 1
+    assert horner(x**2 + 1) == x**2 + 1
+    assert horner(x**2 + x) == (x + 1)*x
+    assert horner(x**2 + x + 1) == (x + 1)*x + 1
+
+    assert horner(
+        9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5) == (((9*x + 8)*x + 7)*x + 6)*x + 5
+    assert horner(
+        a*x**4 + b*x**3 + c*x**2 + d*x + e) == (((a*x + b)*x + c)*x + d)*x + e
+
+    assert horner(4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y, wrt=x) == ((
+        4*y + 2)*x*y + (2*y + 1)*y)*x
+    assert horner(4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y, wrt=y) == ((
+        4*x + 2)*y*x + (2*x + 1)*x)*y
+
+
+def test_interpolate():
+    assert interpolate([1, 4, 9, 16], x) == x**2
+    assert interpolate([1, 4, 9, 25], x) == S(3)*x**3/2 - S(8)*x**2 + S(33)*x/2 - 9
+    assert interpolate([(1, 1), (2, 4), (3, 9)], x) == x**2
+    assert interpolate([(1, 2), (2, 5), (3, 10)], x) == 1 + x**2
+    assert interpolate({1: 2, 2: 5, 3: 10}, x) == 1 + x**2
+    assert interpolate({5: 2, 7: 5, 8: 10, 9: 13}, x) == \
+        -S(13)*x**3/24 + S(12)*x**2 - S(2003)*x/24 + 187
+    assert interpolate([(1, 3), (0, 6), (2, 5), (5, 7), (-2, 4)], x) == \
+        S(-61)*x**4/280 + S(247)*x**3/210 + S(139)*x**2/280 - S(1871)*x/420 + 6
+    assert interpolate((9, 4, 9), 3) == 9
+    assert interpolate((1, 9, 16), 1) is S.One
+    assert interpolate(((x, 1), (2, 3)), x) is S.One
+    assert interpolate({x: 1, 2: 3}, x) is S.One
+    assert interpolate(((2, x), (1, 3)), x) == x**2 - 4*x + 6
+
+
+def test_rational_interpolate():
+    x, y = symbols('x,y')
+    xdata = [1, 2, 3, 4, 5, 6]
+    ydata1 = [120, 150, 200, 255, 312, 370]
+    ydata2 = [-210, -35, 105, 231, 350, 465]
+    assert rational_interpolate(list(zip(xdata, ydata1)), 2) == (
+      (60*x**2 + 60)/x )
+    assert rational_interpolate(list(zip(xdata, ydata1)), 3) == (
+      (60*x**2 + 60)/x )
+    assert rational_interpolate(list(zip(xdata, ydata2)), 2, X=y) == (
+      (105*y**2 - 525)/(y + 1) )
+    xdata = list(range(1,11))
+    ydata = [-1923885361858460, -5212158811973685, -9838050145867125,
+      -15662936261217245, -22469424125057910, -30073793365223685,
+      -38332297297028735, -47132954289530109, -56387719094026320,
+      -66026548943876885]
+    assert rational_interpolate(list(zip(xdata, ydata)), 5) == (
+      (-12986226192544605*x**4 +
+      8657484128363070*x**3 - 30301194449270745*x**2 + 4328742064181535*x
+      - 4328742064181535)/(x**3 + 9*x**2 - 3*x + 11))
+
+
+def test_viete():
+    r1, r2 = symbols('r1, r2')
+
+    assert viete(
+        a*x**2 + b*x + c, [r1, r2], x) == [(r1 + r2, -b/a), (r1*r2, c/a)]
+
+    raises(ValueError, lambda: viete(1, [], x))
+    raises(ValueError, lambda: viete(x**2 + 1, [r1]))
+
+    raises(MultivariatePolynomialError, lambda: viete(x + y, [r1]))

rl/Lib/site-packages/sympy/polys/tests/test_polymatrix.py

@@ -0,0 +1,185 @@
+from sympy.testing.pytest import raises
+
+from sympy.polys.polymatrix import PolyMatrix
+from sympy.polys import Poly
+
+from sympy.core.singleton import S
+from sympy.matrices.dense import Matrix
+from sympy.polys.domains.integerring import ZZ
+from sympy.polys.domains.rationalfield import QQ
+
+from sympy.abc import x, y
+
+
+def _test_polymatrix():
+    pm1 = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(x**3, x), Poly(-1 + x, x)]])
+    v1 = PolyMatrix([[1, 0], [-1, 0]], ring='ZZ[x]')
+    m1 = PolyMatrix([[1, 0], [-1, 0]], ring='ZZ[x]')
+    A = PolyMatrix([[Poly(x**2 + x, x), Poly(0, x)], \
+                    [Poly(x**3 - x + 1, x), Poly(0, x)]])
+    B = PolyMatrix([[Poly(x**2, x), Poly(-x, x)], [Poly(-x**2, x), Poly(x, x)]])
+    assert A.ring == ZZ[x]
+    assert isinstance(pm1*v1, PolyMatrix)
+    assert pm1*v1 == A
+    assert pm1*m1 == A
+    assert v1*pm1 == B
+
+    pm2 = PolyMatrix([[Poly(x**2, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**2, x, domain='QQ'), \
+                    Poly(x**3, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(-x**3, x, domain='QQ')]])
+    assert pm2.ring == QQ[x]
+    v2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]')
+    m2 = PolyMatrix([1, 0, 0, 0, 0, 0], ring='ZZ[x]')
+    C = PolyMatrix([[Poly(x**2, x, domain='QQ')]])
+    assert pm2*v2 == C
+    assert pm2*m2 == C
+
+    pm3 = PolyMatrix([[Poly(x**2, x), S.One]], ring='ZZ[x]')
+    v3 = S.Half*pm3
+    assert v3 == PolyMatrix([[Poly(S.Half*x**2, x, domain='QQ'), S.Half]], ring='QQ[x]')
+    assert pm3*S.Half == v3
+    assert v3.ring == QQ[x]
+
+    pm4 = PolyMatrix([[Poly(x**2, x, domain='ZZ'), Poly(-x**2, x, domain='ZZ')]])
+    v4 = PolyMatrix([1, -1], ring='ZZ[x]')
+    assert pm4*v4 == PolyMatrix([[Poly(2*x**2, x, domain='ZZ')]])
+
+    assert len(PolyMatrix(ring=ZZ[x])) == 0
+    assert PolyMatrix([1, 0, 0, 1], x)/(-1) == PolyMatrix([-1, 0, 0, -1], x)
+
+
+def test_polymatrix_constructor():
+    M1 = PolyMatrix([[x, y]], ring=QQ[x,y])
+    assert M1.ring == QQ[x,y]
+    assert M1.domain == QQ
+    assert M1.gens == (x, y)
+    assert M1.shape == (1, 2)
+    assert M1.rows == 1
+    assert M1.cols == 2
+    assert len(M1) == 2
+    assert list(M1) == [Poly(x, (x, y), domain=QQ), Poly(y, (x, y), domain=QQ)]
+
+    M2 = PolyMatrix([[x, y]], ring=QQ[x][y])
+    assert M2.ring == QQ[x][y]
+    assert M2.domain == QQ[x]
+    assert M2.gens == (y,)
+    assert M2.shape == (1, 2)
+    assert M2.rows == 1
+    assert M2.cols == 2
+    assert len(M2) == 2
+    assert list(M2) == [Poly(x, (y,), domain=QQ[x]), Poly(y, (y,), domain=QQ[x])]
+
+    assert PolyMatrix([[x, y]], y) == PolyMatrix([[x, y]], ring=ZZ.frac_field(x)[y])
+    assert PolyMatrix([[x, y]], ring='ZZ[x,y]') == PolyMatrix([[x, y]], ring=ZZ[x,y])
+
+    assert PolyMatrix([[x, y]], (x, y)) == PolyMatrix([[x, y]], ring=QQ[x,y])
+    assert PolyMatrix([[x, y]], x, y) == PolyMatrix([[x, y]], ring=QQ[x,y])
+    assert PolyMatrix([x, y]) == PolyMatrix([[x], [y]], ring=QQ[x,y])
+    assert PolyMatrix(1, 2, [x, y]) == PolyMatrix([[x, y]], ring=QQ[x,y])
+    assert PolyMatrix(1, 2, lambda i,j: [x,y][j]) == PolyMatrix([[x, y]], ring=QQ[x,y])
+    assert PolyMatrix(0, 2, [], x, y).shape == (0, 2)
+    assert PolyMatrix(2, 0, [], x, y).shape == (2, 0)
+    assert PolyMatrix([[], []], x, y).shape == (2, 0)
+    assert PolyMatrix(ring=QQ[x,y]) == PolyMatrix(0, 0, [], ring=QQ[x,y]) == PolyMatrix([], ring=QQ[x,y])
+    raises(TypeError, lambda: PolyMatrix())
+    raises(TypeError, lambda: PolyMatrix(1))
+
+    assert PolyMatrix([Poly(x), Poly(y)]) == PolyMatrix([[x], [y]], ring=ZZ[x,y])
+
+    # XXX: Maybe a bug in parallel_poly_from_expr (x lost from gens and domain):
+    assert PolyMatrix([Poly(y, x), 1]) == PolyMatrix([[y], [1]], ring=QQ[y])
+
+
+def test_polymatrix_eq():
+    assert (PolyMatrix([x]) == PolyMatrix([x])) is True
+    assert (PolyMatrix([y]) == PolyMatrix([x])) is False
+    assert (PolyMatrix([x]) != PolyMatrix([x])) is False
+    assert (PolyMatrix([y]) != PolyMatrix([x])) is True
+
+    assert PolyMatrix([[x, y]]) != PolyMatrix([x, y]) == PolyMatrix([[x], [y]])
+
+    assert PolyMatrix([x], ring=QQ[x]) != PolyMatrix([x], ring=ZZ[x])
+
+    assert PolyMatrix([x]) != Matrix([x])
+    assert PolyMatrix([x]).to_Matrix() == Matrix([x])
+
+    assert PolyMatrix([1], x) == PolyMatrix([1], x)
+    assert PolyMatrix([1], x) != PolyMatrix([1], y)
+
+
+def test_polymatrix_from_Matrix():
+    assert PolyMatrix.from_Matrix(Matrix([1, 2]), x) == PolyMatrix([1, 2], x, ring=QQ[x])
+    assert PolyMatrix.from_Matrix(Matrix([1]), ring=QQ[x]) == PolyMatrix([1], x)
+    pmx = PolyMatrix([1, 2], x)
+    pmy = PolyMatrix([1, 2], y)
+    assert pmx != pmy
+    assert pmx.set_gens(y) == pmy
+
+
+def test_polymatrix_repr():
+    assert repr(PolyMatrix([[1, 2]], x)) == 'PolyMatrix([[1, 2]], ring=QQ[x])'
+    assert repr(PolyMatrix(0, 2, [], x)) == 'PolyMatrix(0, 2, [], ring=QQ[x])'
+
+
+def test_polymatrix_getitem():
+    M = PolyMatrix([[1, 2], [3, 4]], x)
+    assert M[:, :] == M
+    assert M[0, :] == PolyMatrix([[1, 2]], x)
+    assert M[:, 0] == PolyMatrix([1, 3], x)
+    assert M[0, 0] == Poly(1, x, domain=QQ)
+    assert M[0] == Poly(1, x, domain=QQ)
+    assert M[:2] == [Poly(1, x, domain=QQ), Poly(2, x, domain=QQ)]
+
+
+def test_polymatrix_arithmetic():
+    M = PolyMatrix([[1, 2], [3, 4]], x)
+    assert M + M == PolyMatrix([[2, 4], [6, 8]], x)
+    assert M - M == PolyMatrix([[0, 0], [0, 0]], x)
+    assert -M == PolyMatrix([[-1, -2], [-3, -4]], x)
+    raises(TypeError, lambda: M + 1)
+    raises(TypeError, lambda: M - 1)
+    raises(TypeError, lambda: 1 + M)
+    raises(TypeError, lambda: 1 - M)
+
+    assert M * M == PolyMatrix([[7, 10], [15, 22]], x)
+    assert 2 * M == PolyMatrix([[2, 4], [6, 8]], x)
+    assert M * 2 == PolyMatrix([[2, 4], [6, 8]], x)
+    assert S(2) * M == PolyMatrix([[2, 4], [6, 8]], x)
+    assert M * S(2) == PolyMatrix([[2, 4], [6, 8]], x)
+    raises(TypeError, lambda: [] * M)
+    raises(TypeError, lambda: M * [])
+    M2 = PolyMatrix([[1, 2]], ring=ZZ[x])
+    assert S.Half * M2 == PolyMatrix([[S.Half, 1]], ring=QQ[x])
+    assert M2 * S.Half == PolyMatrix([[S.Half, 1]], ring=QQ[x])
+
+    assert M / 2 == PolyMatrix([[S(1)/2, 1], [S(3)/2, 2]], x)
+    assert M / Poly(2, x) == PolyMatrix([[S(1)/2, 1], [S(3)/2, 2]], x)
+    raises(TypeError, lambda: M / [])
+
+
+def test_polymatrix_manipulations():
+    M1 = PolyMatrix([[1, 2], [3, 4]], x)
+    assert M1.transpose() == PolyMatrix([[1, 3], [2, 4]], x)
+    M2 = PolyMatrix([[5, 6], [7, 8]], x)
+    assert M1.row_join(M2) == PolyMatrix([[1, 2, 5, 6], [3, 4, 7, 8]], x)
+    assert M1.col_join(M2) == PolyMatrix([[1, 2], [3, 4], [5, 6], [7, 8]], x)
+    assert M1.applyfunc(lambda e: 2*e) == PolyMatrix([[2, 4], [6, 8]], x)
+
+
+def test_polymatrix_ones_zeros():
+    assert PolyMatrix.zeros(1, 2, x) == PolyMatrix([[0, 0]], x)
+    assert PolyMatrix.eye(2, x) == PolyMatrix([[1, 0], [0, 1]], x)
+
+
+def test_polymatrix_rref():
+    M = PolyMatrix([[1, 2], [3, 4]], x)
+    assert M.rref() == (PolyMatrix.eye(2, x), (0, 1))
+    raises(ValueError, lambda: PolyMatrix([1, 2], ring=ZZ[x]).rref())
+    raises(ValueError, lambda: PolyMatrix([1, x], ring=QQ[x]).rref())
+
+
+def test_polymatrix_nullspace():
+    M = PolyMatrix([[1, 2], [3, 6]], x)
+    assert M.nullspace() == [PolyMatrix([-2, 1], x)]
+    raises(ValueError, lambda: PolyMatrix([1, 2], ring=ZZ[x]).nullspace())
+    raises(ValueError, lambda: PolyMatrix([1, x], ring=QQ[x]).nullspace())
+    assert M.rank() == 1

rl/Lib/site-packages/sympy/polys/tests/test_polyoptions.py

@@ -0,0 +1,485 @@
+"""Tests for options manager for :class:`Poly` and public API functions. """
+
+from sympy.polys.polyoptions import (
+    Options, Expand, Gens, Wrt, Sort, Order, Field, Greedy, Domain,
+    Split, Gaussian, Extension, Modulus, Symmetric, Strict, Auto,
+    Frac, Formal, Polys, Include, All, Gen, Symbols, Method)
+
+from sympy.polys.orderings import lex
+from sympy.polys.domains import FF, GF, ZZ, QQ, QQ_I, RR, CC, EX
+
+from sympy.polys.polyerrors import OptionError, GeneratorsError
+
+from sympy.core.numbers import (I, Integer)
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.testing.pytest import raises
+from sympy.abc import x, y, z
+
+
+def test_Options_clone():
+    opt = Options((x, y, z), {'domain': 'ZZ'})
+
+    assert opt.gens == (x, y, z)
+    assert opt.domain == ZZ
+    assert ('order' in opt) is False
+
+    new_opt = opt.clone({'gens': (x, y), 'order': 'lex'})
+
+    assert opt.gens == (x, y, z)
+    assert opt.domain == ZZ
+    assert ('order' in opt) is False
+
+    assert new_opt.gens == (x, y)
+    assert new_opt.domain == ZZ
+    assert ('order' in new_opt) is True
+
+
+def test_Expand_preprocess():
+    assert Expand.preprocess(False) is False
+    assert Expand.preprocess(True) is True
+
+    assert Expand.preprocess(0) is False
+    assert Expand.preprocess(1) is True
+
+    raises(OptionError, lambda: Expand.preprocess(x))
+
+
+def test_Expand_postprocess():
+    opt = {'expand': True}
+    Expand.postprocess(opt)
+
+    assert opt == {'expand': True}
+
+
+def test_Gens_preprocess():
+    assert Gens.preprocess((None,)) == ()
+    assert Gens.preprocess((x, y, z)) == (x, y, z)
+    assert Gens.preprocess(((x, y, z),)) == (x, y, z)
+
+    a = Symbol('a', commutative=False)
+
+    raises(GeneratorsError, lambda: Gens.preprocess((x, x, y)))
+    raises(GeneratorsError, lambda: Gens.preprocess((x, y, a)))
+
+
+def test_Gens_postprocess():
+    opt = {'gens': (x, y)}
+    Gens.postprocess(opt)
+
+    assert opt == {'gens': (x, y)}
+
+
+def test_Wrt_preprocess():
+    assert Wrt.preprocess(x) == ['x']
+    assert Wrt.preprocess('') == []
+    assert Wrt.preprocess(' ') == []
+    assert Wrt.preprocess('x,y') == ['x', 'y']
+    assert Wrt.preprocess('x y') == ['x', 'y']
+    assert Wrt.preprocess('x, y') == ['x', 'y']
+    assert Wrt.preprocess('x , y') == ['x', 'y']
+    assert Wrt.preprocess(' x, y') == ['x', 'y']
+    assert Wrt.preprocess(' x,  y') == ['x', 'y']
+    assert Wrt.preprocess([x, y]) == ['x', 'y']
+
+    raises(OptionError, lambda: Wrt.preprocess(','))
+    raises(OptionError, lambda: Wrt.preprocess(0))
+
+
+def test_Wrt_postprocess():
+    opt = {'wrt': ['x']}
+    Wrt.postprocess(opt)
+
+    assert opt == {'wrt': ['x']}
+
+
+def test_Sort_preprocess():
+    assert Sort.preprocess([x, y, z]) == ['x', 'y', 'z']
+    assert Sort.preprocess((x, y, z)) == ['x', 'y', 'z']
+
+    assert Sort.preprocess('x > y > z') == ['x', 'y', 'z']
+    assert Sort.preprocess('x>y>z') == ['x', 'y', 'z']
+
+    raises(OptionError, lambda: Sort.preprocess(0))
+    raises(OptionError, lambda: Sort.preprocess({x, y, z}))
+
+
+def test_Sort_postprocess():
+    opt = {'sort': 'x > y'}
+    Sort.postprocess(opt)
+
+    assert opt == {'sort': 'x > y'}
+
+
+def test_Order_preprocess():
+    assert Order.preprocess('lex') == lex
+
+
+def test_Order_postprocess():
+    opt = {'order': True}
+    Order.postprocess(opt)
+
+    assert opt == {'order': True}
+
+
+def test_Field_preprocess():
+    assert Field.preprocess(False) is False
+    assert Field.preprocess(True) is True
+
+    assert Field.preprocess(0) is False
+    assert Field.preprocess(1) is True
+
+    raises(OptionError, lambda: Field.preprocess(x))
+
+
+def test_Field_postprocess():
+    opt = {'field': True}
+    Field.postprocess(opt)
+
+    assert opt == {'field': True}
+
+
+def test_Greedy_preprocess():
+    assert Greedy.preprocess(False) is False
+    assert Greedy.preprocess(True) is True
+
+    assert Greedy.preprocess(0) is False
+    assert Greedy.preprocess(1) is True
+
+    raises(OptionError, lambda: Greedy.preprocess(x))
+
+
+def test_Greedy_postprocess():
+    opt = {'greedy': True}
+    Greedy.postprocess(opt)
+
+    assert opt == {'greedy': True}
+
+
+def test_Domain_preprocess():
+    assert Domain.preprocess(ZZ) == ZZ
+    assert Domain.preprocess(QQ) == QQ
+    assert Domain.preprocess(EX) == EX
+    assert Domain.preprocess(FF(2)) == FF(2)
+    assert Domain.preprocess(ZZ[x, y]) == ZZ[x, y]
+
+    assert Domain.preprocess('Z') == ZZ
+    assert Domain.preprocess('Q') == QQ
+
+    assert Domain.preprocess('ZZ') == ZZ
+    assert Domain.preprocess('QQ') == QQ
+
+    assert Domain.preprocess('EX') == EX
+
+    assert Domain.preprocess('FF(23)') == FF(23)
+    assert Domain.preprocess('GF(23)') == GF(23)
+
+    raises(OptionError, lambda: Domain.preprocess('Z[]'))
+
+    assert Domain.preprocess('Z[x]') == ZZ[x]
+    assert Domain.preprocess('Q[x]') == QQ[x]
+    assert Domain.preprocess('R[x]') == RR[x]
+    assert Domain.preprocess('C[x]') == CC[x]
+
+    assert Domain.preprocess('ZZ[x]') == ZZ[x]
+    assert Domain.preprocess('QQ[x]') == QQ[x]
+    assert Domain.preprocess('RR[x]') == RR[x]
+    assert Domain.preprocess('CC[x]') == CC[x]
+
+    assert Domain.preprocess('Z[x,y]') == ZZ[x, y]
+    assert Domain.preprocess('Q[x,y]') == QQ[x, y]
+    assert Domain.preprocess('R[x,y]') == RR[x, y]
+    assert Domain.preprocess('C[x,y]') == CC[x, y]
+
+    assert Domain.preprocess('ZZ[x,y]') == ZZ[x, y]
+    assert Domain.preprocess('QQ[x,y]') == QQ[x, y]
+    assert Domain.preprocess('RR[x,y]') == RR[x, y]
+    assert Domain.preprocess('CC[x,y]') == CC[x, y]
+
+    raises(OptionError, lambda: Domain.preprocess('Z()'))
+
+    assert Domain.preprocess('Z(x)') == ZZ.frac_field(x)
+    assert Domain.preprocess('Q(x)') == QQ.frac_field(x)
+
+    assert Domain.preprocess('ZZ(x)') == ZZ.frac_field(x)
+    assert Domain.preprocess('QQ(x)') == QQ.frac_field(x)
+
+    assert Domain.preprocess('Z(x,y)') == ZZ.frac_field(x, y)
+    assert Domain.preprocess('Q(x,y)') == QQ.frac_field(x, y)
+
+    assert Domain.preprocess('ZZ(x,y)') == ZZ.frac_field(x, y)
+    assert Domain.preprocess('QQ(x,y)') == QQ.frac_field(x, y)
+
+    assert Domain.preprocess('Q<I>') == QQ.algebraic_field(I)
+    assert Domain.preprocess('QQ<I>') == QQ.algebraic_field(I)
+
+    assert Domain.preprocess('Q<sqrt(2), I>') == QQ.algebraic_field(sqrt(2), I)
+    assert Domain.preprocess(
+        'QQ<sqrt(2), I>') == QQ.algebraic_field(sqrt(2), I)
+
+    raises(OptionError, lambda: Domain.preprocess('abc'))
+
+
+def test_Domain_postprocess():
+    raises(GeneratorsError, lambda: Domain.postprocess({'gens': (x, y),
+           'domain': ZZ[y, z]}))
+
+    raises(GeneratorsError, lambda: Domain.postprocess({'gens': (),
+           'domain': EX}))
+    raises(GeneratorsError, lambda: Domain.postprocess({'domain': EX}))
+
+
+def test_Split_preprocess():
+    assert Split.preprocess(False) is False
+    assert Split.preprocess(True) is True
+
+    assert Split.preprocess(0) is False
+    assert Split.preprocess(1) is True
+
+    raises(OptionError, lambda: Split.preprocess(x))
+
+
+def test_Split_postprocess():
+    raises(NotImplementedError, lambda: Split.postprocess({'split': True}))
+
+
+def test_Gaussian_preprocess():
+    assert Gaussian.preprocess(False) is False
+    assert Gaussian.preprocess(True) is True
+
+    assert Gaussian.preprocess(0) is False
+    assert Gaussian.preprocess(1) is True
+
+    raises(OptionError, lambda: Gaussian.preprocess(x))
+
+
+def test_Gaussian_postprocess():
+    opt = {'gaussian': True}
+    Gaussian.postprocess(opt)
+
+    assert opt == {
+        'gaussian': True,
+        'domain': QQ_I,
+    }
+
+
+def test_Extension_preprocess():
+    assert Extension.preprocess(True) is True
+    assert Extension.preprocess(1) is True
+
+    assert Extension.preprocess([]) is None
+
+    assert Extension.preprocess(sqrt(2)) == {sqrt(2)}
+    assert Extension.preprocess([sqrt(2)]) == {sqrt(2)}
+
+    assert Extension.preprocess([sqrt(2), I]) == {sqrt(2), I}
+
+    raises(OptionError, lambda: Extension.preprocess(False))
+    raises(OptionError, lambda: Extension.preprocess(0))
+
+
+def test_Extension_postprocess():
+    opt = {'extension': {sqrt(2)}}
+    Extension.postprocess(opt)
+
+    assert opt == {
+        'extension': {sqrt(2)},
+        'domain': QQ.algebraic_field(sqrt(2)),
+    }
+
+    opt = {'extension': True}
+    Extension.postprocess(opt)
+
+    assert opt == {'extension': True}
+
+
+def test_Modulus_preprocess():
+    assert Modulus.preprocess(23) == 23
+    assert Modulus.preprocess(Integer(23)) == 23
+
+    raises(OptionError, lambda: Modulus.preprocess(0))
+    raises(OptionError, lambda: Modulus.preprocess(x))
+
+
+def test_Modulus_postprocess():
+    opt = {'modulus': 5}
+    Modulus.postprocess(opt)
+
+    assert opt == {
+        'modulus': 5,
+        'domain': FF(5),
+    }
+
+    opt = {'modulus': 5, 'symmetric': False}
+    Modulus.postprocess(opt)
+
+    assert opt == {
+        'modulus': 5,
+        'domain': FF(5, False),
+        'symmetric': False,
+    }
+
+
+def test_Symmetric_preprocess():
+    assert Symmetric.preprocess(False) is False
+    assert Symmetric.preprocess(True) is True
+
+    assert Symmetric.preprocess(0) is False
+    assert Symmetric.preprocess(1) is True
+
+    raises(OptionError, lambda: Symmetric.preprocess(x))
+
+
+def test_Symmetric_postprocess():
+    opt = {'symmetric': True}
+    Symmetric.postprocess(opt)
+
+    assert opt == {'symmetric': True}
+
+
+def test_Strict_preprocess():
+    assert Strict.preprocess(False) is False
+    assert Strict.preprocess(True) is True
+
+    assert Strict.preprocess(0) is False
+    assert Strict.preprocess(1) is True
+
+    raises(OptionError, lambda: Strict.preprocess(x))
+
+
+def test_Strict_postprocess():
+    opt = {'strict': True}
+    Strict.postprocess(opt)
+
+    assert opt == {'strict': True}
+
+
+def test_Auto_preprocess():
+    assert Auto.preprocess(False) is False
+    assert Auto.preprocess(True) is True
+
+    assert Auto.preprocess(0) is False
+    assert Auto.preprocess(1) is True
+
+    raises(OptionError, lambda: Auto.preprocess(x))
+
+
+def test_Auto_postprocess():
+    opt = {'auto': True}
+    Auto.postprocess(opt)
+
+    assert opt == {'auto': True}
+
+
+def test_Frac_preprocess():
+    assert Frac.preprocess(False) is False
+    assert Frac.preprocess(True) is True
+
+    assert Frac.preprocess(0) is False
+    assert Frac.preprocess(1) is True
+
+    raises(OptionError, lambda: Frac.preprocess(x))
+
+
+def test_Frac_postprocess():
+    opt = {'frac': True}
+    Frac.postprocess(opt)
+
+    assert opt == {'frac': True}
+
+
+def test_Formal_preprocess():
+    assert Formal.preprocess(False) is False
+    assert Formal.preprocess(True) is True
+
+    assert Formal.preprocess(0) is False
+    assert Formal.preprocess(1) is True
+
+    raises(OptionError, lambda: Formal.preprocess(x))
+
+
+def test_Formal_postprocess():
+    opt = {'formal': True}
+    Formal.postprocess(opt)
+
+    assert opt == {'formal': True}
+
+
+def test_Polys_preprocess():
+    assert Polys.preprocess(False) is False
+    assert Polys.preprocess(True) is True
+
+    assert Polys.preprocess(0) is False
+    assert Polys.preprocess(1) is True
+
+    raises(OptionError, lambda: Polys.preprocess(x))
+
+
+def test_Polys_postprocess():
+    opt = {'polys': True}
+    Polys.postprocess(opt)
+
+    assert opt == {'polys': True}
+
+
+def test_Include_preprocess():
+    assert Include.preprocess(False) is False
+    assert Include.preprocess(True) is True
+
+    assert Include.preprocess(0) is False
+    assert Include.preprocess(1) is True
+
+    raises(OptionError, lambda: Include.preprocess(x))
+
+
+def test_Include_postprocess():
+    opt = {'include': True}
+    Include.postprocess(opt)
+
+    assert opt == {'include': True}
+
+
+def test_All_preprocess():
+    assert All.preprocess(False) is False
+    assert All.preprocess(True) is True
+
+    assert All.preprocess(0) is False
+    assert All.preprocess(1) is True
+
+    raises(OptionError, lambda: All.preprocess(x))
+
+
+def test_All_postprocess():
+    opt = {'all': True}
+    All.postprocess(opt)
+
+    assert opt == {'all': True}
+
+
+def test_Gen_postprocess():
+    opt = {'gen': x}
+    Gen.postprocess(opt)
+
+    assert opt == {'gen': x}
+
+
+def test_Symbols_preprocess():
+    raises(OptionError, lambda: Symbols.preprocess(x))
+
+
+def test_Symbols_postprocess():
+    opt = {'symbols': [x, y, z]}
+    Symbols.postprocess(opt)
+
+    assert opt == {'symbols': [x, y, z]}
+
+
+def test_Method_preprocess():
+    raises(OptionError, lambda: Method.preprocess(10))
+
+
+def test_Method_postprocess():
+    opt = {'method': 'f5b'}
+    Method.postprocess(opt)
+
+    assert opt == {'method': 'f5b'}

rl/Lib/site-packages/sympy/polys/tests/test_polyroots.py

@@ -0,0 +1,758 @@
+"""Tests for algorithms for computing symbolic roots of polynomials. """
+
+from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, Wild, symbols)
+from sympy.functions.elementary.complexes import (conjugate, im, re)
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import (root, sqrt)
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (acos, cos, sin)
+from sympy.polys.domains.integerring import ZZ
+from sympy.sets.sets import Interval
+from sympy.simplify.powsimp import powsimp
+
+from sympy.polys import Poly, cyclotomic_poly, intervals, nroots, rootof
+
+from sympy.polys.polyroots import (root_factors, roots_linear,
+    roots_quadratic, roots_cubic, roots_quartic, roots_quintic,
+    roots_cyclotomic, roots_binomial, preprocess_roots, roots)
+
+from sympy.polys.orthopolys import legendre_poly
+from sympy.polys.polyerrors import PolynomialError, \
+    UnsolvableFactorError
+from sympy.polys.polyutils import _nsort
+
+from sympy.testing.pytest import raises, slow
+from sympy.core.random import verify_numerically
+import mpmath
+from itertools import product
+
+
+
+a, b, c, d, e, q, t, x, y, z = symbols('a,b,c,d,e,q,t,x,y,z')
+
+
+def _check(roots):
+    # this is the desired invariant for roots returned
+    # by all_roots. It is trivially true for linear
+    # polynomials.
+    nreal = sum(1 if i.is_real else 0 for i in roots)
+    assert sorted(roots[:nreal]) == list(roots[:nreal])
+    for ix in range(nreal, len(roots), 2):
+        if not (
+                roots[ix + 1] == roots[ix] or
+                roots[ix + 1] == conjugate(roots[ix])):
+            return False
+    return True
+
+
+def test_roots_linear():
+    assert roots_linear(Poly(2*x + 1, x)) == [Rational(-1, 2)]
+
+
+def test_roots_quadratic():
+    assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
+    assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [Rational(-3, 2), 0]
+    assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
+    assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
+    _check(Poly(2*x**2 + 4*x + 3, x).all_roots())
+
+    f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
+    assert roots_quadratic(Poly(f, x)) == \
+        [-e*(a + c)/(a - c) - sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c),
+         -e*(a + c)/(a - c) + sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)]
+
+    # check for simplification
+    f = Poly(y*x**2 - 2*x - 2*y, x)
+    assert roots_quadratic(f) == \
+        [-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
+    f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
+    assert roots_quadratic(f) == \
+        [1,y**2 + 1]
+
+    f = Poly(sqrt(2)*x**2 - 1, x)
+    r = roots_quadratic(f)
+    assert r == _nsort(r)
+
+    # issue 8255
+    f = Poly(-24*x**2 - 180*x + 264)
+    assert [w.n(2) for w in f.all_roots(radicals=True)] == \
+           [w.n(2) for w in f.all_roots(radicals=False)]
+    for _a, _b, _c in product((-2, 2), (-2, 2), (0, -1)):
+        f = Poly(_a*x**2 + _b*x + _c)
+        roots = roots_quadratic(f)
+        assert roots == _nsort(roots)
+
+
+def test_issue_7724():
+    eq = Poly(x**4*I + x**2 + I, x)
+    assert roots(eq) == {
+        sqrt(I/2 + sqrt(5)*I/2): 1,
+        sqrt(-sqrt(5)*I/2 + I/2): 1,
+        -sqrt(I/2 + sqrt(5)*I/2): 1,
+        -sqrt(-sqrt(5)*I/2 + I/2): 1}
+
+
+def test_issue_8438():
+    p = Poly([1, y, -2, -3], x).as_expr()
+    roots = roots_cubic(Poly(p, x), x)
+    z = Rational(-3, 2) - I*7/2  # this will fail in code given in commit msg
+    post = [r.subs(y, z) for r in roots]
+    assert set(post) == \
+    set(roots_cubic(Poly(p.subs(y, z), x)))
+    # /!\ if p is not made an expression, this is *very* slow
+    assert all(p.subs({y: z, x: i}).n(2, chop=True) == 0 for i in post)
+
+
+def test_issue_8285():
+    roots = (Poly(4*x**8 - 1, x)*Poly(x**2 + 1)).all_roots()
+    assert _check(roots)
+    f = Poly(x**4 + 5*x**2 + 6, x)
+    ro = [rootof(f, i) for i in range(4)]
+    roots = Poly(x**4 + 5*x**2 + 6, x).all_roots()
+    assert roots == ro
+    assert _check(roots)
+    # more than 2 complex roots from which to identify the
+    # imaginary ones
+    roots = Poly(2*x**8 - 1).all_roots()
+    assert _check(roots)
+    assert len(Poly(2*x**10 - 1).all_roots()) == 10  # doesn't fail
+
+
+def test_issue_8289():
+    roots = (Poly(x**2 + 2)*Poly(x**4 + 2)).all_roots()
+    assert _check(roots)
+    roots = Poly(x**6 + 3*x**3 + 2, x).all_roots()
+    assert _check(roots)
+    roots = Poly(x**6 - x + 1).all_roots()
+    assert _check(roots)
+    # all imaginary roots with multiplicity of 2
+    roots = Poly(x**4 + 4*x**2 + 4, x).all_roots()
+    assert _check(roots)
+
+
+def test_issue_14291():
+    assert Poly(((x - 1)**2 + 1)*((x - 1)**2 + 2)*(x - 1)
+        ).all_roots() == [1, 1 - I, 1 + I, 1 - sqrt(2)*I, 1 + sqrt(2)*I]
+    p = x**4 + 10*x**2 + 1
+    ans = [rootof(p, i) for i in range(4)]
+    assert Poly(p).all_roots() == ans
+    _check(ans)
+
+
+def test_issue_13340():
+    eq = Poly(y**3 + exp(x)*y + x, y, domain='EX')
+    roots_d = roots(eq)
+    assert len(roots_d) == 3
+
+
+def test_issue_14522():
+    eq = Poly(x**4 + x**3*(16 + 32*I) + x**2*(-285 + 386*I) + x*(-2824 - 448*I) - 2058 - 6053*I, x)
+    roots_eq = roots(eq)
+    assert all(eq(r) == 0 for r in roots_eq)
+
+
+def test_issue_15076():
+    sol = roots_quartic(Poly(t**4 - 6*t**2 + t/x - 3, t))
+    assert sol[0].has(x)
+
+
+def test_issue_16589():
+    eq = Poly(x**4 - 8*sqrt(2)*x**3 + 4*x**3 - 64*sqrt(2)*x**2 + 1024*x, x)
+    roots_eq = roots(eq)
+    assert 0 in roots_eq
+
+
+def test_roots_cubic():
+    assert roots_cubic(Poly(2*x**3, x)) == [0, 0, 0]
+    assert roots_cubic(Poly(x**3 - 3*x**2 + 3*x - 1, x)) == [1, 1, 1]
+
+    # valid for arbitrary y (issue 21263)
+    r = root(y, 3)
+    assert roots_cubic(Poly(x**3 - y, x)) == [r,
+        r*(-S.Half + sqrt(3)*I/2),
+        r*(-S.Half - sqrt(3)*I/2)]
+    # simpler form when y is negative
+    assert roots_cubic(Poly(x**3 - -1, x)) == \
+        [-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
+    assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \
+         S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2
+    eq = -x**3 + 2*x**2 + 3*x - 2
+    assert roots(eq, trig=True, multiple=True) == \
+           roots_cubic(Poly(eq, x), trig=True) == [
+        Rational(2, 3) + 2*sqrt(13)*cos(acos(8*sqrt(13)/169)/3)/3,
+        -2*sqrt(13)*sin(-acos(8*sqrt(13)/169)/3 + pi/6)/3 + Rational(2, 3),
+        -2*sqrt(13)*cos(-acos(8*sqrt(13)/169)/3 + pi/3)/3 + Rational(2, 3),
+        ]
+
+
+def test_roots_quartic():
+    assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0]
+    assert roots_quartic(Poly(x**4 + x**3, x)) in [
+        [-1, 0, 0, 0],
+        [0, -1, 0, 0],
+        [0, 0, -1, 0],
+        [0, 0, 0, -1]
+    ]
+    assert roots_quartic(Poly(x**4 - x**3, x)) in [
+        [1, 0, 0, 0],
+        [0, 1, 0, 0],
+        [0, 0, 1, 0],
+        [0, 0, 0, 1]
+    ]
+
+    lhs = roots_quartic(Poly(x**4 + x, x))
+    rhs = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
+
+    assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
+
+    # test of all branches of roots quartic
+    for i, (a, b, c, d) in enumerate([(1, 2, 3, 0),
+                                      (3, -7, -9, 9),
+                                      (1, 2, 3, 4),
+                                      (1, 2, 3, 4),
+                                      (-7, -3, 3, -6),
+                                      (-3, 5, -6, -4),
+                                      (6, -5, -10, -3)]):
+        if i == 2:
+            c = -a*(a**2/S(8) - b/S(2))
+        elif i == 3:
+            d = a*(a*(a**2*Rational(3, 256) - b/S(16)) + c/S(4))
+        eq = x**4 + a*x**3 + b*x**2 + c*x + d
+        ans = roots_quartic(Poly(eq, x))
+        assert all(eq.subs(x, ai).n(chop=True) == 0 for ai in ans)
+
+    # not all symbolic quartics are unresolvable
+    eq = Poly(q*x + q/4 + x**4 + x**3 + 2*x**2 - Rational(1, 3), x)
+    sol = roots_quartic(eq)
+    assert all(verify_numerically(eq.subs(x, i), 0) for i in sol)
+    z = symbols('z', negative=True)
+    eq = x**4 + 2*x**3 + 3*x**2 + x*(z + 11) + 5
+    zans = roots_quartic(Poly(eq, x))
+    assert all(verify_numerically(eq.subs(((x, i), (z, -1))), 0) for i in zans)
+    # but some are (see also issue 4989)
+    # it's ok if the solution is not Piecewise, but the tests below should pass
+    eq = Poly(y*x**4 + x**3 - x + z, x)
+    ans = roots_quartic(eq)
+    assert all(type(i) == Piecewise for i in ans)
+    reps = (
+        {"y": Rational(-1, 3), "z": Rational(-1, 4)},  # 4 real
+        {"y": Rational(-1, 3), "z": Rational(-1, 2)},  # 2 real
+        {"y": Rational(-1, 3), "z": -2})  # 0 real
+    for rep in reps:
+        sol = roots_quartic(Poly(eq.subs(rep), x))
+        assert all(verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol))
+
+
+def test_issue_21287():
+    assert not any(isinstance(i, Piecewise) for i in roots_quartic(
+        Poly(x**4 - x**2*(3 + 5*I) + 2*x*(-1 + I) - 1 + 3*I, x)))
+
+
+def test_roots_quintic():
+    eqs = (x**5 - 2,
+            (x/2 + 1)**5 - 5*(x/2 + 1) + 12,
+            x**5 - 110*x**3 - 55*x**2 + 2310*x + 979)
+    for eq in eqs:
+        roots = roots_quintic(Poly(eq))
+        assert len(roots) == 5
+        assert all(eq.subs(x, r.n(10)).n(chop = 1e-5) == 0 for r in roots)
+
+
+def test_roots_cyclotomic():
+    assert roots_cyclotomic(cyclotomic_poly(1, x, polys=True)) == [1]
+    assert roots_cyclotomic(cyclotomic_poly(2, x, polys=True)) == [-1]
+    assert roots_cyclotomic(cyclotomic_poly(
+        3, x, polys=True)) == [Rational(-1, 2) - I*sqrt(3)/2, Rational(-1, 2) + I*sqrt(3)/2]
+    assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True)) == [-I, I]
+    assert roots_cyclotomic(cyclotomic_poly(
+        6, x, polys=True)) == [S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
+
+    assert roots_cyclotomic(cyclotomic_poly(7, x, polys=True)) == [
+        -cos(pi/7) - I*sin(pi/7),
+        -cos(pi/7) + I*sin(pi/7),
+        -cos(pi*Rational(3, 7)) - I*sin(pi*Rational(3, 7)),
+        -cos(pi*Rational(3, 7)) + I*sin(pi*Rational(3, 7)),
+        cos(pi*Rational(2, 7)) - I*sin(pi*Rational(2, 7)),
+        cos(pi*Rational(2, 7)) + I*sin(pi*Rational(2, 7)),
+    ]
+
+    assert roots_cyclotomic(cyclotomic_poly(8, x, polys=True)) == [
+        -sqrt(2)/2 - I*sqrt(2)/2,
+        -sqrt(2)/2 + I*sqrt(2)/2,
+        sqrt(2)/2 - I*sqrt(2)/2,
+        sqrt(2)/2 + I*sqrt(2)/2,
+    ]
+
+    assert roots_cyclotomic(cyclotomic_poly(12, x, polys=True)) == [
+        -sqrt(3)/2 - I/2,
+        -sqrt(3)/2 + I/2,
+        sqrt(3)/2 - I/2,
+        sqrt(3)/2 + I/2,
+    ]
+
+    assert roots_cyclotomic(
+        cyclotomic_poly(1, x, polys=True), factor=True) == [1]
+    assert roots_cyclotomic(
+        cyclotomic_poly(2, x, polys=True), factor=True) == [-1]
+
+    assert roots_cyclotomic(cyclotomic_poly(3, x, polys=True), factor=True) == \
+        [-root(-1, 3), -1 + root(-1, 3)]
+    assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True), factor=True) == \
+        [-I, I]
+    assert roots_cyclotomic(cyclotomic_poly(5, x, polys=True), factor=True) == \
+        [-root(-1, 5), -root(-1, 5)**3, root(-1, 5)**2, -1 - root(-1, 5)**2 + root(-1, 5) + root(-1, 5)**3]
+
+    assert roots_cyclotomic(cyclotomic_poly(6, x, polys=True), factor=True) == \
+        [1 - root(-1, 3), root(-1, 3)]
+
+
+def test_roots_binomial():
+    assert roots_binomial(Poly(5*x, x)) == [0]
+    assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0]
+    assert roots_binomial(Poly(5*x + 2, x)) == [Rational(-2, 5)]
+
+    A = 10**Rational(3, 4)/10
+
+    assert roots_binomial(Poly(5*x**4 + 2, x)) == \
+        [-A - A*I, -A + A*I, A - A*I, A + A*I]
+    _check(roots_binomial(Poly(x**8 - 2)))
+
+    a1 = Symbol('a1', nonnegative=True)
+    b1 = Symbol('b1', nonnegative=True)
+
+    r0 = roots_quadratic(Poly(a1*x**2 + b1, x))
+    r1 = roots_binomial(Poly(a1*x**2 + b1, x))
+
+    assert powsimp(r0[0]) == powsimp(r1[0])
+    assert powsimp(r0[1]) == powsimp(r1[1])
+    for a, b, s, n in product((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
+        if a == b and a != 1:  # a == b == 1 is sufficient
+            continue
+        p = Poly(a*x**n + s*b)
+        ans = roots_binomial(p)
+        assert ans == _nsort(ans)
+
+    # issue 8813
+    assert roots(Poly(2*x**3 - 16*y**3, x)) == {
+        2*y*(Rational(-1, 2) - sqrt(3)*I/2): 1,
+        2*y: 1,
+        2*y*(Rational(-1, 2) + sqrt(3)*I/2): 1}
+
+
+def test_roots_preprocessing():
+    f = a*y*x**2 + y - b
+
+    coeff, poly = preprocess_roots(Poly(f, x))
+
+    assert coeff == 1
+    assert poly == Poly(a*y*x**2 + y - b, x)
+
+    f = c**3*x**3 + c**2*x**2 + c*x + a
+
+    coeff, poly = preprocess_roots(Poly(f, x))
+
+    assert coeff == 1/c
+    assert poly == Poly(x**3 + x**2 + x + a, x)
+
+    f = c**3*x**3 + c**2*x**2 + a
+
+    coeff, poly = preprocess_roots(Poly(f, x))
+
+    assert coeff == 1/c
+    assert poly == Poly(x**3 + x**2 + a, x)
+
+    f = c**3*x**3 + c*x + a
+
+    coeff, poly = preprocess_roots(Poly(f, x))
+
+    assert coeff == 1/c
+    assert poly == Poly(x**3 + x + a, x)
+
+    f = c**3*x**3 + a
+
+    coeff, poly = preprocess_roots(Poly(f, x))
+
+    assert coeff == 1/c
+    assert poly == Poly(x**3 + a, x)
+
+    E, F, J, L = symbols("E,F,J,L")
+
+    f = -21601054687500000000*E**8*J**8/L**16 + \
+        508232812500000000*F*x*E**7*J**7/L**14 - \
+        4269543750000000*E**6*F**2*J**6*x**2/L**12 + \
+        16194716250000*E**5*F**3*J**5*x**3/L**10 - \
+        27633173750*E**4*F**4*J**4*x**4/L**8 + \
+        14840215*E**3*F**5*J**3*x**5/L**6 + \
+        54794*E**2*F**6*J**2*x**6/(5*L**4) - \
+        1153*E*J*F**7*x**7/(80*L**2) + \
+        633*F**8*x**8/160000
+
+    coeff, poly = preprocess_roots(Poly(f, x))
+
+    assert coeff == 20*E*J/(F*L**2)
+    assert poly == 633*x**8 - 115300*x**7 + 4383520*x**6 + 296804300*x**5 - 27633173750*x**4 + \
+        809735812500*x**3 - 10673859375000*x**2 + 63529101562500*x - 135006591796875
+
+    f = Poly(-y**2 + x**2*exp(x), y, domain=ZZ[x, exp(x)])
+    g = Poly(-y**2 + exp(x), y, domain=ZZ[exp(x)])
+
+    assert preprocess_roots(f) == (x, g)
+
+
+def test_roots0():
+    assert roots(1, x) == {}
+    assert roots(x, x) == {S.Zero: 1}
+    assert roots(x**9, x) == {S.Zero: 9}
+    assert roots(((x - 2)*(x + 3)*(x - 4)).expand(), x) == {-S(3): 1, S(2): 1, S(4): 1}
+
+    assert roots(2*x + 1, x) == {Rational(-1, 2): 1}
+    assert roots((2*x + 1)**2, x) == {Rational(-1, 2): 2}
+    assert roots((2*x + 1)**5, x) == {Rational(-1, 2): 5}
+    assert roots((2*x + 1)**10, x) == {Rational(-1, 2): 10}
+
+    assert roots(x**4 - 1, x) == {I: 1, S.One: 1, -S.One: 1, -I: 1}
+    assert roots((x**4 - 1)**2, x) == {I: 2, S.One: 2, -S.One: 2, -I: 2}
+
+    assert roots(((2*x - 3)**2).expand(), x) == {Rational( 3, 2): 2}
+    assert roots(((2*x + 3)**2).expand(), x) == {Rational(-3, 2): 2}
+
+    assert roots(((2*x - 3)**3).expand(), x) == {Rational( 3, 2): 3}
+    assert roots(((2*x + 3)**3).expand(), x) == {Rational(-3, 2): 3}
+
+    assert roots(((2*x - 3)**5).expand(), x) == {Rational( 3, 2): 5}
+    assert roots(((2*x + 3)**5).expand(), x) == {Rational(-3, 2): 5}
+
+    assert roots(((a*x - b)**5).expand(), x) == { b/a: 5}
+    assert roots(((a*x + b)**5).expand(), x) == {-b/a: 5}
+
+    assert roots(x**2 + (-a - 1)*x + a, x) == {a: 1, S.One: 1}
+
+    assert roots(x**4 - 2*x**2 + 1, x) == {S.One: 2, S.NegativeOne: 2}
+
+    assert roots(x**6 - 4*x**4 + 4*x**3 - x**2, x) == \
+        {S.One: 2, -1 - sqrt(2): 1, S.Zero: 2, -1 + sqrt(2): 1}
+
+    assert roots(x**8 - 1, x) == {
+        sqrt(2)/2 + I*sqrt(2)/2: 1,
+        sqrt(2)/2 - I*sqrt(2)/2: 1,
+        -sqrt(2)/2 + I*sqrt(2)/2: 1,
+        -sqrt(2)/2 - I*sqrt(2)/2: 1,
+        S.One: 1, -S.One: 1, I: 1, -I: 1
+    }
+
+    f = -2016*x**2 - 5616*x**3 - 2056*x**4 + 3324*x**5 + 2176*x**6 - \
+        224*x**7 - 384*x**8 - 64*x**9
+
+    assert roots(f) == {S.Zero: 2, -S(2): 2, S(2): 1, Rational(-7, 2): 1,
+                Rational(-3, 2): 1, Rational(-1, 2): 1, Rational(3, 2): 1}
+
+    assert roots((a + b + c)*x - (a + b + c + d), x) == {(a + b + c + d)/(a + b + c): 1}
+
+    assert roots(x**3 + x**2 - x + 1, x, cubics=False) == {}
+    assert roots(((x - 2)*(
+        x + 3)*(x - 4)).expand(), x, cubics=False) == {-S(3): 1, S(2): 1, S(4): 1}
+    assert roots(((x - 2)*(x + 3)*(x - 4)*(x - 5)).expand(), x, cubics=False) == \
+        {-S(3): 1, S(2): 1, S(4): 1, S(5): 1}
+    assert roots(x**3 + 2*x**2 + 4*x + 8, x) == {-S(2): 1, -2*I: 1, 2*I: 1}
+    assert roots(x**3 + 2*x**2 + 4*x + 8, x, cubics=True) == \
+        {-2*I: 1, 2*I: 1, -S(2): 1}
+    assert roots((x**2 - x)*(x**3 + 2*x**2 + 4*x + 8), x ) == \
+        {S.One: 1, S.Zero: 1, -S(2): 1, -2*I: 1, 2*I: 1}
+
+    r1_2, r1_3 = S.Half, Rational(1, 3)
+
+    x0 = (3*sqrt(33) + 19)**r1_3
+    x1 = 4/x0/3
+    x2 = x0/3
+    x3 = sqrt(3)*I/2
+    x4 = x3 - r1_2
+    x5 = -x3 - r1_2
+    assert roots(x**3 + x**2 - x + 1, x, cubics=True) == {
+        -x1 - x2 - r1_3: 1,
+        -x1/x4 - x2*x4 - r1_3: 1,
+        -x1/x5 - x2*x5 - r1_3: 1,
+    }
+
+    f = (x**2 + 2*x + 3).subs(x, 2*x**2 + 3*x).subs(x, 5*x - 4)
+
+    r13_20, r1_20 = [ Rational(*r)
+        for r in ((13, 20), (1, 20)) ]
+
+    s2 = sqrt(2)
+    assert roots(f, x) == {
+        r13_20 + r1_20*sqrt(1 - 8*I*s2): 1,
+        r13_20 - r1_20*sqrt(1 - 8*I*s2): 1,
+        r13_20 + r1_20*sqrt(1 + 8*I*s2): 1,
+        r13_20 - r1_20*sqrt(1 + 8*I*s2): 1,
+    }
+
+    f = x**4 + x**3 + x**2 + x + 1
+
+    r1_4, r1_8, r5_8 = [ Rational(*r) for r in ((1, 4), (1, 8), (5, 8)) ]
+
+    assert roots(f, x) == {
+        -r1_4 + r1_4*5**r1_2 + I*(r5_8 + r1_8*5**r1_2)**r1_2: 1,
+        -r1_4 + r1_4*5**r1_2 - I*(r5_8 + r1_8*5**r1_2)**r1_2: 1,
+        -r1_4 - r1_4*5**r1_2 + I*(r5_8 - r1_8*5**r1_2)**r1_2: 1,
+        -r1_4 - r1_4*5**r1_2 - I*(r5_8 - r1_8*5**r1_2)**r1_2: 1,
+    }
+
+    f = z**3 + (-2 - y)*z**2 + (1 + 2*y - 2*x**2)*z - y + 2*x**2
+
+    assert roots(f, z) == {
+        S.One: 1,
+        S.Half + S.Half*y + S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1,
+        S.Half + S.Half*y - S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1,
+    }
+
+    assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=False) == {}
+    assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=True) != {}
+
+    assert roots(x**4 - 1, x, filter='Z') == {S.One: 1, -S.One: 1}
+    assert roots(x**4 - 1, x, filter='I') == {I: 1, -I: 1}
+
+    assert roots((x - 1)*(x + 1), x) == {S.One: 1, -S.One: 1}
+    assert roots(
+        (x - 1)*(x + 1), x, predicate=lambda r: r.is_positive) == {S.One: 1}
+
+    assert roots(x**4 - 1, x, filter='Z', multiple=True) == [-S.One, S.One]
+    assert roots(x**4 - 1, x, filter='I', multiple=True) == [I, -I]
+
+    ar, br = symbols('a, b', real=True)
+    p = x**2*(ar-br)**2 + 2*x*(br-ar) + 1
+    assert roots(p, x, filter='R') == {1/(ar - br): 2}
+
+    assert roots(x**3, x, multiple=True) == [S.Zero, S.Zero, S.Zero]
+    assert roots(1234, x, multiple=True) == []
+
+    f = x**6 - x**5 + x**4 - x**3 + x**2 - x + 1
+
+    assert roots(f) == {
+        -I*sin(pi/7) + cos(pi/7): 1,
+        -I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1,
+        -I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1,
+        I*sin(pi/7) + cos(pi/7): 1,
+        I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1,
+        I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1,
+    }
+
+    g = ((x**2 + 1)*f**2).expand()
+
+    assert roots(g) == {
+        -I*sin(pi/7) + cos(pi/7): 2,
+        -I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2,
+        -I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2,
+        I*sin(pi/7) + cos(pi/7): 2,
+        I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2,
+        I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2,
+        -I: 1, I: 1,
+    }
+
+    r = roots(x**3 + 40*x + 64)
+    real_root = [rx for rx in r if rx.is_real][0]
+    cr = 108 + 6*sqrt(1074)
+    assert real_root == -2*root(cr, 3)/3 + 20/root(cr, 3)
+
+    eq = Poly((7 + 5*sqrt(2))*x**3 + (-6 - 4*sqrt(2))*x**2 + (-sqrt(2) - 1)*x + 2, x, domain='EX')
+    assert roots(eq) == {-1 + sqrt(2): 1, -2 + 2*sqrt(2): 1, -sqrt(2) + 1: 1}
+
+    eq = Poly(41*x**5 + 29*sqrt(2)*x**5 - 153*x**4 - 108*sqrt(2)*x**4 +
+    175*x**3 + 125*sqrt(2)*x**3 - 45*x**2 - 30*sqrt(2)*x**2 - 26*sqrt(2)*x -
+    26*x + 24, x, domain='EX')
+    assert roots(eq) == {-sqrt(2) + 1: 1, -2 + 2*sqrt(2): 1, -1 + sqrt(2): 1,
+                         -4 + 4*sqrt(2): 1, -3 + 3*sqrt(2): 1}
+
+    eq = Poly(x**3 - 2*x**2 + 6*sqrt(2)*x**2 - 8*sqrt(2)*x + 23*x - 14 +
+            14*sqrt(2), x, domain='EX')
+    assert roots(eq) == {-2*sqrt(2) + 2: 1, -2*sqrt(2) + 1: 1, -2*sqrt(2) - 1: 1}
+
+    assert roots(Poly((x + sqrt(2))**3 - 7, x, domain='EX')) == \
+        {-sqrt(2) + root(7, 3)*(-S.Half - sqrt(3)*I/2): 1,
+         -sqrt(2) + root(7, 3)*(-S.Half + sqrt(3)*I/2): 1,
+         -sqrt(2) + root(7, 3): 1}
+
+def test_roots_slow():
+    """Just test that calculating these roots does not hang. """
+    a, b, c, d, x = symbols("a,b,c,d,x")
+
+    f1 = x**2*c + (a/b) + x*c*d - a
+    f2 = x**2*(a + b*(c - d)*a) + x*a*b*c/(b*d - d) + (a*d - c/d)
+
+    assert list(roots(f1, x).values()) == [1, 1]
+    assert list(roots(f2, x).values()) == [1, 1]
+
+    (zz, yy, xx, zy, zx, yx, k) = symbols("zz,yy,xx,zy,zx,yx,k")
+
+    e1 = (zz - k)*(yy - k)*(xx - k) + zy*yx*zx + zx - zy - yx
+    e2 = (zz - k)*yx*yx + zx*(yy - k)*zx + zy*zy*(xx - k)
+
+    assert list(roots(e1 - e2, k).values()) == [1, 1, 1]
+
+    f = x**3 + 2*x**2 + 8
+    R = list(roots(f).keys())
+
+    assert not any(i for i in [f.subs(x, ri).n(chop=True) for ri in R])
+
+
+def test_roots_inexact():
+    R1 = roots(x**2 + x + 1, x, multiple=True)
+    R2 = roots(x**2 + x + 1.0, x, multiple=True)
+
+    for r1, r2 in zip(R1, R2):
+        assert abs(r1 - r2) < 1e-12
+
+    f = x**4 + 3.0*sqrt(2.0)*x**3 - (78.0 + 24.0*sqrt(3.0))*x**2 \
+        + 144.0*(2*sqrt(3.0) + 9.0)
+
+    R1 = roots(f, multiple=True)
+    R2 = (-12.7530479110482, -3.85012393732929,
+          4.89897948556636, 7.46155167569183)
+
+    for r1, r2 in zip(R1, R2):
+        assert abs(r1 - r2) < 1e-10
+
+
+def test_roots_preprocessed():
+    E, F, J, L = symbols("E,F,J,L")
+
+    f = -21601054687500000000*E**8*J**8/L**16 + \
+        508232812500000000*F*x*E**7*J**7/L**14 - \
+        4269543750000000*E**6*F**2*J**6*x**2/L**12 + \
+        16194716250000*E**5*F**3*J**5*x**3/L**10 - \
+        27633173750*E**4*F**4*J**4*x**4/L**8 + \
+        14840215*E**3*F**5*J**3*x**5/L**6 + \
+        54794*E**2*F**6*J**2*x**6/(5*L**4) - \
+        1153*E*J*F**7*x**7/(80*L**2) + \
+        633*F**8*x**8/160000
+
+    assert roots(f, x) == {}
+
+    R1 = roots(f.evalf(), x, multiple=True)
+    R2 = [-1304.88375606366, 97.1168816800648, 186.946430171876, 245.526792947065,
+          503.441004174773, 791.549343830097, 1273.16678129348, 1850.10650616851]
+
+    w = Wild('w')
+    p = w*E*J/(F*L**2)
+
+    assert len(R1) == len(R2)
+
+    for r1, r2 in zip(R1, R2):
+        match = r1.match(p)
+        assert match is not None and abs(match[w] - r2) < 1e-10
+
+
+def test_roots_strict():
+    assert roots(x**2 - 2*x + 1, strict=False) == {1: 2}
+    assert roots(x**2 - 2*x + 1, strict=True) == {1: 2}
+
+    assert roots(x**6 - 2*x**5 - x**2 + 3*x - 2, strict=False) == {2: 1}
+    raises(UnsolvableFactorError, lambda: roots(x**6 - 2*x**5 - x**2 + 3*x - 2, strict=True))
+
+
+def test_roots_mixed():
+    f = -1936 - 5056*x - 7592*x**2 + 2704*x**3 - 49*x**4
+
+    _re, _im = intervals(f, all=True)
+    _nroots = nroots(f)
+    _sroots = roots(f, multiple=True)
+
+    _re = [ Interval(a, b) for (a, b), _ in _re ]
+    _im = [ Interval(re(a), re(b))*Interval(im(a), im(b)) for (a, b),
+            _ in _im ]
+
+    _intervals = _re + _im
+    _sroots = [ r.evalf() for r in _sroots ]
+
+    _nroots = sorted(_nroots, key=lambda x: x.sort_key())
+    _sroots = sorted(_sroots, key=lambda x: x.sort_key())
+
+    for _roots in (_nroots, _sroots):
+        for i, r in zip(_intervals, _roots):
+            if r.is_real:
+                assert r in i
+            else:
+                assert (re(r), im(r)) in i
+
+
+def test_root_factors():
+    assert root_factors(Poly(1, x)) == [Poly(1, x)]
+    assert root_factors(Poly(x, x)) == [Poly(x, x)]
+
+    assert root_factors(x**2 - 1, x) == [x + 1, x - 1]
+    assert root_factors(x**2 - y, x) == [x - sqrt(y), x + sqrt(y)]
+
+    assert root_factors((x**4 - 1)**2) == \
+        [x + 1, x + 1, x - 1, x - 1, x - I, x - I, x + I, x + I]
+
+    assert root_factors(Poly(x**4 - 1, x), filter='Z') == \
+        [Poly(x + 1, x), Poly(x - 1, x), Poly(x**2 + 1, x)]
+    assert root_factors(8*x**2 + 12*x**4 + 6*x**6 + x**8, x, filter='Q') == \
+        [x, x, x**6 + 6*x**4 + 12*x**2 + 8]
+
+
+@slow
+def test_nroots1():
+    n = 64
+    p = legendre_poly(n, x, polys=True)
+
+    raises(mpmath.mp.NoConvergence, lambda: p.nroots(n=3, maxsteps=5))
+
+    roots = p.nroots(n=3)
+    # The order of roots matters. They are ordered from smallest to the
+    # largest.
+    assert [str(r) for r in roots] == \
+            ['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961',
+            '-0.946', '-0.930', '-0.911', '-0.889', '-0.866', '-0.841',
+            '-0.813', '-0.784', '-0.753', '-0.720', '-0.685', '-0.649',
+            '-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402',
+            '-0.357', '-0.311', '-0.265', '-0.217', '-0.170', '-0.121',
+            '-0.0730', '-0.0243', '0.0243', '0.0730', '0.121', '0.170',
+            '0.217', '0.265', '0.311', '0.357', '0.402', '0.446', '0.489',
+            '0.531', '0.572', '0.611', '0.649', '0.685', '0.720', '0.753',
+            '0.784', '0.813', '0.841', '0.866', '0.889', '0.911', '0.930',
+            '0.946', '0.961', '0.973', '0.983', '0.991', '0.996', '0.999']
+
+def test_nroots2():
+    p = Poly(x**5 + 3*x + 1, x)
+
+    roots = p.nroots(n=3)
+    # The order of roots matters. The roots are ordered by their real
+    # components (if they agree, then by their imaginary components),
+    # with real roots appearing first.
+    assert [str(r) for r in roots] == \
+            ['-0.332', '-0.839 - 0.944*I', '-0.839 + 0.944*I',
+                '1.01 - 0.937*I', '1.01 + 0.937*I']
+
+    roots = p.nroots(n=5)
+    assert [str(r) for r in roots] == \
+            ['-0.33199', '-0.83907 - 0.94385*I', '-0.83907 + 0.94385*I',
+              '1.0051 - 0.93726*I', '1.0051 + 0.93726*I']
+
+
+def test_roots_composite():
+    assert len(roots(Poly(y**3 + y**2*sqrt(x) + y + x, y, composite=True))) == 3
+
+
+def test_issue_19113():
+    eq = cos(x)**3 - cos(x) + 1
+    raises(PolynomialError, lambda: roots(eq))
+
+
+def test_issue_17454():
+    assert roots([1, -3*(-4 - 4*I)**2/8 + 12*I, 0], multiple=True) == [0, 0]
+
+
+def test_issue_20913():
+    assert Poly(x + 9671406556917067856609794, x).real_roots() == [-9671406556917067856609794]
+    assert Poly(x**3 + 4, x).real_roots() == [-2**(S(2)/3)]
+
+
+def test_issue_22768():
+    e = Rational(1, 3)
+    r = (-1/a)**e*(a + 1)**(5*e)
+    assert roots(Poly(a*x**3 + (a + 1)**5, x)) == {
+        r: 1,
+        -r*(1 + sqrt(3)*I)/2: 1,
+        r*(-1 + sqrt(3)*I)/2: 1}

rl/Lib/site-packages/sympy/polys/tests/test_polyutils.py

@@ -0,0 +1,300 @@
+"""Tests for useful utilities for higher level polynomial classes. """
+
+from sympy.core.mul import Mul
+from sympy.core.numbers import (Integer, pi)
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.integrals.integrals import Integral
+from sympy.testing.pytest import raises
+
+from sympy.polys.polyutils import (
+    _nsort,
+    _sort_gens,
+    _unify_gens,
+    _analyze_gens,
+    _sort_factors,
+    parallel_dict_from_expr,
+    dict_from_expr,
+)
+
+from sympy.polys.polyerrors import PolynomialError
+
+from sympy.polys.domains import ZZ
+
+x, y, z, p, q, r, s, t, u, v, w = symbols('x,y,z,p,q,r,s,t,u,v,w')
+A, B = symbols('A,B', commutative=False)
+
+
+def test__nsort():
+    # issue 6137
+    r = S('''[3/2 + sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) - 4/sqrt(-7/3 +
+    61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3)) -
+    61/(18*(-415/216 + 13*I/12)**(1/3)))/2 - sqrt(-7/3 + 61/(18*(-415/216
+    + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3))/2, 3/2 - sqrt(-7/3
+    + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
+    13*I/12)**(1/3))/2 - sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) -
+    4/sqrt(-7/3 + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
+    13*I/12)**(1/3)) - 61/(18*(-415/216 + 13*I/12)**(1/3)))/2, 3/2 +
+    sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) + 4/sqrt(-7/3 +
+    61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3)) -
+    61/(18*(-415/216 + 13*I/12)**(1/3)))/2 + sqrt(-7/3 + 61/(18*(-415/216
+    + 13*I/12)**(1/3)) + 2*(-415/216 + 13*I/12)**(1/3))/2, 3/2 + sqrt(-7/3
+    + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
+    13*I/12)**(1/3))/2 - sqrt(-14/3 - 2*(-415/216 + 13*I/12)**(1/3) +
+    4/sqrt(-7/3 + 61/(18*(-415/216 + 13*I/12)**(1/3)) + 2*(-415/216 +
+    13*I/12)**(1/3)) - 61/(18*(-415/216 + 13*I/12)**(1/3)))/2]''')
+    ans = [r[1], r[0], r[-1], r[-2]]
+    assert _nsort(r) == ans
+    assert len(_nsort(r, separated=True)[0]) == 0
+    b, c, a = exp(-1000), exp(-999), exp(-1001)
+    assert _nsort((b, c, a)) == [a, b, c]
+    # issue 12560
+    a = cos(1)**2 + sin(1)**2 - 1
+    assert _nsort([a]) == [a]
+
+
+def test__sort_gens():
+    assert _sort_gens([]) == ()
+
+    assert _sort_gens([x]) == (x,)
+    assert _sort_gens([p]) == (p,)
+    assert _sort_gens([q]) == (q,)
+
+    assert _sort_gens([x, p]) == (x, p)
+    assert _sort_gens([p, x]) == (x, p)
+    assert _sort_gens([q, p]) == (p, q)
+
+    assert _sort_gens([q, p, x]) == (x, p, q)
+
+    assert _sort_gens([x, p, q], wrt=x) == (x, p, q)
+    assert _sort_gens([x, p, q], wrt=p) == (p, x, q)
+    assert _sort_gens([x, p, q], wrt=q) == (q, x, p)
+
+    assert _sort_gens([x, p, q], wrt='x') == (x, p, q)
+    assert _sort_gens([x, p, q], wrt='p') == (p, x, q)
+    assert _sort_gens([x, p, q], wrt='q') == (q, x, p)
+
+    assert _sort_gens([x, p, q], wrt='x,q') == (x, q, p)
+    assert _sort_gens([x, p, q], wrt='q,x') == (q, x, p)
+    assert _sort_gens([x, p, q], wrt='p,q') == (p, q, x)
+    assert _sort_gens([x, p, q], wrt='q,p') == (q, p, x)
+
+    assert _sort_gens([x, p, q], wrt='x, q') == (x, q, p)
+    assert _sort_gens([x, p, q], wrt='q, x') == (q, x, p)
+    assert _sort_gens([x, p, q], wrt='p, q') == (p, q, x)
+    assert _sort_gens([x, p, q], wrt='q, p') == (q, p, x)
+
+    assert _sort_gens([x, p, q], wrt=[x, 'q']) == (x, q, p)
+    assert _sort_gens([x, p, q], wrt=[q, 'x']) == (q, x, p)
+    assert _sort_gens([x, p, q], wrt=[p, 'q']) == (p, q, x)
+    assert _sort_gens([x, p, q], wrt=[q, 'p']) == (q, p, x)
+
+    assert _sort_gens([x, p, q], wrt=['x', 'q']) == (x, q, p)
+    assert _sort_gens([x, p, q], wrt=['q', 'x']) == (q, x, p)
+    assert _sort_gens([x, p, q], wrt=['p', 'q']) == (p, q, x)
+    assert _sort_gens([x, p, q], wrt=['q', 'p']) == (q, p, x)
+
+    assert _sort_gens([x, p, q], sort='x > p > q') == (x, p, q)
+    assert _sort_gens([x, p, q], sort='p > x > q') == (p, x, q)
+    assert _sort_gens([x, p, q], sort='p > q > x') == (p, q, x)
+
+    assert _sort_gens([x, p, q], wrt='x', sort='q > p') == (x, q, p)
+    assert _sort_gens([x, p, q], wrt='p', sort='q > x') == (p, q, x)
+    assert _sort_gens([x, p, q], wrt='q', sort='p > x') == (q, p, x)
+
+    # https://github.com/sympy/sympy/issues/19353
+    n1 = Symbol('\n1')
+    assert _sort_gens([n1]) == (n1,)
+    assert _sort_gens([x, n1]) == (x, n1)
+
+    X = symbols('x0,x1,x2,x10,x11,x12,x20,x21,x22')
+
+    assert _sort_gens(X) == X
+
+
+def test__unify_gens():
+    assert _unify_gens([], []) == ()
+
+    assert _unify_gens([x], [x]) == (x,)
+    assert _unify_gens([y], [y]) == (y,)
+
+    assert _unify_gens([x, y], [x]) == (x, y)
+    assert _unify_gens([x], [x, y]) == (x, y)
+
+    assert _unify_gens([x, y], [x, y]) == (x, y)
+    assert _unify_gens([y, x], [y, x]) == (y, x)
+
+    assert _unify_gens([x], [y]) == (x, y)
+    assert _unify_gens([y], [x]) == (y, x)
+
+    assert _unify_gens([x], [y, x]) == (y, x)
+    assert _unify_gens([y, x], [x]) == (y, x)
+
+    assert _unify_gens([x, y, z], [x, y, z]) == (x, y, z)
+    assert _unify_gens([z, y, x], [x, y, z]) == (z, y, x)
+    assert _unify_gens([x, y, z], [z, y, x]) == (x, y, z)
+    assert _unify_gens([z, y, x], [z, y, x]) == (z, y, x)
+
+    assert _unify_gens([x, y, z], [t, x, p, q, z]) == (t, x, y, p, q, z)
+
+
+def test__analyze_gens():
+    assert _analyze_gens((x, y, z)) == (x, y, z)
+    assert _analyze_gens([x, y, z]) == (x, y, z)
+
+    assert _analyze_gens(([x, y, z],)) == (x, y, z)
+    assert _analyze_gens(((x, y, z),)) == (x, y, z)
+
+
+def test__sort_factors():
+    assert _sort_factors([], multiple=True) == []
+    assert _sort_factors([], multiple=False) == []
+
+    F = [[1, 2, 3], [1, 2], [1]]
+    G = [[1], [1, 2], [1, 2, 3]]
+
+    assert _sort_factors(F, multiple=False) == G
+
+    F = [[1, 2], [1, 2, 3], [1, 2], [1]]
+    G = [[1], [1, 2], [1, 2], [1, 2, 3]]
+
+    assert _sort_factors(F, multiple=False) == G
+
+    F = [[2, 2], [1, 2, 3], [1, 2], [1]]
+    G = [[1], [1, 2], [2, 2], [1, 2, 3]]
+
+    assert _sort_factors(F, multiple=False) == G
+
+    F = [([1, 2, 3], 1), ([1, 2], 1), ([1], 1)]
+    G = [([1], 1), ([1, 2], 1), ([1, 2, 3], 1)]
+
+    assert _sort_factors(F, multiple=True) == G
+
+    F = [([1, 2], 1), ([1, 2, 3], 1), ([1, 2], 1), ([1], 1)]
+    G = [([1], 1), ([1, 2], 1), ([1, 2], 1), ([1, 2, 3], 1)]
+
+    assert _sort_factors(F, multiple=True) == G
+
+    F = [([2, 2], 1), ([1, 2, 3], 1), ([1, 2], 1), ([1], 1)]
+    G = [([1], 1), ([1, 2], 1), ([2, 2], 1), ([1, 2, 3], 1)]
+
+    assert _sort_factors(F, multiple=True) == G
+
+    F = [([2, 2], 1), ([1, 2, 3], 1), ([1, 2], 2), ([1], 1)]
+    G = [([1], 1), ([2, 2], 1), ([1, 2], 2), ([1, 2, 3], 1)]
+
+    assert _sort_factors(F, multiple=True) == G
+
+
+def test__dict_from_expr_if_gens():
+    assert dict_from_expr(
+        Integer(17), gens=(x,)) == ({(0,): Integer(17)}, (x,))
+    assert dict_from_expr(
+        Integer(17), gens=(x, y)) == ({(0, 0): Integer(17)}, (x, y))
+    assert dict_from_expr(
+        Integer(17), gens=(x, y, z)) == ({(0, 0, 0): Integer(17)}, (x, y, z))
+
+    assert dict_from_expr(
+        Integer(-17), gens=(x,)) == ({(0,): Integer(-17)}, (x,))
+    assert dict_from_expr(
+        Integer(-17), gens=(x, y)) == ({(0, 0): Integer(-17)}, (x, y))
+    assert dict_from_expr(Integer(
+        -17), gens=(x, y, z)) == ({(0, 0, 0): Integer(-17)}, (x, y, z))
+
+    assert dict_from_expr(
+        Integer(17)*x, gens=(x,)) == ({(1,): Integer(17)}, (x,))
+    assert dict_from_expr(
+        Integer(17)*x, gens=(x, y)) == ({(1, 0): Integer(17)}, (x, y))
+    assert dict_from_expr(Integer(
+        17)*x, gens=(x, y, z)) == ({(1, 0, 0): Integer(17)}, (x, y, z))
+
+    assert dict_from_expr(
+        Integer(17)*x**7, gens=(x,)) == ({(7,): Integer(17)}, (x,))
+    assert dict_from_expr(
+        Integer(17)*x**7*y, gens=(x, y)) == ({(7, 1): Integer(17)}, (x, y))
+    assert dict_from_expr(Integer(17)*x**7*y*z**12, gens=(
+        x, y, z)) == ({(7, 1, 12): Integer(17)}, (x, y, z))
+
+    assert dict_from_expr(x + 2*y + 3*z, gens=(x,)) == \
+        ({(1,): Integer(1), (0,): 2*y + 3*z}, (x,))
+    assert dict_from_expr(x + 2*y + 3*z, gens=(x, y)) == \
+        ({(1, 0): Integer(1), (0, 1): Integer(2), (0, 0): 3*z}, (x, y))
+    assert dict_from_expr(x + 2*y + 3*z, gens=(x, y, z)) == \
+        ({(1, 0, 0): Integer(
+            1), (0, 1, 0): Integer(2), (0, 0, 1): Integer(3)}, (x, y, z))
+
+    assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x,)) == \
+        ({(1,): y + 2*z, (0,): 3*y*z}, (x,))
+    assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x, y)) == \
+        ({(1, 1): Integer(1), (1, 0): 2*z, (0, 1): 3*z}, (x, y))
+    assert dict_from_expr(x*y + 2*x*z + 3*y*z, gens=(x, y, z)) == \
+        ({(1, 1, 0): Integer(
+            1), (1, 0, 1): Integer(2), (0, 1, 1): Integer(3)}, (x, y, z))
+
+    assert dict_from_expr(2**y*x, gens=(x,)) == ({(1,): 2**y}, (x,))
+    assert dict_from_expr(Integral(x, (x, 1, 2)) + x) == (
+        {(0, 1): 1, (1, 0): 1}, (x, Integral(x, (x, 1, 2))))
+    raises(PolynomialError, lambda: dict_from_expr(2**y*x, gens=(x, y)))
+
+
+def test__dict_from_expr_no_gens():
+    assert dict_from_expr(Integer(17)) == ({(): Integer(17)}, ())
+
+    assert dict_from_expr(x) == ({(1,): Integer(1)}, (x,))
+    assert dict_from_expr(y) == ({(1,): Integer(1)}, (y,))
+
+    assert dict_from_expr(x*y) == ({(1, 1): Integer(1)}, (x, y))
+    assert dict_from_expr(
+        x + y) == ({(1, 0): Integer(1), (0, 1): Integer(1)}, (x, y))
+
+    assert dict_from_expr(sqrt(2)) == ({(1,): Integer(1)}, (sqrt(2),))
+    assert dict_from_expr(sqrt(2), greedy=False) == ({(): sqrt(2)}, ())
+
+    assert dict_from_expr(x*y, domain=ZZ[x]) == ({(1,): x}, (y,))
+    assert dict_from_expr(x*y, domain=ZZ[y]) == ({(1,): y}, (x,))
+
+    assert dict_from_expr(3*sqrt(
+        2)*pi*x*y, extension=None) == ({(1, 1, 1, 1): 3}, (x, y, pi, sqrt(2)))
+    assert dict_from_expr(3*sqrt(
+        2)*pi*x*y, extension=True) == ({(1, 1, 1): 3*sqrt(2)}, (x, y, pi))
+
+    assert dict_from_expr(3*sqrt(
+        2)*pi*x*y, extension=True) == ({(1, 1, 1): 3*sqrt(2)}, (x, y, pi))
+
+    f = cos(x)*sin(x) + cos(x)*sin(y) + cos(y)*sin(x) + cos(y)*sin(y)
+
+    assert dict_from_expr(f) == ({(0, 1, 0, 1): 1, (0, 1, 1, 0): 1,
+        (1, 0, 0, 1): 1, (1, 0, 1, 0): 1}, (cos(x), cos(y), sin(x), sin(y)))
+
+
+def test__parallel_dict_from_expr_if_gens():
+    assert parallel_dict_from_expr([x + 2*y + 3*z, Integer(7)], gens=(x,)) == \
+        ([{(1,): Integer(1), (0,): 2*y + 3*z}, {(0,): Integer(7)}], (x,))
+
+
+def test__parallel_dict_from_expr_no_gens():
+    assert parallel_dict_from_expr([x*y, Integer(3)]) == \
+        ([{(1, 1): Integer(1)}, {(0, 0): Integer(3)}], (x, y))
+    assert parallel_dict_from_expr([x*y, 2*z, Integer(3)]) == \
+        ([{(1, 1, 0): Integer(
+            1)}, {(0, 0, 1): Integer(2)}, {(0, 0, 0): Integer(3)}], (x, y, z))
+    assert parallel_dict_from_expr((Mul(x, x**2, evaluate=False),)) == \
+        ([{(3,): 1}], (x,))
+
+
+def test_parallel_dict_from_expr():
+    assert parallel_dict_from_expr([Eq(x, 1), Eq(
+        x**2, 2)]) == ([{(0,): -Integer(1), (1,): Integer(1)},
+                        {(0,): -Integer(2), (2,): Integer(1)}], (x,))
+    raises(PolynomialError, lambda: parallel_dict_from_expr([A*B - B*A]))
+
+
+def test_dict_from_expr():
+    assert dict_from_expr(Eq(x, 1)) == \
+        ({(0,): -Integer(1), (1,): Integer(1)}, (x,))
+    raises(PolynomialError, lambda: dict_from_expr(A*B - B*A))
+    raises(PolynomialError, lambda: dict_from_expr(S.true))

rl/Lib/site-packages/sympy/polys/tests/test_pythonrational.py

@@ -0,0 +1,139 @@
+"""Tests for PythonRational type. """
+
+from sympy.polys.domains import PythonRational as QQ
+from sympy.testing.pytest import raises
+
+def test_PythonRational__init__():
+    assert QQ(0).numerator == 0
+    assert QQ(0).denominator == 1
+    assert QQ(0, 1).numerator == 0
+    assert QQ(0, 1).denominator == 1
+    assert QQ(0, -1).numerator == 0
+    assert QQ(0, -1).denominator == 1
+
+    assert QQ(1).numerator == 1
+    assert QQ(1).denominator == 1
+    assert QQ(1, 1).numerator == 1
+    assert QQ(1, 1).denominator == 1
+    assert QQ(-1, -1).numerator == 1
+    assert QQ(-1, -1).denominator == 1
+
+    assert QQ(-1).numerator == -1
+    assert QQ(-1).denominator == 1
+    assert QQ(-1, 1).numerator == -1
+    assert QQ(-1, 1).denominator == 1
+    assert QQ( 1, -1).numerator == -1
+    assert QQ( 1, -1).denominator == 1
+
+    assert QQ(1, 2).numerator == 1
+    assert QQ(1, 2).denominator == 2
+    assert QQ(3, 4).numerator == 3
+    assert QQ(3, 4).denominator == 4
+
+    assert QQ(2, 2).numerator == 1
+    assert QQ(2, 2).denominator == 1
+    assert QQ(2, 4).numerator == 1
+    assert QQ(2, 4).denominator == 2
+
+def test_PythonRational__hash__():
+    assert hash(QQ(0)) == hash(0)
+    assert hash(QQ(1)) == hash(1)
+    assert hash(QQ(117)) == hash(117)
+
+def test_PythonRational__int__():
+    assert int(QQ(-1, 4)) == 0
+    assert int(QQ( 1, 4)) == 0
+    assert int(QQ(-5, 4)) == -1
+    assert int(QQ( 5, 4)) == 1
+
+def test_PythonRational__float__():
+    assert float(QQ(-1, 2)) == -0.5
+    assert float(QQ( 1, 2)) == 0.5
+
+def test_PythonRational__abs__():
+    assert abs(QQ(-1, 2)) == QQ(1, 2)
+    assert abs(QQ( 1, 2)) == QQ(1, 2)
+
+def test_PythonRational__pos__():
+    assert +QQ(-1, 2) == QQ(-1, 2)
+    assert +QQ( 1, 2) == QQ( 1, 2)
+
+def test_PythonRational__neg__():
+    assert -QQ(-1, 2) == QQ( 1, 2)
+    assert -QQ( 1, 2) == QQ(-1, 2)
+
+def test_PythonRational__add__():
+    assert QQ(-1, 2) + QQ( 1, 2) == QQ(0)
+    assert QQ( 1, 2) + QQ(-1, 2) == QQ(0)
+
+    assert QQ(1, 2) + QQ(1, 2) == QQ(1)
+    assert QQ(1, 2) + QQ(3, 2) == QQ(2)
+    assert QQ(3, 2) + QQ(1, 2) == QQ(2)
+    assert QQ(3, 2) + QQ(3, 2) == QQ(3)
+
+    assert 1 + QQ(1, 2) == QQ(3, 2)
+    assert QQ(1, 2) + 1 == QQ(3, 2)
+
+def test_PythonRational__sub__():
+    assert QQ(-1, 2) - QQ( 1, 2) == QQ(-1)
+    assert QQ( 1, 2) - QQ(-1, 2) == QQ( 1)
+
+    assert QQ(1, 2) - QQ(1, 2) == QQ( 0)
+    assert QQ(1, 2) - QQ(3, 2) == QQ(-1)
+    assert QQ(3, 2) - QQ(1, 2) == QQ( 1)
+    assert QQ(3, 2) - QQ(3, 2) == QQ( 0)
+
+    assert 1 - QQ(1, 2) == QQ( 1, 2)
+    assert QQ(1, 2) - 1 == QQ(-1, 2)
+
+def test_PythonRational__mul__():
+    assert QQ(-1, 2) * QQ( 1, 2) == QQ(-1, 4)
+    assert QQ( 1, 2) * QQ(-1, 2) == QQ(-1, 4)
+
+    assert QQ(1, 2) * QQ(1, 2) == QQ(1, 4)
+    assert QQ(1, 2) * QQ(3, 2) == QQ(3, 4)
+    assert QQ(3, 2) * QQ(1, 2) == QQ(3, 4)
+    assert QQ(3, 2) * QQ(3, 2) == QQ(9, 4)
+
+    assert 2 * QQ(1, 2) == QQ(1)
+    assert QQ(1, 2) * 2 == QQ(1)
+
+def test_PythonRational__truediv__():
+    assert QQ(-1, 2) / QQ( 1, 2) == QQ(-1)
+    assert QQ( 1, 2) / QQ(-1, 2) == QQ(-1)
+
+    assert QQ(1, 2) / QQ(1, 2) == QQ(1)
+    assert QQ(1, 2) / QQ(3, 2) == QQ(1, 3)
+    assert QQ(3, 2) / QQ(1, 2) == QQ(3)
+    assert QQ(3, 2) / QQ(3, 2) == QQ(1)
+
+    assert 2 / QQ(1, 2) == QQ(4)
+    assert QQ(1, 2) / 2 == QQ(1, 4)
+
+    raises(ZeroDivisionError, lambda: QQ(1, 2) / QQ(0))
+    raises(ZeroDivisionError, lambda: QQ(1, 2) / 0)
+
+def test_PythonRational__pow__():
+    assert QQ(1)**10 == QQ(1)
+    assert QQ(2)**10 == QQ(1024)
+
+    assert QQ(1)**(-10) == QQ(1)
+    assert QQ(2)**(-10) == QQ(1, 1024)
+
+def test_PythonRational__eq__():
+    assert (QQ(1, 2) == QQ(1, 2)) is True
+    assert (QQ(1, 2) != QQ(1, 2)) is False
+
+    assert (QQ(1, 2) == QQ(1, 3)) is False
+    assert (QQ(1, 2) != QQ(1, 3)) is True
+
+def test_PythonRational__lt_le_gt_ge__():
+    assert (QQ(1, 2) < QQ(1, 4)) is False
+    assert (QQ(1, 2) <= QQ(1, 4)) is False
+    assert (QQ(1, 2) > QQ(1, 4)) is True
+    assert (QQ(1, 2) >= QQ(1, 4)) is True
+
+    assert (QQ(1, 4) < QQ(1, 2)) is True
+    assert (QQ(1, 4) <= QQ(1, 2)) is True
+    assert (QQ(1, 4) > QQ(1, 2)) is False
+    assert (QQ(1, 4) >= QQ(1, 2)) is False

rl/Lib/site-packages/sympy/polys/tests/test_rationaltools.py

@@ -0,0 +1,63 @@
+"""Tests for tools for manipulation of rational expressions. """
+
+from sympy.polys.rationaltools import together
+
+from sympy.core.mul import Mul
+from sympy.core.numbers import Rational
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.trigonometric import sin
+from sympy.integrals.integrals import Integral
+from sympy.abc import x, y, z
+
+A, B = symbols('A,B', commutative=False)
+
+
+def test_together():
+    assert together(0) == 0
+    assert together(1) == 1
+
+    assert together(x*y*z) == x*y*z
+    assert together(x + y) == x + y
+
+    assert together(1/x) == 1/x
+
+    assert together(1/x + 1) == (x + 1)/x
+    assert together(1/x + 3) == (3*x + 1)/x
+    assert together(1/x + x) == (x**2 + 1)/x
+
+    assert together(1/x + S.Half) == (x + 2)/(2*x)
+    assert together(S.Half + x/2) == Mul(S.Half, x + 1, evaluate=False)
+
+    assert together(1/x + 2/y) == (2*x + y)/(y*x)
+    assert together(1/(1 + 1/x)) == x/(1 + x)
+    assert together(x/(1 + 1/x)) == x**2/(1 + x)
+
+    assert together(1/x + 1/y + 1/z) == (x*y + x*z + y*z)/(x*y*z)
+    assert together(1/(1 + x + 1/y + 1/z)) == y*z/(y + z + y*z + x*y*z)
+
+    assert together(1/(x*y) + 1/(x*y)**2) == y**(-2)*x**(-2)*(1 + x*y)
+    assert together(1/(x*y) + 1/(x*y)**4) == y**(-4)*x**(-4)*(1 + x**3*y**3)
+    assert together(1/(x**7*y) + 1/(x*y)**4) == y**(-4)*x**(-7)*(x**3 + y**3)
+
+    assert together(5/(2 + 6/(3 + 7/(4 + 8/(5 + 9/x))))) == \
+        Rational(5, 2)*((171 + 119*x)/(279 + 203*x))
+
+    assert together(1 + 1/(x + 1)**2) == (1 + (x + 1)**2)/(x + 1)**2
+    assert together(1 + 1/(x*(1 + x))) == (1 + x*(1 + x))/(x*(1 + x))
+    assert together(
+        1/(x*(x + 1)) + 1/(x*(x + 2))) == (3 + 2*x)/(x*(1 + x)*(2 + x))
+    assert together(1 + 1/(2*x + 2)**2) == (4*(x + 1)**2 + 1)/(4*(x + 1)**2)
+
+    assert together(sin(1/x + 1/y)) == sin(1/x + 1/y)
+    assert together(sin(1/x + 1/y), deep=True) == sin((x + y)/(x*y))
+
+    assert together(1/exp(x) + 1/(x*exp(x))) == (1 + x)/(x*exp(x))
+    assert together(1/exp(2*x) + 1/(x*exp(3*x))) == (1 + exp(x)*x)/(x*exp(3*x))
+
+    assert together(Integral(1/x + 1/y, x)) == Integral((x + y)/(x*y), x)
+    assert together(Eq(1/x + 1/y, 1 + 1/z)) == Eq((x + y)/(x*y), (z + 1)/z)
+
+    assert together((A*B)**-1 + (B*A)**-1) == (A*B)**-1 + (B*A)**-1

rl/Lib/site-packages/sympy/polys/tests/test_ring_series.py

@@ -0,0 +1,635 @@
+from sympy.polys.domains import QQ, EX, RR
+from sympy.polys.rings import ring
+from sympy.polys.ring_series import (_invert_monoms, rs_integrate,
+    rs_trunc, rs_mul, rs_square, rs_pow, _has_constant_term, rs_hadamard_exp,
+    rs_series_from_list, rs_exp, rs_log, rs_newton, rs_series_inversion,
+    rs_compose_add, rs_asin, rs_atan, rs_atanh, rs_tan, rs_cot, rs_sin, rs_cos,
+    rs_cos_sin, rs_sinh, rs_cosh, rs_tanh, _tan1, rs_fun, rs_nth_root,
+    rs_LambertW, rs_series_reversion, rs_is_puiseux, rs_series)
+from sympy.testing.pytest import raises, slow
+from sympy.core.symbol import symbols
+from sympy.functions import (sin, cos, exp, tan, cot, atan, atanh,
+    tanh, log, sqrt)
+from sympy.core.numbers import Rational
+from sympy.core import expand, S
+
+def is_close(a, b):
+    tol = 10**(-10)
+    assert abs(a - b) < tol
+
+def test_ring_series1():
+    R, x = ring('x', QQ)
+    p = x**4 + 2*x**3 + 3*x + 4
+    assert _invert_monoms(p) == 4*x**4 + 3*x**3 + 2*x + 1
+    assert rs_hadamard_exp(p) == x**4/24 + x**3/3 + 3*x + 4
+    R, x = ring('x', QQ)
+    p = x**4 + 2*x**3 + 3*x + 4
+    assert rs_integrate(p, x) == x**5/5 + x**4/2 + 3*x**2/2 + 4*x
+    R, x, y = ring('x, y', QQ)
+    p = x**2*y**2 + x + 1
+    assert rs_integrate(p, x) == x**3*y**2/3 + x**2/2 + x
+    assert rs_integrate(p, y) == x**2*y**3/3 + x*y + y
+
+def test_trunc():
+    R, x, y, t = ring('x, y, t', QQ)
+    p = (y + t*x)**4
+    p1 = rs_trunc(p, x, 3)
+    assert p1 == y**4 + 4*y**3*t*x + 6*y**2*t**2*x**2
+
+def test_mul_trunc():
+    R, x, y, t = ring('x, y, t', QQ)
+    p = 1 + t*x + t*y
+    for i in range(2):
+        p = rs_mul(p, p, t, 3)
+
+    assert p == 6*x**2*t**2 + 12*x*y*t**2 + 6*y**2*t**2 + 4*x*t + 4*y*t + 1
+    p = 1 + t*x + t*y + t**2*x*y
+    p1 = rs_mul(p, p, t, 2)
+    assert p1 == 1 + 2*t*x + 2*t*y
+    R1, z = ring('z', QQ)
+    raises(ValueError, lambda: rs_mul(p, z, x, 2))
+
+    p1 = 2 + 2*x + 3*x**2
+    p2 = 3 + x**2
+    assert rs_mul(p1, p2, x, 4) == 2*x**3 + 11*x**2 + 6*x + 6
+
+def test_square_trunc():
+    R, x, y, t = ring('x, y, t', QQ)
+    p = (1 + t*x + t*y)*2
+    p1 = rs_mul(p, p, x, 3)
+    p2 = rs_square(p, x, 3)
+    assert p1 == p2
+    p = 1 + x + x**2 + x**3
+    assert rs_square(p, x, 4) == 4*x**3 + 3*x**2 + 2*x + 1
+
+def test_pow_trunc():
+    R, x, y, z = ring('x, y, z', QQ)
+    p0 = y + x*z
+    p = p0**16
+    for xx in (x, y, z):
+        p1 = rs_trunc(p, xx, 8)
+        p2 = rs_pow(p0, 16, xx, 8)
+        assert p1 == p2
+
+    p = 1 + x
+    p1 = rs_pow(p, 3, x, 2)
+    assert p1 == 1 + 3*x
+    assert rs_pow(p, 0, x, 2) == 1
+    assert rs_pow(p, -2, x, 2) == 1 - 2*x
+    p = x + y
+    assert rs_pow(p, 3, y, 3) == x**3 + 3*x**2*y + 3*x*y**2
+    assert rs_pow(1 + x, Rational(2, 3), x, 4) == 4*x**3/81 - x**2/9 + x*Rational(2, 3) + 1
+
+def test_has_constant_term():
+    R, x, y, z = ring('x, y, z', QQ)
+    p = y + x*z
+    assert _has_constant_term(p, x)
+    p = x + x**4
+    assert not _has_constant_term(p, x)
+    p = 1 + x + x**4
+    assert _has_constant_term(p, x)
+    p = x + y + x*z
+
+def test_inversion():
+    R, x = ring('x', QQ)
+    p = 2 + x + 2*x**2
+    n = 5
+    p1 = rs_series_inversion(p, x, n)
+    assert rs_trunc(p*p1, x, n) == 1
+    R, x, y = ring('x, y', QQ)
+    p = 2 + x + 2*x**2 + y*x + x**2*y
+    p1 = rs_series_inversion(p, x, n)
+    assert rs_trunc(p*p1, x, n) == 1
+
+    R, x, y = ring('x, y', QQ)
+    p = 1 + x + y
+    raises(NotImplementedError, lambda: rs_series_inversion(p, x, 4))
+    p = R.zero
+    raises(ZeroDivisionError, lambda: rs_series_inversion(p, x, 3))
+
+
+def test_series_reversion():
+    R, x, y = ring('x, y', QQ)
+
+    p = rs_tan(x, x, 10)
+    assert rs_series_reversion(p, x, 8, y) == rs_atan(y, y, 8)
+
+    p = rs_sin(x, x, 10)
+    assert rs_series_reversion(p, x, 8, y) == 5*y**7/112 + 3*y**5/40 + \
+        y**3/6 + y
+
+def test_series_from_list():
+    R, x = ring('x', QQ)
+    p = 1 + 2*x + x**2 + 3*x**3
+    c = [1, 2, 0, 4, 4]
+    r = rs_series_from_list(p, c, x, 5)
+    pc = R.from_list(list(reversed(c)))
+    r1 = rs_trunc(pc.compose(x, p), x, 5)
+    assert r == r1
+    R, x, y = ring('x, y', QQ)
+    c = [1, 3, 5, 7]
+    p1 = rs_series_from_list(x + y, c, x, 3, concur=0)
+    p2 = rs_trunc((1 + 3*(x+y) + 5*(x+y)**2 + 7*(x+y)**3), x, 3)
+    assert p1 == p2
+
+    R, x = ring('x', QQ)
+    h = 25
+    p = rs_exp(x, x, h) - 1
+    p1 = rs_series_from_list(p, c, x, h)
+    p2 = 0
+    for i, cx in enumerate(c):
+        p2 += cx*rs_pow(p, i, x, h)
+    assert p1 == p2
+
+def test_log():
+    R, x = ring('x', QQ)
+    p = 1 + x
+    p1 = rs_log(p, x, 4)/x**2
+    assert p1 == Rational(1, 3)*x - S.Half + x**(-1)
+    p = 1 + x +2*x**2/3
+    p1 = rs_log(p, x, 9)
+    assert p1 == -17*x**8/648 + 13*x**7/189 - 11*x**6/162 - x**5/45 + \
+      7*x**4/36 - x**3/3 + x**2/6 + x
+    p2 = rs_series_inversion(p, x, 9)
+    p3 = rs_log(p2, x, 9)
+    assert p3 == -p1
+
+    R, x, y = ring('x, y', QQ)
+    p = 1 + x + 2*y*x**2
+    p1 = rs_log(p, x, 6)
+    assert p1 == (4*x**5*y**2 - 2*x**5*y - 2*x**4*y**2 + x**5/5 + 2*x**4*y -
+                  x**4/4 - 2*x**3*y + x**3/3 + 2*x**2*y - x**2/2 + x)
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', EX)
+    assert rs_log(x + a, x, 5) == -EX(1/(4*a**4))*x**4 + EX(1/(3*a**3))*x**3 \
+        - EX(1/(2*a**2))*x**2 + EX(1/a)*x + EX(log(a))
+    assert rs_log(x + x**2*y + a, x, 4) == -EX(a**(-2))*x**3*y + \
+        EX(1/(3*a**3))*x**3 + EX(1/a)*x**2*y - EX(1/(2*a**2))*x**2 + \
+        EX(1/a)*x + EX(log(a))
+
+    p = x + x**2 + 3
+    assert rs_log(p, x, 10).compose(x, 5) == EX(log(3) + Rational(19281291595, 9920232))
+
+def test_exp():
+    R, x = ring('x', QQ)
+    p = x + x**4
+    for h in [10, 30]:
+        q = rs_series_inversion(1 + p, x, h) - 1
+        p1 = rs_exp(q, x, h)
+        q1 = rs_log(p1, x, h)
+        assert q1 == q
+    p1 = rs_exp(p, x, 30)
+    assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
+    prec = 21
+    p = rs_log(1 + x, x, prec)
+    p1 = rs_exp(p, x, prec)
+    assert p1 == x + 1
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', QQ[exp(a), a])
+    assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
+        exp(a)*x**2/2 + exp(a)*x + exp(a)
+    assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
+            exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
+            exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)
+
+    R, x, y = ring('x, y', EX)
+    assert rs_exp(x + a, x, 5) ==  EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
+        EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
+    assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
+        EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
+        EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
+        EX(exp(a))*x + EX(exp(a))
+
+def test_newton():
+    R, x = ring('x', QQ)
+    p = x**2 - 2
+    r = rs_newton(p, x, 4)
+    assert r == 8*x**4 + 4*x**2 + 2
+
+def test_compose_add():
+    R, x = ring('x', QQ)
+    p1 = x**3 - 1
+    p2 = x**2 - 2
+    assert rs_compose_add(p1, p2) == x**6 - 6*x**4 - 2*x**3 + 12*x**2 - 12*x - 7
+
+def test_fun():
+    R, x, y = ring('x, y', QQ)
+    p = x*y + x**2*y**3 + x**5*y
+    assert rs_fun(p, rs_tan, x, 10) == rs_tan(p, x, 10)
+    assert rs_fun(p, _tan1, x, 10) == _tan1(p, x, 10)
+
+def test_nth_root():
+    R, x, y = ring('x, y', QQ)
+    assert rs_nth_root(1 + x**2*y, 4, x, 10) == -77*x**8*y**4/2048 + \
+        7*x**6*y**3/128 - 3*x**4*y**2/32 + x**2*y/4 + 1
+    assert rs_nth_root(1 + x*y + x**2*y**3, 3, x, 5) == -x**4*y**6/9 + \
+        5*x**4*y**5/27 - 10*x**4*y**4/243 - 2*x**3*y**4/9 + 5*x**3*y**3/81 + \
+        x**2*y**3/3 - x**2*y**2/9 + x*y/3 + 1
+    assert rs_nth_root(8*x, 3, x, 3) == 2*x**QQ(1, 3)
+    assert rs_nth_root(8*x + x**2 + x**3, 3, x, 3) == x**QQ(4,3)/12 + 2*x**QQ(1,3)
+    r = rs_nth_root(8*x + x**2*y + x**3, 3, x, 4)
+    assert r == -x**QQ(7,3)*y**2/288 + x**QQ(7,3)/12 + x**QQ(4,3)*y/12 + 2*x**QQ(1,3)
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', EX)
+    assert rs_nth_root(x + a, 3, x, 4) == EX(5/(81*a**QQ(8, 3)))*x**3 - \
+        EX(1/(9*a**QQ(5, 3)))*x**2 + EX(1/(3*a**QQ(2, 3)))*x + EX(a**QQ(1, 3))
+    assert rs_nth_root(x**QQ(2, 3) + x**2*y + 5, 2, x, 3) == -EX(sqrt(5)/100)*\
+        x**QQ(8, 3)*y - EX(sqrt(5)/16000)*x**QQ(8, 3) + EX(sqrt(5)/10)*x**2*y + \
+        EX(sqrt(5)/2000)*x**2 - EX(sqrt(5)/200)*x**QQ(4, 3) + \
+        EX(sqrt(5)/10)*x**QQ(2, 3) + EX(sqrt(5))
+
+def test_atan():
+    R, x, y = ring('x, y', QQ)
+    assert rs_atan(x, x, 9) == -x**7/7 + x**5/5 - x**3/3 + x
+    assert rs_atan(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 - x**8*y**9 + \
+        2*x**7*y**9 - x**7*y**7/7 - x**6*y**9/3 + x**6*y**7 - x**5*y**7 + \
+        x**5*y**5/5 - x**4*y**5 - x**3*y**3/3 + x**2*y**3 + x*y
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', EX)
+    assert rs_atan(x + a, x, 5) == -EX((a**3 - a)/(a**8 + 4*a**6 + 6*a**4 + \
+        4*a**2 + 1))*x**4 + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + \
+        9*a**2 + 3))*x**3 - EX(a/(a**4 + 2*a**2 + 1))*x**2 + \
+        EX(1/(a**2 + 1))*x + EX(atan(a))
+    assert rs_atan(x + x**2*y + a, x, 4) == -EX(2*a/(a**4 + 2*a**2 + 1)) \
+        *x**3*y + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + 9*a**2 + 3))*x**3 + \
+        EX(1/(a**2 + 1))*x**2*y - EX(a/(a**4 + 2*a**2 + 1))*x**2 + EX(1/(a**2 \
+        + 1))*x + EX(atan(a))
+
+def test_asin():
+    R, x, y = ring('x, y', QQ)
+    assert rs_asin(x + x*y, x, 5) == x**3*y**3/6 + x**3*y**2/2 + x**3*y/2 + \
+        x**3/6 + x*y + x
+    assert rs_asin(x*y + x**2*y**3, x, 6) == x**5*y**7/2 + 3*x**5*y**5/40 + \
+        x**4*y**5/2 + x**3*y**3/6 + x**2*y**3 + x*y
+
+def test_tan():
+    R, x, y = ring('x, y', QQ)
+    assert rs_tan(x, x, 9)/x**5 == \
+        Rational(17, 315)*x**2 + Rational(2, 15) + Rational(1, 3)*x**(-2) + x**(-4)
+    assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
+        4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
+        x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', QQ[tan(a), a])
+    assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 +
+        2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + Rational(1, 3))*x**3 + \
+        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
+    assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
+        (tan(a)**4 + Rational(4, 3)*tan(a)**2 + Rational(1, 3))*x**3 + (tan(a)**2 + 1)*x**2*y + \
+        (tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
+
+    R, x, y = ring('x, y', EX)
+    assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 +
+        2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
+        EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
+    assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 +
+        2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
+        EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
+        EX(tan(a)**2 + 1)*x + EX(tan(a))
+
+    p = x + x**2 + 5
+    assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \
+        668083460499)
+
+def test_cot():
+    R, x, y = ring('x, y', QQ)
+    assert rs_cot(x**6 + x**7, x, 8) == x**(-6) - x**(-5) + x**(-4) - \
+        x**(-3) + x**(-2) - x**(-1) + 1 - x + x**2 - x**3 + x**4 - x**5 + \
+        2*x**6/3 - 4*x**7/3
+    assert rs_cot(x + x**2*y, x, 5) == -x**4*y**5 - x**4*y/15 + x**3*y**4 - \
+        x**3/45 - x**2*y**3 - x**2*y/3 + x*y**2 - x/3 - y + x**(-1)
+
+def test_sin():
+    R, x, y = ring('x, y', QQ)
+    assert rs_sin(x, x, 9)/x**5 == \
+        Rational(-1, 5040)*x**2 + Rational(1, 120) - Rational(1, 6)*x**(-2) + x**(-4)
+    assert rs_sin(x*y + x**2*y**3, x, 9) == x**8*y**11/12 - \
+        x**8*y**9/720 + x**7*y**9/12 - x**7*y**7/5040 - x**6*y**9/6 + \
+        x**6*y**7/24 - x**5*y**7/2 + x**5*y**5/120 - x**4*y**5/2 - \
+        x**3*y**3/6 + x**2*y**3 + x*y
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
+    assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \
+        sin(a)*x**2/2 + cos(a)*x + sin(a)
+    assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \
+        cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \
+        cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a)
+
+    R, x, y = ring('x, y', EX)
+    assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \
+        EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a))
+    assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \
+        EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \
+        EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \
+        EX(cos(a))*x + EX(sin(a))
+
+def test_cos():
+    R, x, y = ring('x, y', QQ)
+    assert rs_cos(x, x, 9)/x**5 == \
+        Rational(1, 40320)*x**3 - Rational(1, 720)*x + Rational(1, 24)*x**(-1) - S.Half*x**(-3) + x**(-5)
+    assert rs_cos(x*y + x**2*y**3, x, 9) == x**8*y**12/24 - \
+        x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 - \
+        x**7*y**8/120 + x**6*y**8/4 - x**6*y**6/720 + x**5*y**6/6 - \
+        x**4*y**6/2 + x**4*y**4/24 - x**3*y**4 - x**2*y**2/2 + 1
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
+    assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \
+        cos(a)*x**2/2 - sin(a)*x + cos(a)
+    assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \
+        sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \
+        sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a)
+
+    R, x, y = ring('x, y', EX)
+    assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \
+        EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a))
+    assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \
+        EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \
+        EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \
+        EX(sin(a))*x + EX(cos(a))
+
+def test_cos_sin():
+    R, x, y = ring('x, y', QQ)
+    cos, sin = rs_cos_sin(x, x, 9)
+    assert cos == rs_cos(x, x, 9)
+    assert sin == rs_sin(x, x, 9)
+    cos, sin = rs_cos_sin(x + x*y, x, 5)
+    assert cos == rs_cos(x + x*y, x, 5)
+    assert sin == rs_sin(x + x*y, x, 5)
+
+def test_atanh():
+    R, x, y = ring('x, y', QQ)
+    assert rs_atanh(x, x, 9)/x**5 == Rational(1, 7)*x**2 + Rational(1, 5) + Rational(1, 3)*x**(-2) + x**(-4)
+    assert rs_atanh(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 + x**8*y**9 + \
+        2*x**7*y**9 + x**7*y**7/7 + x**6*y**9/3 + x**6*y**7 + x**5*y**7 + \
+        x**5*y**5/5 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', EX)
+    assert rs_atanh(x + a, x, 5) == EX((a**3 + a)/(a**8 - 4*a**6 + 6*a**4 - \
+        4*a**2 + 1))*x**4 - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + \
+        9*a**2 - 3))*x**3 + EX(a/(a**4 - 2*a**2 + 1))*x**2 - EX(1/(a**2 - \
+        1))*x + EX(atanh(a))
+    assert rs_atanh(x + x**2*y + a, x, 4) == EX(2*a/(a**4 - 2*a**2 + \
+        1))*x**3*y - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + 9*a**2 - 3))*x**3 - \
+        EX(1/(a**2 - 1))*x**2*y + EX(a/(a**4 - 2*a**2 + 1))*x**2 - \
+        EX(1/(a**2 - 1))*x + EX(atanh(a))
+
+    p = x + x**2 + 5
+    assert rs_atanh(p, x, 10).compose(x, 10) == EX(Rational(-733442653682135, 5079158784) \
+        + atanh(5))
+
+def test_sinh():
+    R, x, y = ring('x, y', QQ)
+    assert rs_sinh(x, x, 9)/x**5 == Rational(1, 5040)*x**2 + Rational(1, 120) + Rational(1, 6)*x**(-2) + x**(-4)
+    assert rs_sinh(x*y + x**2*y**3, x, 9) == x**8*y**11/12 + \
+        x**8*y**9/720 + x**7*y**9/12 + x**7*y**7/5040 + x**6*y**9/6 + \
+        x**6*y**7/24 + x**5*y**7/2 + x**5*y**5/120 + x**4*y**5/2 + \
+        x**3*y**3/6 + x**2*y**3 + x*y
+
+def test_cosh():
+    R, x, y = ring('x, y', QQ)
+    assert rs_cosh(x, x, 9)/x**5 == Rational(1, 40320)*x**3 + Rational(1, 720)*x + Rational(1, 24)*x**(-1) + \
+        S.Half*x**(-3) + x**(-5)
+    assert rs_cosh(x*y + x**2*y**3, x, 9) == x**8*y**12/24 + \
+        x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 + \
+        x**7*y**8/120 + x**6*y**8/4 + x**6*y**6/720 + x**5*y**6/6 + \
+        x**4*y**6/2 + x**4*y**4/24 + x**3*y**4 + x**2*y**2/2 + 1
+
+def test_tanh():
+    R, x, y = ring('x, y', QQ)
+    assert rs_tanh(x, x, 9)/x**5 == Rational(-17, 315)*x**2 + Rational(2, 15) - Rational(1, 3)*x**(-2) + x**(-4)
+    assert rs_tanh(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 - \
+        17*x**8*y**9/45 + 4*x**7*y**9/3 - 17*x**7*y**7/315 - x**6*y**9/3 + \
+        2*x**6*y**7/3 - x**5*y**7 + 2*x**5*y**5/15 - x**4*y**5 - \
+        x**3*y**3/3 + x**2*y**3 + x*y
+
+    # Constant term in series
+    a = symbols('a')
+    R, x, y = ring('x, y', EX)
+    assert rs_tanh(x + a, x, 5) == EX(tanh(a)**5 - 5*tanh(a)**3/3 +
+        2*tanh(a)/3)*x**4 + EX(-tanh(a)**4 + 4*tanh(a)**2/3 - QQ(1, 3))*x**3 + \
+        EX(tanh(a)**3 - tanh(a))*x**2 + EX(-tanh(a)**2 + 1)*x + EX(tanh(a))
+
+    p = rs_tanh(x + x**2*y + a, x, 4)
+    assert (p.compose(x, 10)).compose(y, 5) == EX(-1000*tanh(a)**4 + \
+        10100*tanh(a)**3 + 2470*tanh(a)**2/3 - 10099*tanh(a) + QQ(530, 3))
+
+def test_RR():
+    rs_funcs = [rs_sin, rs_cos, rs_tan, rs_cot, rs_atan, rs_tanh]
+    sympy_funcs = [sin, cos, tan, cot, atan, tanh]
+    R, x, y = ring('x, y', RR)
+    a = symbols('a')
+    for rs_func, sympy_func in zip(rs_funcs, sympy_funcs):
+        p = rs_func(2 + x, x, 5).compose(x, 5)
+        q = sympy_func(2 + a).series(a, 0, 5).removeO()
+        is_close(p.as_expr(), q.subs(a, 5).n())
+
+    p = rs_nth_root(2 + x, 5, x, 5).compose(x, 5)
+    q = ((2 + a)**QQ(1, 5)).series(a, 0, 5).removeO()
+    is_close(p.as_expr(), q.subs(a, 5).n())
+
+def test_is_regular():
+    R, x, y = ring('x, y', QQ)
+    p = 1 + 2*x + x**2 + 3*x**3
+    assert not rs_is_puiseux(p, x)
+
+    p = x + x**QQ(1,5)*y
+    assert rs_is_puiseux(p, x)
+    assert not rs_is_puiseux(p, y)
+
+    p = x + x**2*y**QQ(1,5)*y
+    assert not rs_is_puiseux(p, x)
+
+def test_puiseux():
+    R, x, y = ring('x, y', QQ)
+    p = x**QQ(2,5) + x**QQ(2,3) + x
+
+    r = rs_series_inversion(p, x, 1)
+    r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \
+        2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \
+        + x**QQ(-2,5)
+    assert r == r1
+
+    r = rs_nth_root(1 + p, 3, x, 1)
+    assert r == -x**QQ(4,5)/9 + x**QQ(2,3)/3 + x**QQ(2,5)/3 + 1
+
+    r = rs_log(1 + p, x, 1)
+    assert r == -x**QQ(4,5)/2 + x**QQ(2,3) + x**QQ(2,5)
+
+    r = rs_LambertW(p, x, 1)
+    assert r == -x**QQ(4,5) + x**QQ(2,3) + x**QQ(2,5)
+
+    p1 = x + x**QQ(1,5)*y
+    r = rs_exp(p1, x, 1)
+    assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \
+        x**QQ(1,5)*y + 1
+
+    r = rs_atan(p, x, 2)
+    assert r ==  -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
+        x + x**QQ(2,3) + x**QQ(2,5)
+
+    r = rs_atan(p1, x, 2)
+    assert r ==  x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \
+        x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y
+
+    r = rs_asin(p, x, 2)
+    assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
+        x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
+
+    r = rs_cot(p, x, 1)
+    assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \
+        2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \
+        x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5)
+
+    r = rs_cos_sin(p, x, 2)
+    assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \
+        x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1
+    assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \
+        x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
+
+    r = rs_atanh(p, x, 2)
+    assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \
+        x**QQ(2,3) + x**QQ(2,5)
+
+    r = rs_sinh(p, x, 2)
+    assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
+        x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
+
+    r = rs_cosh(p, x, 2)
+    assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \
+        x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1
+
+    r = rs_tanh(p, x, 2)
+    assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
+        x + x**QQ(2,3) + x**QQ(2,5)
+
+def test_puiseux_algebraic(): # https://github.com/sympy/sympy/issues/24395
+
+    K = QQ.algebraic_field(sqrt(2))
+    sqrt2 = K.from_sympy(sqrt(2))
+    x, y = symbols('x, y')
+    R, xr, yr = ring([x, y], K)
+    p = (1+sqrt2)*xr**QQ(1,2) + (1-sqrt2)*yr**QQ(2,3)
+
+    assert dict(p) == {(QQ(1,2),QQ(0)):1+sqrt2, (QQ(0),QQ(2,3)):1-sqrt2}
+    assert p.as_expr() == (1 + sqrt(2))*x**(S(1)/2) + (1 - sqrt(2))*y**(S(2)/3)
+
+
+def test1():
+    R, x = ring('x', QQ)
+    r = rs_sin(x, x, 15)*x**(-5)
+    assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \
+        QQ(1,120) - x**-2/6 + x**-4
+
+    p = rs_sin(x, x, 10)
+    r = rs_nth_root(p, 2, x, 10)
+    assert  r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \
+        x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2)
+
+    p = rs_sin(x, x, 10)
+    r = rs_nth_root(p, 7, x, 10)
+    r = rs_pow(r, 5, x, 10)
+    assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \
+        11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7)
+
+    r = rs_exp(x**QQ(1,2), x, 10)
+    assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \
+        x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \
+        x**QQ(15,2)/1307674368000 + x**7/87178291200 + \
+        x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \
+        x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \
+        x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \
+        x**QQ(1,2) + 1
+
+def test_puiseux2():
+    R, y = ring('y', QQ)
+    S, x = ring('x', R)
+
+    p = x + x**QQ(1,5)*y
+    r = rs_atan(p, x, 3)
+    assert r == (y**13/13 + y**8 + 2*y**3)*x**QQ(13,5) - (y**11/11 + y**6 +
+        y)*x**QQ(11,5) + (y**9/9 + y**4)*x**QQ(9,5) - (y**7/7 +
+        y**2)*x**QQ(7,5) + (y**5/5 + 1)*x - y**3*x**QQ(3,5)/3 + y*x**QQ(1,5)
+
+
+@slow
+def test_rs_series():
+    x, a, b, c = symbols('x, a, b, c')
+
+    assert rs_series(a, a, 5).as_expr() == a
+    assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
+        5)).removeO()
+    assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) +
+        cos(a)).series(a, 0, 5)).removeO()
+    assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)*
+        cos(a)).series(a, 0, 5)).removeO()
+
+    p = (sin(a) - a)*(cos(a**2) + a**4/2)
+    assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
+        10).removeO())
+
+    p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3
+    assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
+        5).removeO())
+
+    p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2)
+    assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
+        5).removeO())
+
+    p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2)
+    assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
+        10).removeO())
+
+    p = sin(a + b + c)
+    assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
+        5).removeO())
+
+    p = tan(sin(a**2 + 4) + b + c)
+    assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0,
+        6).removeO())
+
+    p = a**QQ(2,5) + a**QQ(2,3) + a
+
+    r = rs_series(tan(p), a, 2)
+    assert r.as_expr() == a**QQ(9,5) + a**QQ(26,15) + a**QQ(22,15) + a**QQ(6,5)/3 + \
+        a + a**QQ(2,3) + a**QQ(2,5)
+
+    r = rs_series(exp(p), a, 1)
+    assert r.as_expr() == a**QQ(4,5)/2 + a**QQ(2,3) + a**QQ(2,5) + 1
+
+    r = rs_series(sin(p), a, 2)
+    assert r.as_expr() == -a**QQ(9,5)/2 - a**QQ(26,15)/2 - a**QQ(22,15)/2 - \
+        a**QQ(6,5)/6 + a + a**QQ(2,3) + a**QQ(2,5)
+
+    r = rs_series(cos(p), a, 2)
+    assert r.as_expr() == a**QQ(28,15)/6 - a**QQ(5,3) + a**QQ(8,5)/24 - a**QQ(7,5) - \
+        a**QQ(4,3)/2 - a**QQ(16,15) - a**QQ(4,5)/2 + 1
+
+    assert rs_series(sin(a)/7, a, 5).as_expr() == (sin(a)/7).series(a, 0,
+            5).removeO()
+
+    assert rs_series(log(1 + x), x, 5).as_expr() == -x**4/4 + x**3/3 - \
+                    x**2/2 + x
+    assert rs_series(log(1 + 4*x), x, 5).as_expr() == -64*x**4 + 64*x**3/3 - \
+                    8*x**2 + 4*x
+    assert rs_series(log(1 + x + x**2), x, 10).as_expr() == -2*x**9/9 + \
+                    x**8/8 + x**7/7 - x**6/3 + x**5/5 + x**4/4 - 2*x**3/3 + \
+                    x**2/2 + x
+    assert rs_series(log(1 + x*a**2), x, 7).as_expr() == -x**6*a**12/6 + \
+                    x**5*a**10/5 - x**4*a**8/4 + x**3*a**6/3 - \
+                    x**2*a**4/2 + x*a**2

rl/Lib/site-packages/sympy/polys/tests/test_rootisolation.py

@@ -0,0 +1,823 @@
+"""Tests for real and complex root isolation and refinement algorithms. """
+
+from sympy.polys.rings import ring
+from sympy.polys.domains import ZZ, QQ, ZZ_I, EX
+from sympy.polys.polyerrors import DomainError, RefinementFailed, PolynomialError
+from sympy.polys.rootisolation import (
+    dup_cauchy_upper_bound, dup_cauchy_lower_bound,
+    dup_mignotte_sep_bound_squared,
+)
+from sympy.testing.pytest import raises
+
+def test_dup_sturm():
+    R, x = ring("x", QQ)
+
+    assert R.dup_sturm(5) == [1]
+    assert R.dup_sturm(x) == [x, 1]
+
+    f = x**3 - 2*x**2 + 3*x - 5
+    assert R.dup_sturm(f) == [f, 3*x**2 - 4*x + 3, -QQ(10,9)*x + QQ(13,3), -QQ(3303,100)]
+
+
+def test_dup_cauchy_upper_bound():
+    raises(PolynomialError, lambda: dup_cauchy_upper_bound([], QQ))
+    raises(PolynomialError, lambda: dup_cauchy_upper_bound([QQ(1)], QQ))
+    raises(DomainError, lambda: dup_cauchy_upper_bound([ZZ_I(1), ZZ_I(1)], ZZ_I))
+
+    assert dup_cauchy_upper_bound([QQ(1), QQ(0), QQ(0)], QQ) == QQ.zero
+    assert dup_cauchy_upper_bound([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(3)
+
+
+def test_dup_cauchy_lower_bound():
+    raises(PolynomialError, lambda: dup_cauchy_lower_bound([], QQ))
+    raises(PolynomialError, lambda: dup_cauchy_lower_bound([QQ(1)], QQ))
+    raises(PolynomialError, lambda: dup_cauchy_lower_bound([QQ(1), QQ(0), QQ(0)], QQ))
+    raises(DomainError, lambda: dup_cauchy_lower_bound([ZZ_I(1), ZZ_I(1)], ZZ_I))
+
+    assert dup_cauchy_lower_bound([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(2, 3)
+
+
+def test_dup_mignotte_sep_bound_squared():
+    raises(PolynomialError, lambda: dup_mignotte_sep_bound_squared([], QQ))
+    raises(PolynomialError, lambda: dup_mignotte_sep_bound_squared([QQ(1)], QQ))
+
+    assert dup_mignotte_sep_bound_squared([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(3, 5)
+
+
+def test_dup_refine_real_root():
+    R, x = ring("x", ZZ)
+    f = x**2 - 2
+
+    assert R.dup_refine_real_root(f, QQ(1), QQ(1), steps=1) == (QQ(1), QQ(1))
+    assert R.dup_refine_real_root(f, QQ(1), QQ(1), steps=9) == (QQ(1), QQ(1))
+
+    raises(ValueError, lambda: R.dup_refine_real_root(f, QQ(-2), QQ(2)))
+
+    s, t = QQ(1, 1), QQ(2, 1)
+
+    assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(2, 1))
+    assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(1, 1), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(4, 3), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(10, 7))
+
+    s, t = QQ(1, 1), QQ(3, 2)
+
+    assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(4, 3), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(7, 5), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(10, 7))
+    assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(17, 12))
+
+    s, t = QQ(1, 1), QQ(5, 3)
+
+    assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(5, 3))
+    assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(1, 1), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(7, 5), QQ(3, 2))
+    assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(13, 9))
+    assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(27, 19))
+
+    s, t = QQ(-1, 1), QQ(-2, 1)
+
+    assert R.dup_refine_real_root(f, s, t, steps=0) == (-QQ(2, 1), -QQ(1, 1))
+    assert R.dup_refine_real_root(f, s, t, steps=1) == (-QQ(3, 2), -QQ(1, 1))
+    assert R.dup_refine_real_root(f, s, t, steps=2) == (-QQ(3, 2), -QQ(4, 3))
+    assert R.dup_refine_real_root(f, s, t, steps=3) == (-QQ(3, 2), -QQ(7, 5))
+    assert R.dup_refine_real_root(f, s, t, steps=4) == (-QQ(10, 7), -QQ(7, 5))
+
+    raises(RefinementFailed, lambda: R.dup_refine_real_root(f, QQ(0), QQ(1)))
+
+    s, t, u, v, w = QQ(1), QQ(2), QQ(24, 17), QQ(17, 12), QQ(7, 5)
+
+    assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100)) == (u, v)
+    assert R.dup_refine_real_root(f, s, t, steps=6) == (u, v)
+
+    assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=5) == (w, v)
+    assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=6) == (u, v)
+    assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=7) == (u, v)
+
+    s, t, u, v = QQ(-2), QQ(-1), QQ(-3, 2), QQ(-4, 3)
+
+    assert R.dup_refine_real_root(f, s, t, disjoint=QQ(-5)) == (s, t)
+    assert R.dup_refine_real_root(f, s, t, disjoint=-v) == (s, t)
+    assert R.dup_refine_real_root(f, s, t, disjoint=v) == (u, v)
+
+    s, t, u, v = QQ(1), QQ(2), QQ(4, 3), QQ(3, 2)
+
+    assert R.dup_refine_real_root(f, s, t, disjoint=QQ(5)) == (s, t)
+    assert R.dup_refine_real_root(f, s, t, disjoint=-u) == (s, t)
+    assert R.dup_refine_real_root(f, s, t, disjoint=u) == (u, v)
+
+
+def test_dup_isolate_real_roots_sqf():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_isolate_real_roots_sqf(0) == []
+    assert R.dup_isolate_real_roots_sqf(5) == []
+
+    assert R.dup_isolate_real_roots_sqf(x**2 + x) == [(-1, -1), (0, 0)]
+    assert R.dup_isolate_real_roots_sqf(x**2 - x) == [( 0,  0), (1, 1)]
+
+    assert R.dup_isolate_real_roots_sqf(x**4 + x + 1) == []
+
+    I = [(-2, -1), (1, 2)]
+
+    assert R.dup_isolate_real_roots_sqf(x**2 - 2) == I
+    assert R.dup_isolate_real_roots_sqf(-x**2 + 2) == I
+
+    assert R.dup_isolate_real_roots_sqf(x - 1) == \
+        [(1, 1)]
+    assert R.dup_isolate_real_roots_sqf(x**2 - 3*x + 2) == \
+        [(1, 1), (2, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**3 - 6*x**2 + 11*x - 6) == \
+        [(1, 1), (2, 2), (3, 3)]
+    assert R.dup_isolate_real_roots_sqf(x**4 - 10*x**3 + 35*x**2 - 50*x + 24) == \
+        [(1, 1), (2, 2), (3, 3), (4, 4)]
+    assert R.dup_isolate_real_roots_sqf(x**5 - 15*x**4 + 85*x**3 - 225*x**2 + 274*x - 120) == \
+        [(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)]
+
+    assert R.dup_isolate_real_roots_sqf(x - 10) == \
+        [(10, 10)]
+    assert R.dup_isolate_real_roots_sqf(x**2 - 30*x + 200) == \
+        [(10, 10), (20, 20)]
+    assert R.dup_isolate_real_roots_sqf(x**3 - 60*x**2 + 1100*x - 6000) == \
+        [(10, 10), (20, 20), (30, 30)]
+    assert R.dup_isolate_real_roots_sqf(x**4 - 100*x**3 + 3500*x**2 - 50000*x + 240000) == \
+        [(10, 10), (20, 20), (30, 30), (40, 40)]
+    assert R.dup_isolate_real_roots_sqf(x**5 - 150*x**4 + 8500*x**3 - 225000*x**2 + 2740000*x - 12000000) == \
+        [(10, 10), (20, 20), (30, 30), (40, 40), (50, 50)]
+
+    assert R.dup_isolate_real_roots_sqf(x + 1) == \
+        [(-1, -1)]
+    assert R.dup_isolate_real_roots_sqf(x**2 + 3*x + 2) == \
+        [(-2, -2), (-1, -1)]
+    assert R.dup_isolate_real_roots_sqf(x**3 + 6*x**2 + 11*x + 6) == \
+        [(-3, -3), (-2, -2), (-1, -1)]
+    assert R.dup_isolate_real_roots_sqf(x**4 + 10*x**3 + 35*x**2 + 50*x + 24) == \
+        [(-4, -4), (-3, -3), (-2, -2), (-1, -1)]
+    assert R.dup_isolate_real_roots_sqf(x**5 + 15*x**4 + 85*x**3 + 225*x**2 + 274*x + 120) == \
+        [(-5, -5), (-4, -4), (-3, -3), (-2, -2), (-1, -1)]
+
+    assert R.dup_isolate_real_roots_sqf(x + 10) == \
+        [(-10, -10)]
+    assert R.dup_isolate_real_roots_sqf(x**2 + 30*x + 200) == \
+        [(-20, -20), (-10, -10)]
+    assert R.dup_isolate_real_roots_sqf(x**3 + 60*x**2 + 1100*x + 6000) == \
+        [(-30, -30), (-20, -20), (-10, -10)]
+    assert R.dup_isolate_real_roots_sqf(x**4 + 100*x**3 + 3500*x**2 + 50000*x + 240000) == \
+        [(-40, -40), (-30, -30), (-20, -20), (-10, -10)]
+    assert R.dup_isolate_real_roots_sqf(x**5 + 150*x**4 + 8500*x**3 + 225000*x**2 + 2740000*x + 12000000) == \
+        [(-50, -50), (-40, -40), (-30, -30), (-20, -20), (-10, -10)]
+
+    assert R.dup_isolate_real_roots_sqf(x**2 - 5) == [(-3, -2), (2, 3)]
+    assert R.dup_isolate_real_roots_sqf(x**3 - 5) == [(1, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**4 - 5) == [(-2, -1), (1, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**5 - 5) == [(1, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**6 - 5) == [(-2, -1), (1, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**7 - 5) == [(1, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**8 - 5) == [(-2, -1), (1, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**9 - 5) == [(1, 2)]
+
+    assert R.dup_isolate_real_roots_sqf(x**2 - 1) == \
+        [(-1, -1), (1, 1)]
+    assert R.dup_isolate_real_roots_sqf(x**3 + 2*x**2 - x - 2) == \
+        [(-2, -2), (-1, -1), (1, 1)]
+    assert R.dup_isolate_real_roots_sqf(x**4 - 5*x**2 + 4) == \
+        [(-2, -2), (-1, -1), (1, 1), (2, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**5 + 3*x**4 - 5*x**3 - 15*x**2 + 4*x + 12) == \
+        [(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2)]
+    assert R.dup_isolate_real_roots_sqf(x**6 - 14*x**4 + 49*x**2 - 36) == \
+        [(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2), (3, 3)]
+    assert R.dup_isolate_real_roots_sqf(2*x**7 + x**6 - 28*x**5 - 14*x**4 + 98*x**3 + 49*x**2 - 72*x - 36) == \
+        [(-3, -3), (-2, -2), (-1, -1), (-1, 0), (1, 1), (2, 2), (3, 3)]
+    assert R.dup_isolate_real_roots_sqf(4*x**8 - 57*x**6 + 210*x**4 - 193*x**2 + 36) == \
+        [(-3, -3), (-2, -2), (-1, -1), (-1, 0), (0, 1), (1, 1), (2, 2), (3, 3)]
+
+    f = 9*x**2 - 2
+
+    assert R.dup_isolate_real_roots_sqf(f) == \
+        [(-1, 0), (0, 1)]
+
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10)) == \
+        [(QQ(-1, 2), QQ(-3, 7)), (QQ(3, 7), QQ(1, 2))]
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \
+        [(QQ(-9, 19), QQ(-8, 17)), (QQ(8, 17), QQ(9, 19))]
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000)) == \
+        [(QQ(-33, 70), QQ(-8, 17)), (QQ(8, 17), QQ(33, 70))]
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10000)) == \
+        [(QQ(-33, 70), QQ(-107, 227)), (QQ(107, 227), QQ(33, 70))]
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000)) == \
+        [(QQ(-305, 647), QQ(-272, 577)), (QQ(272, 577), QQ(305, 647))]
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000000)) == \
+        [(QQ(-1121, 2378), QQ(-272, 577)), (QQ(272, 577), QQ(1121, 2378))]
+
+    f = 200100012*x**5 - 700390052*x**4 + 700490079*x**3 - 200240054*x**2 + 40017*x - 2
+
+    assert R.dup_isolate_real_roots_sqf(f) == \
+        [(QQ(0), QQ(1, 10002)), (QQ(1, 10002), QQ(1, 10002)),
+         (QQ(1, 2), QQ(1, 2)), (QQ(1), QQ(1)), (QQ(2), QQ(2))]
+
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000)) == \
+        [(QQ(1, 10003), QQ(1, 10003)), (QQ(1, 10002), QQ(1, 10002)),
+         (QQ(1, 2), QQ(1, 2)), (QQ(1), QQ(1)), (QQ(2), QQ(2))]
+
+    a, b, c, d = 10000090000001, 2000100003, 10000300007, 10000005000008
+
+    f = 20001600074001600021*x**4 \
+      + 1700135866278935491773999857*x**3 \
+      - 2000179008931031182161141026995283662899200197*x**2 \
+      - 800027600594323913802305066986600025*x \
+      + 100000950000540000725000008
+
+    assert R.dup_isolate_real_roots_sqf(f) == \
+        [(-a, -a), (-1, 0), (0, 1), (d, d)]
+
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000000000)) == \
+        [(-QQ(a), -QQ(a)), (-QQ(1, b), -QQ(1, b)), (QQ(1, c), QQ(1, c)), (QQ(d), QQ(d))]
+
+    (u, v), B, C, (s, t) = R.dup_isolate_real_roots_sqf(f, fast=True)
+
+    assert u < -a < v and B == (-QQ(1), QQ(0)) and C == (QQ(0), QQ(1)) and s < d < t
+
+    assert R.dup_isolate_real_roots_sqf(f, fast=True, eps=QQ(1, 100000000000000000000000000000)) == \
+        [(-QQ(a), -QQ(a)), (-QQ(1, b), -QQ(1, b)), (QQ(1, c), QQ(1, c)), (QQ(d), QQ(d))]
+
+    f = -10*x**4 + 8*x**3 + 80*x**2 - 32*x - 160
+
+    assert R.dup_isolate_real_roots_sqf(f) == \
+        [(-2, -2), (-2, -1), (2, 2), (2, 3)]
+
+    assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \
+        [(-QQ(2), -QQ(2)), (-QQ(23, 14), -QQ(18, 11)), (QQ(2), QQ(2)), (QQ(39, 16), QQ(22, 9))]
+
+    f = x - 1
+
+    assert R.dup_isolate_real_roots_sqf(f, inf=2) == []
+    assert R.dup_isolate_real_roots_sqf(f, sup=0) == []
+
+    assert R.dup_isolate_real_roots_sqf(f) == [(1, 1)]
+    assert R.dup_isolate_real_roots_sqf(f, inf=1) == [(1, 1)]
+    assert R.dup_isolate_real_roots_sqf(f, sup=1) == [(1, 1)]
+    assert R.dup_isolate_real_roots_sqf(f, inf=1, sup=1) == [(1, 1)]
+
+    f = x**2 - 2
+
+    assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 4)) == []
+    assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 5)) == [(QQ(7, 5), QQ(3, 2))]
+    assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 5)) == [(-2, -1)]
+    assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 4)) == [(-2, -1), (1, QQ(3, 2))]
+    assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 4)) == []
+    assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 5)) == [(-QQ(3, 2), -QQ(7, 5))]
+    assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 5)) == [(1, 2)]
+    assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 4)) == [(-QQ(3, 2), -1), (1, 2)]
+
+    I = [(-2, -1), (1, 2)]
+
+    assert R.dup_isolate_real_roots_sqf(f, inf=-2) == I
+    assert R.dup_isolate_real_roots_sqf(f, sup=+2) == I
+
+    assert R.dup_isolate_real_roots_sqf(f, inf=-2, sup=2) == I
+
+    R, x = ring("x", QQ)
+    f = QQ(8, 5)*x**2 - QQ(87374, 3855)*x - QQ(17, 771)
+
+    assert R.dup_isolate_real_roots_sqf(f) == [(-1, 0), (14, 15)]
+
+    R, x = ring("x", EX)
+    raises(DomainError, lambda: R.dup_isolate_real_roots_sqf(x + 3))
+
+def test_dup_isolate_real_roots():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_isolate_real_roots(0) == []
+    assert R.dup_isolate_real_roots(3) == []
+
+    assert R.dup_isolate_real_roots(5*x) == [((0, 0), 1)]
+    assert R.dup_isolate_real_roots(7*x**4) == [((0, 0), 4)]
+
+    assert R.dup_isolate_real_roots(x**2 + x) == [((-1, -1), 1), ((0, 0), 1)]
+    assert R.dup_isolate_real_roots(x**2 - x) == [((0, 0), 1), ((1, 1), 1)]
+
+    assert R.dup_isolate_real_roots(x**4 + x + 1) == []
+
+    I = [((-2, -1), 1), ((1, 2), 1)]
+
+    assert R.dup_isolate_real_roots(x**2 - 2) == I
+    assert R.dup_isolate_real_roots(-x**2 + 2) == I
+
+    f = 16*x**14 - 96*x**13 + 24*x**12 + 936*x**11 - 1599*x**10 - 2880*x**9 + 9196*x**8 \
+      + 552*x**7 - 21831*x**6 + 13968*x**5 + 21690*x**4 - 26784*x**3 - 2916*x**2 + 15552*x - 5832
+    g = R.dup_sqf_part(f)
+
+    assert R.dup_isolate_real_roots(f) == \
+        [((-QQ(2), -QQ(3, 2)), 2), ((-QQ(3, 2), -QQ(1, 1)), 3), ((QQ(1), QQ(3, 2)), 3),
+         ((QQ(3, 2), QQ(3, 2)), 4), ((QQ(5, 3), QQ(2)), 2)]
+
+    assert R.dup_isolate_real_roots_sqf(g) == \
+        [(-QQ(2), -QQ(3, 2)), (-QQ(3, 2), -QQ(1, 1)), (QQ(1), QQ(3, 2)),
+         (QQ(3, 2), QQ(3, 2)), (QQ(3, 2), QQ(2))]
+    assert R.dup_isolate_real_roots(g) == \
+        [((-QQ(2), -QQ(3, 2)), 1), ((-QQ(3, 2), -QQ(1, 1)), 1), ((QQ(1), QQ(3, 2)), 1),
+         ((QQ(3, 2), QQ(3, 2)), 1), ((QQ(3, 2), QQ(2)), 1)]
+
+    f = x - 1
+
+    assert R.dup_isolate_real_roots(f, inf=2) == []
+    assert R.dup_isolate_real_roots(f, sup=0) == []
+
+    assert R.dup_isolate_real_roots(f) == [((1, 1), 1)]
+    assert R.dup_isolate_real_roots(f, inf=1) == [((1, 1), 1)]
+    assert R.dup_isolate_real_roots(f, sup=1) == [((1, 1), 1)]
+    assert R.dup_isolate_real_roots(f, inf=1, sup=1) == [((1, 1), 1)]
+
+    f = x**4 - 4*x**2 + 4
+
+    assert R.dup_isolate_real_roots(f, inf=QQ(7, 4)) == []
+    assert R.dup_isolate_real_roots(f, inf=QQ(7, 5)) == [((QQ(7, 5), QQ(3, 2)), 2)]
+    assert R.dup_isolate_real_roots(f, sup=QQ(7, 5)) == [((-2, -1), 2)]
+    assert R.dup_isolate_real_roots(f, sup=QQ(7, 4)) == [((-2, -1), 2), ((1, QQ(3, 2)), 2)]
+    assert R.dup_isolate_real_roots(f, sup=-QQ(7, 4)) == []
+    assert R.dup_isolate_real_roots(f, sup=-QQ(7, 5)) == [((-QQ(3, 2), -QQ(7, 5)), 2)]
+    assert R.dup_isolate_real_roots(f, inf=-QQ(7, 5)) == [((1, 2), 2)]
+    assert R.dup_isolate_real_roots(f, inf=-QQ(7, 4)) == [((-QQ(3, 2), -1), 2), ((1, 2), 2)]
+
+    I = [((-2, -1), 2), ((1, 2), 2)]
+
+    assert R.dup_isolate_real_roots(f, inf=-2) == I
+    assert R.dup_isolate_real_roots(f, sup=+2) == I
+
+    assert R.dup_isolate_real_roots(f, inf=-2, sup=2) == I
+
+    f = x**11 - 3*x**10 - x**9 + 11*x**8 - 8*x**7 - 8*x**6 + 12*x**5 - 4*x**4
+
+    assert R.dup_isolate_real_roots(f, basis=False) == \
+        [((-2, -1), 2), ((0, 0), 4), ((1, 1), 3), ((1, 2), 2)]
+    assert R.dup_isolate_real_roots(f, basis=True) == \
+        [((-2, -1), 2, [1, 0, -2]), ((0, 0), 4, [1, 0]), ((1, 1), 3, [1, -1]), ((1, 2), 2, [1, 0, -2])]
+
+    f = (x**45 - 45*x**44 + 990*x**43 - 1)
+    g = (x**46 - 15180*x**43 + 9366819*x**40 - 53524680*x**39 + 260932815*x**38 - 1101716330*x**37 + 4076350421*x**36 - 13340783196*x**35 + 38910617655*x**34 - 101766230790*x**33 + 239877544005*x**32 - 511738760544*x**31 + 991493848554*x**30 - 1749695026860*x**29 + 2818953098830*x**28 - 4154246671960*x**27 + 5608233007146*x**26 - 6943526580276*x**25 + 7890371113950*x**24 - 8233430727600*x**23 + 7890371113950*x**22 - 6943526580276*x**21 + 5608233007146*x**20 - 4154246671960*x**19 + 2818953098830*x**18 - 1749695026860*x**17 + 991493848554*x**16 - 511738760544*x**15 + 239877544005*x**14 - 101766230790*x**13 + 38910617655*x**12 - 13340783196*x**11 + 4076350421*x**10 - 1101716330*x**9 + 260932815*x**8 - 53524680*x**7 + 9366819*x**6 - 1370754*x**5 + 163185*x**4 - 15180*x**3 + 1035*x**2 - 47*x + 1)
+
+    assert R.dup_isolate_real_roots(f*g) == \
+        [((0, QQ(1, 2)), 1), ((QQ(2, 3), QQ(3, 4)), 1), ((QQ(3, 4), 1), 1), ((6, 7), 1), ((24, 25), 1)]
+
+    R, x = ring("x", EX)
+    raises(DomainError, lambda: R.dup_isolate_real_roots(x + 3))
+
+
+def test_dup_isolate_real_roots_list():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_isolate_real_roots_list([x**2 + x, x]) == \
+        [((-1, -1), {0: 1}), ((0, 0), {0: 1, 1: 1})]
+    assert R.dup_isolate_real_roots_list([x**2 - x, x]) == \
+        [((0, 0), {0: 1, 1: 1}), ((1, 1), {0: 1})]
+
+    assert R.dup_isolate_real_roots_list([x + 1, x + 2, x - 1, x + 1, x - 1, x - 1]) == \
+        [((-QQ(2), -QQ(2)), {1: 1}), ((-QQ(1), -QQ(1)), {0: 1, 3: 1}), ((QQ(1), QQ(1)), {2: 1, 4: 1, 5: 1})]
+
+    assert R.dup_isolate_real_roots_list([x + 1, x + 2, x - 1, x + 1, x - 1, x + 2]) == \
+        [((-QQ(2), -QQ(2)), {1: 1, 5: 1}), ((-QQ(1), -QQ(1)), {0: 1, 3: 1}), ((QQ(1), QQ(1)), {2: 1, 4: 1})]
+
+    f, g = x**4 - 4*x**2 + 4, x - 1
+
+    assert R.dup_isolate_real_roots_list([f, g], inf=QQ(7, 4)) == []
+    assert R.dup_isolate_real_roots_list([f, g], inf=QQ(7, 5)) == \
+        [((QQ(7, 5), QQ(3, 2)), {0: 2})]
+    assert R.dup_isolate_real_roots_list([f, g], sup=QQ(7, 5)) == \
+        [((-2, -1), {0: 2}), ((1, 1), {1: 1})]
+    assert R.dup_isolate_real_roots_list([f, g], sup=QQ(7, 4)) == \
+        [((-2, -1), {0: 2}), ((1, 1), {1: 1}), ((1, QQ(3, 2)), {0: 2})]
+    assert R.dup_isolate_real_roots_list([f, g], sup=-QQ(7, 4)) == []
+    assert R.dup_isolate_real_roots_list([f, g], sup=-QQ(7, 5)) == \
+        [((-QQ(3, 2), -QQ(7, 5)), {0: 2})]
+    assert R.dup_isolate_real_roots_list([f, g], inf=-QQ(7, 5)) == \
+        [((1, 1), {1: 1}), ((1, 2), {0: 2})]
+    assert R.dup_isolate_real_roots_list([f, g], inf=-QQ(7, 4)) == \
+        [((-QQ(3, 2), -1), {0: 2}), ((1, 1), {1: 1}), ((1, 2), {0: 2})]
+
+    f, g = 2*x**2 - 1, x**2 - 2
+
+    assert R.dup_isolate_real_roots_list([f, g]) == \
+        [((-QQ(2), -QQ(1)), {1: 1}), ((-QQ(1), QQ(0)), {0: 1}),
+         ((QQ(0), QQ(1)), {0: 1}), ((QQ(1), QQ(2)), {1: 1})]
+    assert R.dup_isolate_real_roots_list([f, g], strict=True) == \
+        [((-QQ(3, 2), -QQ(4, 3)), {1: 1}), ((-QQ(1), -QQ(2, 3)), {0: 1}),
+         ((QQ(2, 3), QQ(1)), {0: 1}), ((QQ(4, 3), QQ(3, 2)), {1: 1})]
+
+    f, g = x**2 - 2, x**3 - x**2 - 2*x + 2
+
+    assert R.dup_isolate_real_roots_list([f, g]) == \
+        [((-QQ(2), -QQ(1)), {1: 1, 0: 1}), ((QQ(1), QQ(1)), {1: 1}), ((QQ(1), QQ(2)), {1: 1, 0: 1})]
+
+    f, g = x**3 - 2*x, x**5 - x**4 - 2*x**3 + 2*x**2
+
+    assert R.dup_isolate_real_roots_list([f, g]) == \
+        [((-QQ(2), -QQ(1)), {1: 1, 0: 1}), ((QQ(0), QQ(0)), {0: 1, 1: 2}),
+         ((QQ(1), QQ(1)), {1: 1}), ((QQ(1), QQ(2)), {1: 1, 0: 1})]
+
+    f, g = x**9 - 3*x**8 - x**7 + 11*x**6 - 8*x**5 - 8*x**4 + 12*x**3 - 4*x**2, x**5 - 2*x**4 + 3*x**3 - 4*x**2 + 2*x
+
+    assert R.dup_isolate_real_roots_list([f, g], basis=False) == \
+        [((-2, -1), {0: 2}), ((0, 0), {0: 2, 1: 1}), ((1, 1), {0: 3, 1: 2}), ((1, 2), {0: 2})]
+    assert R.dup_isolate_real_roots_list([f, g], basis=True) == \
+        [((-2, -1), {0: 2}, [1, 0, -2]), ((0, 0), {0: 2, 1: 1}, [1, 0]),
+         ((1, 1), {0: 3, 1: 2}, [1, -1]), ((1, 2), {0: 2}, [1, 0, -2])]
+
+    R, x = ring("x", EX)
+    raises(DomainError, lambda: R.dup_isolate_real_roots_list([x + 3]))
+
+
+def test_dup_isolate_real_roots_list_QQ():
+    R, x = ring("x", ZZ)
+
+    f = x**5 - 200
+    g = x**5 - 201
+
+    assert R.dup_isolate_real_roots_list([f, g]) == \
+        [((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})]
+
+    R, x = ring("x", QQ)
+
+    f = -QQ(1, 200)*x**5 + 1
+    g = -QQ(1, 201)*x**5 + 1
+
+    assert R.dup_isolate_real_roots_list([f, g]) == \
+        [((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})]
+
+
+def test_dup_count_real_roots():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_count_real_roots(0) == 0
+    assert R.dup_count_real_roots(7) == 0
+
+    f = x - 1
+    assert R.dup_count_real_roots(f) == 1
+    assert R.dup_count_real_roots(f, inf=1) == 1
+    assert R.dup_count_real_roots(f, sup=0) == 0
+    assert R.dup_count_real_roots(f, sup=1) == 1
+    assert R.dup_count_real_roots(f, inf=0, sup=1) == 1
+    assert R.dup_count_real_roots(f, inf=0, sup=2) == 1
+    assert R.dup_count_real_roots(f, inf=1, sup=2) == 1
+
+    f = x**2 - 2
+    assert R.dup_count_real_roots(f) == 2
+    assert R.dup_count_real_roots(f, sup=0) == 1
+    assert R.dup_count_real_roots(f, inf=-1, sup=1) == 0
+
+
+# parameters for test_dup_count_complex_roots_n(): n = 1..8
+a, b = (-QQ(1), -QQ(1)), (QQ(1), QQ(1))
+c, d = ( QQ(0),  QQ(0)), (QQ(1), QQ(1))
+
+def test_dup_count_complex_roots_1():
+    R, x = ring("x", ZZ)
+
+    # z-1
+    f = x - 1
+    assert R.dup_count_complex_roots(f, a, b) == 1
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # z+1
+    f = x + 1
+    assert R.dup_count_complex_roots(f, a, b) == 1
+    assert R.dup_count_complex_roots(f, c, d) == 0
+
+
+def test_dup_count_complex_roots_2():
+    R, x = ring("x", ZZ)
+
+    # (z-1)*(z)
+    f = x**2 - x
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-1)*(-z)
+    f = -x**2 + x
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z+1)*(z)
+    f = x**2 + x
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z+1)*(-z)
+    f = -x**2 - x
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+
+def test_dup_count_complex_roots_3():
+    R, x = ring("x", ZZ)
+
+    # (z-1)*(z+1)
+    f = x**2 - 1
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-1)*(z+1)*(z)
+    f = x**3 - x
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-1)*(z+1)*(-z)
+    f = -x**3 + x
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+
+def test_dup_count_complex_roots_4():
+    R, x = ring("x", ZZ)
+
+    # (z-I)*(z+I)
+    f = x**2 + 1
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I)*(z+I)*(z)
+    f = x**3 + x
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I)*(z+I)*(-z)
+    f = -x**3 - x
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I)*(z+I)*(z-1)
+    f = x**3 - x**2 + x - 1
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I)*(z+I)*(z-1)*(z)
+    f = x**4 - x**3 + x**2 - x
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+    # (z-I)*(z+I)*(z-1)*(-z)
+    f = -x**4 + x**3 - x**2 + x
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+    # (z-I)*(z+I)*(z-1)*(z+1)
+    f = x**4 - 1
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I)*(z+I)*(z-1)*(z+1)*(z)
+    f = x**5 - x
+    assert R.dup_count_complex_roots(f, a, b) == 5
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+    # (z-I)*(z+I)*(z-1)*(z+1)*(-z)
+    f = -x**5 + x
+    assert R.dup_count_complex_roots(f, a, b) == 5
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+
+def test_dup_count_complex_roots_5():
+    R, x = ring("x", ZZ)
+
+    # (z-I+1)*(z+I+1)
+    f = x**2 + 2*x + 2
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 0
+
+    # (z-I+1)*(z+I+1)*(z-1)
+    f = x**3 + x**2 - 2
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I+1)*(z+I+1)*(z-1)*z
+    f = x**4 + x**3 - 2*x
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I+1)*(z+I+1)*(z+1)
+    f = x**3 + 3*x**2 + 4*x + 2
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 0
+
+    # (z-I+1)*(z+I+1)*(z+1)*z
+    f = x**4 + 3*x**3 + 4*x**2 + 2*x
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I+1)*(z+I+1)*(z-1)*(z+1)
+    f = x**4 + 2*x**3 + x**2 - 2*x - 2
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I+1)*(z+I+1)*(z-1)*(z+1)*z
+    f = x**5 + 2*x**4 + x**3 - 2*x**2 - 2*x
+    assert R.dup_count_complex_roots(f, a, b) == 5
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+
+def test_dup_count_complex_roots_6():
+    R, x = ring("x", ZZ)
+
+    # (z-I-1)*(z+I-1)
+    f = x**2 - 2*x + 2
+    assert R.dup_count_complex_roots(f, a, b) == 2
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I-1)*(z+I-1)*(z-1)
+    f = x**3 - 3*x**2 + 4*x - 2
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I-1)*(z+I-1)*(z-1)*z
+    f = x**4 - 3*x**3 + 4*x**2 - 2*x
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+    # (z-I-1)*(z+I-1)*(z+1)
+    f = x**3 - x**2 + 2
+    assert R.dup_count_complex_roots(f, a, b) == 3
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I-1)*(z+I-1)*(z+1)*z
+    f = x**4 - x**3 + 2*x
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I-1)*(z+I-1)*(z-1)*(z+1)
+    f = x**4 - 2*x**3 + x**2 + 2*x - 2
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I-1)*(z+I-1)*(z-1)*(z+1)*z
+    f = x**5 - 2*x**4 + x**3 + 2*x**2 - 2*x
+    assert R.dup_count_complex_roots(f, a, b) == 5
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+
+def test_dup_count_complex_roots_7():
+    R, x = ring("x", ZZ)
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)
+    f = x**4 + 4
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-2)
+    f = x**5 - 2*x**4 + 4*x - 8
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z**2-2)
+    f = x**6 - 2*x**4 + 4*x**2 - 8
+    assert R.dup_count_complex_roots(f, a, b) == 4
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)
+    f = x**5 - x**4 + 4*x - 4
+    assert R.dup_count_complex_roots(f, a, b) == 5
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*z
+    f = x**6 - x**5 + 4*x**2 - 4*x
+    assert R.dup_count_complex_roots(f, a, b) == 6
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z+1)
+    f = x**5 + x**4 + 4*x + 4
+    assert R.dup_count_complex_roots(f, a, b) == 5
+    assert R.dup_count_complex_roots(f, c, d) == 1
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z+1)*z
+    f = x**6 + x**5 + 4*x**2 + 4*x
+    assert R.dup_count_complex_roots(f, a, b) == 6
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)
+    f = x**6 - x**4 + 4*x**2 - 4
+    assert R.dup_count_complex_roots(f, a, b) == 6
+    assert R.dup_count_complex_roots(f, c, d) == 2
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*z
+    f = x**7 - x**5 + 4*x**3 - 4*x
+    assert R.dup_count_complex_roots(f, a, b) == 7
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)
+    f = x**8 + 3*x**4 - 4
+    assert R.dup_count_complex_roots(f, a, b) == 8
+    assert R.dup_count_complex_roots(f, c, d) == 3
+
+
+def test_dup_count_complex_roots_8():
+    R, x = ring("x", ZZ)
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)*z
+    f = x**9 + 3*x**5 - 4*x
+    assert R.dup_count_complex_roots(f, a, b) == 9
+    assert R.dup_count_complex_roots(f, c, d) == 4
+
+    # (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)*(z**2-2)*z
+    f = x**11 - 2*x**9 + 3*x**7 - 6*x**5 - 4*x**3 + 8*x
+    assert R.dup_count_complex_roots(f, a, b) == 9
+    assert R.dup_count_complex_roots(f, c, d) == 4
+
+
+def test_dup_count_complex_roots_implicit():
+    R, x = ring("x", ZZ)
+
+    # z*(z-1)*(z+1)*(z-I)*(z+I)
+    f = x**5 - x
+
+    assert R.dup_count_complex_roots(f) == 5
+
+    assert R.dup_count_complex_roots(f, sup=(0, 0)) == 3
+    assert R.dup_count_complex_roots(f, inf=(0, 0)) == 3
+
+
+def test_dup_count_complex_roots_exclude():
+    R, x = ring("x", ZZ)
+
+    # z*(z-1)*(z+1)*(z-I)*(z+I)
+    f = x**5 - x
+
+    a, b = (-QQ(1), QQ(0)), (QQ(1), QQ(1))
+
+    assert R.dup_count_complex_roots(f, a, b) == 4
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=['S']) == 3
+    assert R.dup_count_complex_roots(f, a, b, exclude=['N']) == 3
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=['S', 'N']) == 2
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=['E']) == 4
+    assert R.dup_count_complex_roots(f, a, b, exclude=['W']) == 4
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=['E', 'W']) == 4
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=['N', 'S', 'E', 'W']) == 2
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=['SW']) == 3
+    assert R.dup_count_complex_roots(f, a, b, exclude=['SE']) == 3
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE']) == 2
+    assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S']) == 1
+    assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S', 'N']) == 0
+
+    a, b = (QQ(0), QQ(0)), (QQ(1), QQ(1))
+
+    assert R.dup_count_complex_roots(f, a, b, exclude=True) == 1
+
+
+def test_dup_isolate_complex_roots_sqf():
+    R, x = ring("x", ZZ)
+    f = x**2 - 2*x + 3
+
+    assert R.dup_isolate_complex_roots_sqf(f) == \
+        [((0, -6), (6, 0)), ((0, 0), (6, 6))]
+    assert [ r.as_tuple() for r in R.dup_isolate_complex_roots_sqf(f, blackbox=True) ] == \
+        [((0, -6), (6, 0)), ((0, 0), (6, 6))]
+
+    assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 10)) == \
+        [((QQ(15, 16), -QQ(3, 2)), (QQ(33, 32), -QQ(45, 32))),
+         ((QQ(15, 16), QQ(45, 32)), (QQ(33, 32), QQ(3, 2)))]
+    assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 100)) == \
+        [((QQ(255, 256), -QQ(363, 256)), (QQ(513, 512), -QQ(723, 512))),
+         ((QQ(255, 256), QQ(723, 512)), (QQ(513, 512), QQ(363, 256)))]
+
+    f = 7*x**4 - 19*x**3 + 20*x**2 + 17*x + 20
+
+    assert R.dup_isolate_complex_roots_sqf(f) == \
+        [((-QQ(40, 7), -QQ(40, 7)), (0, 0)), ((-QQ(40, 7), 0), (0, QQ(40, 7))),
+         ((0, -QQ(40, 7)), (QQ(40, 7), 0)), ((0, 0), (QQ(40, 7), QQ(40, 7)))]
+
+
+def test_dup_isolate_all_roots_sqf():
+    R, x = ring("x", ZZ)
+    f = 4*x**4 - x**3 + 2*x**2 + 5*x
+
+    assert R.dup_isolate_all_roots_sqf(f) == \
+        ([(-1, 0), (0, 0)],
+         [((0, -QQ(5, 2)), (QQ(5, 2), 0)), ((0, 0), (QQ(5, 2), QQ(5, 2)))])
+
+    assert R.dup_isolate_all_roots_sqf(f, eps=QQ(1, 10)) == \
+        ([(QQ(-7, 8), QQ(-6, 7)), (0, 0)],
+         [((QQ(35, 64), -QQ(35, 32)), (QQ(5, 8), -QQ(65, 64))), ((QQ(35, 64), QQ(65, 64)), (QQ(5, 8), QQ(35, 32)))])
+
+
+def test_dup_isolate_all_roots():
+    R, x = ring("x", ZZ)
+    f = 4*x**4 - x**3 + 2*x**2 + 5*x
+
+    assert R.dup_isolate_all_roots(f) == \
+        ([((-1, 0), 1), ((0, 0), 1)],
+         [(((0, -QQ(5, 2)), (QQ(5, 2), 0)), 1),
+          (((0, 0), (QQ(5, 2), QQ(5, 2))), 1)])
+
+    assert R.dup_isolate_all_roots(f, eps=QQ(1, 10)) == \
+        ([((QQ(-7, 8), QQ(-6, 7)), 1), ((0, 0), 1)],
+         [(((QQ(35, 64), -QQ(35, 32)), (QQ(5, 8), -QQ(65, 64))), 1),
+          (((QQ(35, 64), QQ(65, 64)), (QQ(5, 8), QQ(35, 32))), 1)])
+
+    f = x**5 + x**4 - 2*x**3 - 2*x**2 + x + 1
+    raises(NotImplementedError, lambda: R.dup_isolate_all_roots(f))

rl/Lib/site-packages/sympy/polys/tests/test_rootoftools.py

@@ -0,0 +1,653 @@
+"""Tests for the implementation of RootOf class and related tools. """
+
+from sympy.polys.polytools import Poly
+import sympy.polys.rootoftools as rootoftools
+from sympy.polys.rootoftools import (rootof, RootOf, CRootOf, RootSum,
+    _pure_key_dict as D)
+
+from sympy.polys.polyerrors import (
+    MultivariatePolynomialError,
+    GeneratorsNeeded,
+    PolynomialError,
+)
+
+from sympy.core.function import (Function, Lambda)
+from sympy.core.numbers import (Float, I, Rational)
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import tan
+from sympy.integrals.integrals import Integral
+from sympy.polys.orthopolys import legendre_poly
+from sympy.solvers.solvers import solve
+
+
+from sympy.testing.pytest import raises, slow
+from sympy.core.expr import unchanged
+
+from sympy.abc import a, b, x, y, z, r
+
+
+def test_CRootOf___new__():
+    assert rootof(x, 0) == 0
+    assert rootof(x, -1) == 0
+
+    assert rootof(x, S.Zero) == 0
+
+    assert rootof(x - 1, 0) == 1
+    assert rootof(x - 1, -1) == 1
+
+    assert rootof(x + 1, 0) == -1
+    assert rootof(x + 1, -1) == -1
+
+    assert rootof(x**2 + 2*x + 3, 0) == -1 - I*sqrt(2)
+    assert rootof(x**2 + 2*x + 3, 1) == -1 + I*sqrt(2)
+    assert rootof(x**2 + 2*x + 3, -1) == -1 + I*sqrt(2)
+    assert rootof(x**2 + 2*x + 3, -2) == -1 - I*sqrt(2)
+
+    r = rootof(x**2 + 2*x + 3, 0, radicals=False)
+    assert isinstance(r, RootOf) is True
+
+    r = rootof(x**2 + 2*x + 3, 1, radicals=False)
+    assert isinstance(r, RootOf) is True
+
+    r = rootof(x**2 + 2*x + 3, -1, radicals=False)
+    assert isinstance(r, RootOf) is True
+
+    r = rootof(x**2 + 2*x + 3, -2, radicals=False)
+    assert isinstance(r, RootOf) is True
+
+    assert rootof((x - 1)*(x + 1), 0, radicals=False) == -1
+    assert rootof((x - 1)*(x + 1), 1, radicals=False) == 1
+    assert rootof((x - 1)*(x + 1), -1, radicals=False) == 1
+    assert rootof((x - 1)*(x + 1), -2, radicals=False) == -1
+
+    assert rootof((x - 1)*(x + 1), 0, radicals=True) == -1
+    assert rootof((x - 1)*(x + 1), 1, radicals=True) == 1
+    assert rootof((x - 1)*(x + 1), -1, radicals=True) == 1
+    assert rootof((x - 1)*(x + 1), -2, radicals=True) == -1
+
+    assert rootof((x - 1)*(x**3 + x + 3), 0) == rootof(x**3 + x + 3, 0)
+    assert rootof((x - 1)*(x**3 + x + 3), 1) == 1
+    assert rootof((x - 1)*(x**3 + x + 3), 2) == rootof(x**3 + x + 3, 1)
+    assert rootof((x - 1)*(x**3 + x + 3), 3) == rootof(x**3 + x + 3, 2)
+    assert rootof((x - 1)*(x**3 + x + 3), -1) == rootof(x**3 + x + 3, 2)
+    assert rootof((x - 1)*(x**3 + x + 3), -2) == rootof(x**3 + x + 3, 1)
+    assert rootof((x - 1)*(x**3 + x + 3), -3) == 1
+    assert rootof((x - 1)*(x**3 + x + 3), -4) == rootof(x**3 + x + 3, 0)
+
+    assert rootof(x**4 + 3*x**3, 0) == -3
+    assert rootof(x**4 + 3*x**3, 1) == 0
+    assert rootof(x**4 + 3*x**3, 2) == 0
+    assert rootof(x**4 + 3*x**3, 3) == 0
+
+    raises(GeneratorsNeeded, lambda: rootof(0, 0))
+    raises(GeneratorsNeeded, lambda: rootof(1, 0))
+
+    raises(PolynomialError, lambda: rootof(Poly(0, x), 0))
+    raises(PolynomialError, lambda: rootof(Poly(1, x), 0))
+    raises(PolynomialError, lambda: rootof(x - y, 0))
+    # issue 8617
+    raises(PolynomialError, lambda: rootof(exp(x), 0))
+
+    raises(NotImplementedError, lambda: rootof(x**3 - x + sqrt(2), 0))
+    raises(NotImplementedError, lambda: rootof(x**3 - x + I, 0))
+
+    raises(IndexError, lambda: rootof(x**2 - 1, -4))
+    raises(IndexError, lambda: rootof(x**2 - 1, -3))
+    raises(IndexError, lambda: rootof(x**2 - 1, 2))
+    raises(IndexError, lambda: rootof(x**2 - 1, 3))
+    raises(ValueError, lambda: rootof(x**2 - 1, x))
+
+    assert rootof(Poly(x - y, x), 0) == y
+
+    assert rootof(Poly(x**2 - y, x), 0) == -sqrt(y)
+    assert rootof(Poly(x**2 - y, x), 1) == sqrt(y)
+
+    assert rootof(Poly(x**3 - y, x), 0) == y**Rational(1, 3)
+
+    assert rootof(y*x**3 + y*x + 2*y, x, 0) == -1
+    raises(NotImplementedError, lambda: rootof(x**3 + x + 2*y, x, 0))
+
+    assert rootof(x**3 + x + 1, 0).is_commutative is True
+
+
+def test_CRootOf_attributes():
+    r = rootof(x**3 + x + 3, 0)
+    assert r.is_number
+    assert r.free_symbols == set()
+    # if the following assertion fails then multivariate polynomials
+    # are apparently supported and the RootOf.free_symbols routine
+    # should be changed to return whatever symbols would not be
+    # the PurePoly dummy symbol
+    raises(NotImplementedError, lambda: rootof(Poly(x**3 + y*x + 1, x), 0))
+
+
+def test_CRootOf___eq__():
+    assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 0)) is True
+    assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 1)) is False
+    assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 1)) is True
+    assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 2)) is False
+    assert (rootof(x**3 + x + 3, 2) == rootof(x**3 + x + 3, 2)) is True
+
+    assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 0)) is True
+    assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 1)) is False
+    assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 1)) is True
+    assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 2)) is False
+    assert (rootof(x**3 + x + 3, 2) == rootof(y**3 + y + 3, 2)) is True
+
+
+def test_CRootOf___eval_Eq__():
+    f = Function('f')
+    eq = x**3 + x + 3
+    r = rootof(eq, 2)
+    r1 = rootof(eq, 1)
+    assert Eq(r, r1) is S.false
+    assert Eq(r, r) is S.true
+    assert unchanged(Eq, r, x)
+    assert Eq(r, 0) is S.false
+    assert Eq(r, S.Infinity) is S.false
+    assert Eq(r, I) is S.false
+    assert unchanged(Eq, r, f(0))
+    sol = solve(eq)
+    for s in sol:
+        if s.is_real:
+            assert Eq(r, s) is S.false
+    r = rootof(eq, 0)
+    for s in sol:
+        if s.is_real:
+            assert Eq(r, s) is S.true
+    eq = x**3 + x + 1
+    sol = solve(eq)
+    assert [Eq(rootof(eq, i), j) for i in range(3) for j in sol
+        ].count(True) == 3
+    assert Eq(rootof(eq, 0), 1 + S.ImaginaryUnit) == False
+
+
+def test_CRootOf_is_real():
+    assert rootof(x**3 + x + 3, 0).is_real is True
+    assert rootof(x**3 + x + 3, 1).is_real is False
+    assert rootof(x**3 + x + 3, 2).is_real is False
+
+
+def test_CRootOf_is_complex():
+    assert rootof(x**3 + x + 3, 0).is_complex is True
+
+
+def test_CRootOf_subs():
+    assert rootof(x**3 + x + 1, 0).subs(x, y) == rootof(y**3 + y + 1, 0)
+
+
+def test_CRootOf_diff():
+    assert rootof(x**3 + x + 1, 0).diff(x) == 0
+    assert rootof(x**3 + x + 1, 0).diff(y) == 0
+
+
+@slow
+def test_CRootOf_evalf():
+    real = rootof(x**3 + x + 3, 0).evalf(n=20)
+
+    assert real.epsilon_eq(Float("-1.2134116627622296341"))
+
+    re, im = rootof(x**3 + x + 3, 1).evalf(n=20).as_real_imag()
+
+    assert re.epsilon_eq( Float("0.60670583138111481707"))
+    assert im.epsilon_eq(-Float("1.45061224918844152650"))
+
+    re, im = rootof(x**3 + x + 3, 2).evalf(n=20).as_real_imag()
+
+    assert re.epsilon_eq(Float("0.60670583138111481707"))
+    assert im.epsilon_eq(Float("1.45061224918844152650"))
+
+    p = legendre_poly(4, x, polys=True)
+    roots = [str(r.n(17)) for r in p.real_roots()]
+    # magnitudes are given by
+    # sqrt(3/S(7) - 2*sqrt(6/S(5))/7)
+    #   and
+    # sqrt(3/S(7) + 2*sqrt(6/S(5))/7)
+    assert roots == [
+            "-0.86113631159405258",
+            "-0.33998104358485626",
+             "0.33998104358485626",
+             "0.86113631159405258",
+             ]
+
+    re = rootof(x**5 - 5*x + 12, 0).evalf(n=20)
+    assert re.epsilon_eq(Float("-1.84208596619025438271"))
+
+    re, im = rootof(x**5 - 5*x + 12, 1).evalf(n=20).as_real_imag()
+    assert re.epsilon_eq(Float("-0.351854240827371999559"))
+    assert im.epsilon_eq(Float("-1.709561043370328882010"))
+
+    re, im = rootof(x**5 - 5*x + 12, 2).evalf(n=20).as_real_imag()
+    assert re.epsilon_eq(Float("-0.351854240827371999559"))
+    assert im.epsilon_eq(Float("+1.709561043370328882010"))
+
+    re, im = rootof(x**5 - 5*x + 12, 3).evalf(n=20).as_real_imag()
+    assert re.epsilon_eq(Float("+1.272897223922499190910"))
+    assert im.epsilon_eq(Float("-0.719798681483861386681"))
+
+    re, im = rootof(x**5 - 5*x + 12, 4).evalf(n=20).as_real_imag()
+    assert re.epsilon_eq(Float("+1.272897223922499190910"))
+    assert im.epsilon_eq(Float("+0.719798681483861386681"))
+
+    # issue 6393
+    assert str(rootof(x**5 + 2*x**4 + x**3 - 68719476736, 0).n(3)) == '147.'
+    eq = (531441*x**11 + 3857868*x**10 + 13730229*x**9 + 32597882*x**8 +
+        55077472*x**7 + 60452000*x**6 + 32172064*x**5 - 4383808*x**4 -
+        11942912*x**3 - 1506304*x**2 + 1453312*x + 512)
+    a, b = rootof(eq, 1).n(2).as_real_imag()
+    c, d = rootof(eq, 2).n(2).as_real_imag()
+    assert a == c
+    assert b < d
+    assert b == -d
+    # issue 6451
+    r = rootof(legendre_poly(64, x), 7)
+    assert r.n(2) == r.n(100).n(2)
+    # issue 9019
+    r0 = rootof(x**2 + 1, 0, radicals=False)
+    r1 = rootof(x**2 + 1, 1, radicals=False)
+    assert r0.n(4) == Float(-1.0, 4) * I
+    assert r1.n(4) == Float(1.0, 4) * I
+
+    # make sure verification is used in case a max/min traps the "root"
+    assert str(rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0).n(3)) == '-0.976'
+
+    # watch out for UnboundLocalError
+    c = CRootOf(90720*x**6 - 4032*x**4 + 84*x**2 - 1, 0)
+    assert c._eval_evalf(2)  # doesn't fail
+
+    # watch out for imaginary parts that don't want to evaluate
+    assert str(RootOf(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 +
+        39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 +
+        877969, 10).n(2)) == '-3.4*I'
+    assert abs(RootOf(x**4 + 10*x**2 + 1, 0).n(2)) < 0.4
+
+    # check reset and args
+    r = [RootOf(x**3 + x + 3, i) for i in range(3)]
+    r[0]._reset()
+    for ri in r:
+        i = ri._get_interval()
+        ri.n(2)
+        assert i != ri._get_interval()
+        ri._reset()
+        assert i == ri._get_interval()
+        assert i == i.func(*i.args)
+
+
+def test_issue_24978():
+    # Irreducible poly with negative leading coeff is normalized
+    # (factor of -1 is extracted), before being stored as CRootOf.poly.
+    f = -x**2 + 2
+    r = CRootOf(f, 0)
+    assert r.poly.as_expr() == x**2 - 2
+    # An action that prompts calculation of an interval puts r.poly in
+    # the cache.
+    r.n()
+    assert r.poly in rootoftools._reals_cache
+
+
+def test_CRootOf_evalf_caching_bug():
+    r = rootof(x**5 - 5*x + 12, 1)
+    r.n()
+    a = r._get_interval()
+    r = rootof(x**5 - 5*x + 12, 1)
+    r.n()
+    b = r._get_interval()
+    assert a == b
+
+
+def test_CRootOf_real_roots():
+    assert Poly(x**5 + x + 1).real_roots() == [rootof(x**3 - x**2 + 1, 0)]
+    assert Poly(x**5 + x + 1).real_roots(radicals=False) == [rootof(
+        x**3 - x**2 + 1, 0)]
+
+    # https://github.com/sympy/sympy/issues/20902
+    p = Poly(-3*x**4 - 10*x**3 - 12*x**2 - 6*x - 1, x, domain='ZZ')
+    assert CRootOf.real_roots(p) == [S(-1), S(-1), S(-1), S(-1)/3]
+
+
+def test_CRootOf_all_roots():
+    assert Poly(x**5 + x + 1).all_roots() == [
+        rootof(x**3 - x**2 + 1, 0),
+        Rational(-1, 2) - sqrt(3)*I/2,
+        Rational(-1, 2) + sqrt(3)*I/2,
+        rootof(x**3 - x**2 + 1, 1),
+        rootof(x**3 - x**2 + 1, 2),
+    ]
+
+    assert Poly(x**5 + x + 1).all_roots(radicals=False) == [
+        rootof(x**3 - x**2 + 1, 0),
+        rootof(x**2 + x + 1, 0, radicals=False),
+        rootof(x**2 + x + 1, 1, radicals=False),
+        rootof(x**3 - x**2 + 1, 1),
+        rootof(x**3 - x**2 + 1, 2),
+    ]
+
+
+def test_CRootOf_eval_rational():
+    p = legendre_poly(4, x, polys=True)
+    roots = [r.eval_rational(n=18) for r in p.real_roots()]
+    for root in roots:
+        assert isinstance(root, Rational)
+    roots = [str(root.n(17)) for root in roots]
+    assert roots == [
+            "-0.86113631159405258",
+            "-0.33998104358485626",
+             "0.33998104358485626",
+             "0.86113631159405258",
+             ]
+
+
+def test_CRootOf_lazy():
+    # irreducible poly with both real and complex roots:
+    f = Poly(x**3 + 2*x + 2)
+
+    # real root:
+    CRootOf.clear_cache()
+    r = CRootOf(f, 0)
+    # Not yet in cache, after construction:
+    assert r.poly not in rootoftools._reals_cache
+    assert r.poly not in rootoftools._complexes_cache
+    r.evalf()
+    # In cache after evaluation:
+    assert r.poly in rootoftools._reals_cache
+    assert r.poly not in rootoftools._complexes_cache
+
+    # complex root:
+    CRootOf.clear_cache()
+    r = CRootOf(f, 1)
+    # Not yet in cache, after construction:
+    assert r.poly not in rootoftools._reals_cache
+    assert r.poly not in rootoftools._complexes_cache
+    r.evalf()
+    # In cache after evaluation:
+    assert r.poly in rootoftools._reals_cache
+    assert r.poly in rootoftools._complexes_cache
+
+    # composite poly with both real and complex roots:
+    f = Poly((x**2 - 2)*(x**2 + 1))
+
+    # real root:
+    CRootOf.clear_cache()
+    r = CRootOf(f, 0)
+    # In cache immediately after construction:
+    assert r.poly in rootoftools._reals_cache
+    assert r.poly not in rootoftools._complexes_cache
+
+    # complex root:
+    CRootOf.clear_cache()
+    r = CRootOf(f, 2)
+    # In cache immediately after construction:
+    assert r.poly in rootoftools._reals_cache
+    assert r.poly in rootoftools._complexes_cache
+
+
+def test_RootSum___new__():
+    f = x**3 + x + 3
+
+    g = Lambda(r, log(r*x))
+    s = RootSum(f, g)
+
+    assert isinstance(s, RootSum) is True
+
+    assert RootSum(f**2, g) == 2*RootSum(f, g)
+    assert RootSum((x - 7)*f**3, g) == log(7*x) + 3*RootSum(f, g)
+
+    # issue 5571
+    assert hash(RootSum((x - 7)*f**3, g)) == hash(log(7*x) + 3*RootSum(f, g))
+
+    raises(MultivariatePolynomialError, lambda: RootSum(x**3 + x + y))
+    raises(ValueError, lambda: RootSum(x**2 + 3, lambda x: x))
+
+    assert RootSum(f, exp) == RootSum(f, Lambda(x, exp(x)))
+    assert RootSum(f, log) == RootSum(f, Lambda(x, log(x)))
+
+    assert isinstance(RootSum(f, auto=False), RootSum) is True
+
+    assert RootSum(f) == 0
+    assert RootSum(f, Lambda(x, x)) == 0
+    assert RootSum(f, Lambda(x, x**2)) == -2
+
+    assert RootSum(f, Lambda(x, 1)) == 3
+    assert RootSum(f, Lambda(x, 2)) == 6
+
+    assert RootSum(f, auto=False).is_commutative is True
+
+    assert RootSum(f, Lambda(x, 1/(x + x**2))) == Rational(11, 3)
+    assert RootSum(f, Lambda(x, y/(x + x**2))) == Rational(11, 3)*y
+
+    assert RootSum(x**2 - 1, Lambda(x, 3*x**2), x) == 6
+    assert RootSum(x**2 - y, Lambda(x, 3*x**2), x) == 6*y
+
+    assert RootSum(x**2 - 1, Lambda(x, z*x**2), x) == 2*z
+    assert RootSum(x**2 - y, Lambda(x, z*x**2), x) == 2*z*y
+
+    assert RootSum(
+        x**2 - 1, Lambda(x, exp(x)), quadratic=True) == exp(-1) + exp(1)
+
+    assert RootSum(x**3 + a*x + a**3, tan, x) == \
+        RootSum(x**3 + x + 1, Lambda(x, tan(a*x)))
+    assert RootSum(a**3*x**3 + a*x + 1, tan, x) == \
+        RootSum(x**3 + x + 1, Lambda(x, tan(x/a)))
+
+
+def test_RootSum_free_symbols():
+    assert RootSum(x**3 + x + 3, Lambda(r, exp(r))).free_symbols == set()
+    assert RootSum(x**3 + x + 3, Lambda(r, exp(a*r))).free_symbols == {a}
+    assert RootSum(
+        x**3 + x + y, Lambda(r, exp(a*r)), x).free_symbols == {a, y}
+
+
+def test_RootSum___eq__():
+    f = Lambda(x, exp(x))
+
+    assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 1, f)) is True
+    assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 1, f)) is True
+
+    assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 2, f)) is False
+    assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 2, f)) is False
+
+
+def test_RootSum_doit():
+    rs = RootSum(x**2 + 1, exp)
+
+    assert isinstance(rs, RootSum) is True
+    assert rs.doit() == exp(-I) + exp(I)
+
+    rs = RootSum(x**2 + a, exp, x)
+
+    assert isinstance(rs, RootSum) is True
+    assert rs.doit() == exp(-sqrt(-a)) + exp(sqrt(-a))
+
+
+def test_RootSum_evalf():
+    rs = RootSum(x**2 + 1, exp)
+
+    assert rs.evalf(n=20, chop=True).epsilon_eq(Float("1.0806046117362794348"))
+    assert rs.evalf(n=15, chop=True).epsilon_eq(Float("1.08060461173628"))
+
+    rs = RootSum(x**2 + a, exp, x)
+
+    assert rs.evalf() == rs
+
+
+def test_RootSum_diff():
+    f = x**3 + x + 3
+
+    g = Lambda(r, exp(r*x))
+    h = Lambda(r, r*exp(r*x))
+
+    assert RootSum(f, g).diff(x) == RootSum(f, h)
+
+
+def test_RootSum_subs():
+    f = x**3 + x + 3
+    g = Lambda(r, exp(r*x))
+
+    F = y**3 + y + 3
+    G = Lambda(r, exp(r*y))
+
+    assert RootSum(f, g).subs(y, 1) == RootSum(f, g)
+    assert RootSum(f, g).subs(x, y) == RootSum(F, G)
+
+
+def test_RootSum_rational():
+    assert RootSum(
+        z**5 - z + 1, Lambda(z, z/(x - z))) == (4*x - 5)/(x**5 - x + 1)
+
+    f = 161*z**3 + 115*z**2 + 19*z + 1
+    g = Lambda(z, z*log(
+        -3381*z**4/4 - 3381*z**3/4 - 625*z**2/2 - z*Rational(125, 2) - 5 + exp(x)))
+
+    assert RootSum(f, g).diff(x) == -(
+        (5*exp(2*x) - 6*exp(x) + 4)*exp(x)/(exp(3*x) - exp(2*x) + 1))/7
+
+
+def test_RootSum_independent():
+    f = (x**3 - a)**2*(x**4 - b)**3
+
+    g = Lambda(x, 5*tan(x) + 7)
+    h = Lambda(x, tan(x))
+
+    r0 = RootSum(x**3 - a, h, x)
+    r1 = RootSum(x**4 - b, h, x)
+
+    assert RootSum(f, g, x).as_ordered_terms() == [10*r0, 15*r1, 126]
+
+
+def test_issue_7876():
+    l1 = Poly(x**6 - x + 1, x).all_roots()
+    l2 = [rootof(x**6 - x + 1, i) for i in range(6)]
+    assert frozenset(l1) == frozenset(l2)
+
+
+def test_issue_8316():
+    f = Poly(7*x**8 - 9)
+    assert len(f.all_roots()) == 8
+    f = Poly(7*x**8 - 10)
+    assert len(f.all_roots()) == 8
+
+
+def test__imag_count():
+    from sympy.polys.rootoftools import _imag_count_of_factor
+    def imag_count(p):
+        return sum(_imag_count_of_factor(f)*m for f, m in
+        p.factor_list()[1])
+    assert imag_count(Poly(x**6 + 10*x**2 + 1)) == 2
+    assert imag_count(Poly(x**2)) == 0
+    assert imag_count(Poly([1]*3 + [-1], x)) == 0
+    assert imag_count(Poly(x**3 + 1)) == 0
+    assert imag_count(Poly(x**2 + 1)) == 2
+    assert imag_count(Poly(x**2 - 1)) == 0
+    assert imag_count(Poly(x**4 - 1)) == 2
+    assert imag_count(Poly(x**4 + 1)) == 0
+    assert imag_count(Poly([1, 2, 3], x)) == 0
+    assert imag_count(Poly(x**3 + x + 1)) == 0
+    assert imag_count(Poly(x**4 + x + 1)) == 0
+    def q(r1, r2, p):
+        return Poly(((x - r1)*(x - r2)).subs(x, x**p), x)
+    assert imag_count(q(-1, -2, 2)) == 4
+    assert imag_count(q(-1, 2, 2)) == 2
+    assert imag_count(q(1, 2, 2)) == 0
+    assert imag_count(q(1, 2, 4)) == 4
+    assert imag_count(q(-1, 2, 4)) == 2
+    assert imag_count(q(-1, -2, 4)) == 0
+
+
+def test_RootOf_is_imaginary():
+    r = RootOf(x**4 + 4*x**2 + 1, 1)
+    i = r._get_interval()
+    assert r.is_imaginary and i.ax*i.bx <= 0
+
+
+def test_is_disjoint():
+    eq = x**3 + 5*x + 1
+    ir = rootof(eq, 0)._get_interval()
+    ii = rootof(eq, 1)._get_interval()
+    assert ir.is_disjoint(ii)
+    assert ii.is_disjoint(ir)
+
+
+def test_pure_key_dict():
+    p = D()
+    assert (x in p) is False
+    assert (1 in p) is False
+    p[x] = 1
+    assert x in p
+    assert y in p
+    assert p[y] == 1
+    raises(KeyError, lambda: p[1])
+    def dont(k):
+        p[k] = 2
+    raises(ValueError, lambda: dont(1))
+
+
+@slow
+def test_eval_approx_relative():
+    CRootOf.clear_cache()
+    t = [CRootOf(x**3 + 10*x + 1, i) for i in range(3)]
+    assert [i.eval_rational(1e-1) for i in t] == [
+        Rational(-21, 220), Rational(15, 256) - I*805/256,
+        Rational(15, 256) + I*805/256]
+    t[0]._reset()
+    assert [i.eval_rational(1e-1, 1e-4) for i in t] == [
+        Rational(-21, 220), Rational(3275, 65536) - I*414645/131072,
+        Rational(3275, 65536) + I*414645/131072]
+    assert S(t[0]._get_interval().dx) < 1e-1
+    assert S(t[1]._get_interval().dx) < 1e-1
+    assert S(t[1]._get_interval().dy) < 1e-4
+    assert S(t[2]._get_interval().dx) < 1e-1
+    assert S(t[2]._get_interval().dy) < 1e-4
+    t[0]._reset()
+    assert [i.eval_rational(1e-4, 1e-4) for i in t] == [
+        Rational(-2001, 20020), Rational(6545, 131072) - I*414645/131072,
+        Rational(6545, 131072) + I*414645/131072]
+    assert S(t[0]._get_interval().dx) < 1e-4
+    assert S(t[1]._get_interval().dx) < 1e-4
+    assert S(t[1]._get_interval().dy) < 1e-4
+    assert S(t[2]._get_interval().dx) < 1e-4
+    assert S(t[2]._get_interval().dy) < 1e-4
+    # in the following, the actual relative precision is
+    # less than tested, but it should never be greater
+    t[0]._reset()
+    assert [i.eval_rational(n=2) for i in t] == [
+        Rational(-202201, 2024022), Rational(104755, 2097152) - I*6634255/2097152,
+        Rational(104755, 2097152) + I*6634255/2097152]
+    assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-2
+    assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-2
+    assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-2
+    assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-2
+    assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-2
+    t[0]._reset()
+    assert [i.eval_rational(n=3) for i in t] == [
+        Rational(-202201, 2024022), Rational(1676045, 33554432) - I*106148135/33554432,
+        Rational(1676045, 33554432) + I*106148135/33554432]
+    assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-3
+    assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-3
+    assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-3
+    assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-3
+    assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-3
+
+    t[0]._reset()
+    a = [i.eval_approx(2) for i in t]
+    assert [str(i) for i in a] == [
+        '-0.10', '0.05 - 3.2*I', '0.05 + 3.2*I']
+    assert all(abs(((a[i] - t[i])/t[i]).n()) < 1e-2 for i in range(len(a)))
+
+
+def test_issue_15920():
+    r = rootof(x**5 - x + 1, 0)
+    p = Integral(x, (x, 1, y))
+    assert unchanged(Eq, r, p)
+
+
+def test_issue_19113():
+    eq = y**3 - y + 1
+    # generator is a canonical x in RootOf
+    assert str(Poly(eq).real_roots()) == '[CRootOf(x**3 - x + 1, 0)]'
+    assert str(Poly(eq.subs(y, tan(y))).real_roots()
+        ) == '[CRootOf(x**3 - x + 1, 0)]'
+    assert str(Poly(eq.subs(y, tan(x))).real_roots()
+        ) == '[CRootOf(x**3 - x + 1, 0)]'

rl/Lib/site-packages/sympy/polys/tests/test_solvers.py

@@ -0,0 +1,112 @@
+"""Tests for low-level linear systems solver. """
+
+from sympy.matrices import Matrix
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.fields import field
+from sympy.polys.rings import ring
+from sympy.polys.solvers import solve_lin_sys, eqs_to_matrix
+
+
+def test_solve_lin_sys_2x2_one():
+    domain, x1,x2 = ring("x1,x2", QQ)
+    eqs = [x1 + x2 - 5,
+           2*x1 - x2]
+    sol = {x1: QQ(5, 3), x2: QQ(10, 3)}
+    _sol = solve_lin_sys(eqs, domain)
+    assert _sol == sol and all(isinstance(s, domain.dtype) for s in _sol)
+
+def test_solve_lin_sys_2x4_none():
+    domain, x1,x2 = ring("x1,x2", QQ)
+    eqs = [x1 - 1,
+           x1 - x2,
+           x1 - 2*x2,
+           x2 - 1]
+    assert solve_lin_sys(eqs, domain) is None
+
+
+def test_solve_lin_sys_3x4_one():
+    domain, x1,x2,x3 = ring("x1,x2,x3", QQ)
+    eqs = [x1 + 2*x2 + 3*x3,
+           2*x1 - x2 + x3,
+           3*x1 + x2 + x3,
+           5*x2 + 2*x3]
+    sol = {x1: 0, x2: 0, x3: 0}
+    assert solve_lin_sys(eqs, domain) == sol
+
+def test_solve_lin_sys_3x3_inf():
+    domain, x1,x2,x3 = ring("x1,x2,x3", QQ)
+    eqs = [x1 - x2 + 2*x3 - 1,
+           2*x1 + x2 + x3 - 8,
+           x1 + x2 - 5]
+    sol = {x1: -x3 + 3, x2: x3 + 2}
+    assert solve_lin_sys(eqs, domain) == sol
+
+def test_solve_lin_sys_3x4_none():
+    domain, x1,x2,x3,x4 = ring("x1,x2,x3,x4", QQ)
+    eqs = [2*x1 + x2 + 7*x3 - 7*x4 - 2,
+           -3*x1 + 4*x2 - 5*x3 - 6*x4 - 3,
+           x1 + x2 + 4*x3 - 5*x4 - 2]
+    assert solve_lin_sys(eqs, domain) is None
+
+
+def test_solve_lin_sys_4x7_inf():
+    domain, x1,x2,x3,x4,x5,x6,x7 = ring("x1,x2,x3,x4,x5,x6,x7", QQ)
+    eqs = [x1 + 4*x2 - x4 + 7*x6 - 9*x7 - 3,
+           2*x1 + 8*x2 - x3 + 3*x4 + 9*x5 - 13*x6 + 7*x7 - 9,
+           2*x3 - 3*x4 - 4*x5 + 12*x6 - 8*x7 - 1,
+           -x1 - 4*x2 + 2*x3 + 4*x4 + 8*x5 - 31*x6 + 37*x7 - 4]
+    sol = {x1: 4 - 4*x2 - 2*x5 - x6 + 3*x7,
+           x3: 2 - x5 + 3*x6 - 5*x7,
+           x4: 1 - 2*x5 + 6*x6 - 6*x7}
+    assert solve_lin_sys(eqs, domain) == sol
+
+def test_solve_lin_sys_5x5_inf():
+    domain, x1,x2,x3,x4,x5 = ring("x1,x2,x3,x4,x5", QQ)
+    eqs = [x1 - x2 - 2*x3 + x4 + 11*x5 - 13,
+           x1 - x2 + x3 + x4 + 5*x5 - 16,
+           2*x1 - 2*x2 + x4 + 10*x5 - 21,
+           2*x1 - 2*x2 - x3 + 3*x4 + 20*x5 - 38,
+           2*x1 - 2*x2 + x3 + x4 + 8*x5 - 22]
+    sol = {x1: 6 + x2 - 3*x5,
+           x3: 1 + 2*x5,
+           x4: 9 - 4*x5}
+    assert solve_lin_sys(eqs, domain) == sol
+
+def test_solve_lin_sys_6x6_1():
+    ground, d,r,e,g,i,j,l,o,m,p,q = field("d,r,e,g,i,j,l,o,m,p,q", ZZ)
+    domain, c,f,h,k,n,b = ring("c,f,h,k,n,b", ground)
+
+    eqs = [b + q/d - c/d, c*(1/d + 1/e + 1/g) - f/g - q/d, f*(1/g + 1/i + 1/j) - c/g - h/i, h*(1/i + 1/l + 1/m) - f/i - k/m, k*(1/m + 1/o + 1/p) - h/m - n/p, n/p - k/p]
+    sol = {
+         b: (e*i*l*q + e*i*m*q + e*i*o*q + e*j*l*q + e*j*m*q + e*j*o*q + e*l*m*q + e*l*o*q + g*i*l*q + g*i*m*q + g*i*o*q + g*j*l*q + g*j*m*q + g*j*o*q + g*l*m*q + g*l*o*q + i*j*l*q + i*j*m*q + i*j*o*q + j*l*m*q + j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
+         c: (-e*g*i*l*q - e*g*i*m*q - e*g*i*o*q - e*g*j*l*q - e*g*j*m*q - e*g*j*o*q - e*g*l*m*q - e*g*l*o*q - e*i*j*l*q - e*i*j*m*q - e*i*j*o*q - e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
+         f: (-e*i*j*l*q - e*i*j*m*q - e*i*j*o*q - e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
+         h: (-e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
+         k: e*j*l*o*q/(d*e*i*l + d*e*i*m + d*e*i*o + d*e*j*l + d*e*j*m + d*e*j*o + d*e*l*m + d*e*l*o + d*g*i*l + d*g*i*m + d*g*i*o + d*g*j*l + d*g*j*m + d*g*j*o + d*g*l*m + d*g*l*o + d*i*j*l + d*i*j*m + d*i*j*o + d*j*l*m + d*j*l*o + e*g*i*l + e*g*i*m + e*g*i*o + e*g*j*l + e*g*j*m + e*g*j*o + e*g*l*m + e*g*l*o + e*i*j*l + e*i*j*m + e*i*j*o + e*j*l*m + e*j*l*o),
+         n: e*j*l*o*q/(d*e*i*l + d*e*i*m + d*e*i*o + d*e*j*l + d*e*j*m + d*e*j*o + d*e*l*m + d*e*l*o + d*g*i*l + d*g*i*m + d*g*i*o + d*g*j*l + d*g*j*m + d*g*j*o + d*g*l*m + d*g*l*o + d*i*j*l + d*i*j*m + d*i*j*o + d*j*l*m + d*j*l*o + e*g*i*l + e*g*i*m + e*g*i*o + e*g*j*l + e*g*j*m + e*g*j*o + e*g*l*m + e*g*l*o + e*i*j*l + e*i*j*m + e*i*j*o + e*j*l*m + e*j*l*o),
+    }
+
+    assert solve_lin_sys(eqs, domain) == sol
+
+def test_solve_lin_sys_6x6_2():
+    ground, d,r,e,g,i,j,l,o,m,p,q = field("d,r,e,g,i,j,l,o,m,p,q", ZZ)
+    domain, c,f,h,k,n,b = ring("c,f,h,k,n,b", ground)
+
+    eqs = [b + r/d - c/d, c*(1/d + 1/e + 1/g) - f/g - r/d, f*(1/g + 1/i + 1/j) - c/g - h/i, h*(1/i + 1/l + 1/m) - f/i - k/m, k*(1/m + 1/o + 1/p) - h/m - n/p, n*(1/p + 1/q) - k/p]
+    sol = {
+        b: -((l*q*e*o + l*q*g*o + i*m*q*e + i*l*q*e + i*l*p*e + i*j*o*q + j*e*o*q + g*j*o*q + i*e*o*q + g*i*o*q + e*l*o*p + e*l*m*p + e*l*m*o + e*i*o*p + e*i*m*p + e*i*m*o + e*i*l*o + j*e*o*p + j*e*m*q + j*e*m*p + j*e*m*o + j*l*m*q + j*l*m*p + j*l*m*o + i*j*m*p + i*j*m*o + i*j*l*q + i*j*l*o + i*j*m*q + j*l*o*p + j*e*l*o + g*j*o*p + g*j*m*q + g*j*m*p + i*j*l*p + i*j*o*p + j*e*l*q + j*e*l*p + j*l*o*q + g*j*m*o + g*j*l*q + g*j*l*p + g*j*l*o + g*l*o*p + g*l*m*p + g*l*m*o + g*i*m*o + g*i*o*p + g*i*m*q + g*i*m*p + g*i*l*q + g*i*l*p + g*i*l*o + l*m*q*e + l*m*q*g)*r)/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
+        c: (r*e*(l*q*g*o + i*j*o*q + g*j*o*q + g*i*o*q + j*l*m*q + j*l*m*p + j*l*m*o + i*j*m*p + i*j*m*o + i*j*l*q + i*j*l*o + i*j*m*q + j*l*o*p + g*j*o*p + g*j*m*q + g*j*m*p + i*j*l*p + i*j*o*p + j*l*o*q + g*j*m*o + g*j*l*q + g*j*l*p + g*j*l*o + g*l*o*p + g*l*m*p + g*l*m*o + g*i*m*o + g*i*o*p + g*i*m*q + g*i*m*p + g*i*l*q + g*i*l*p + g*i*l*o + l*m*q*g))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
+        f: (r*e*j*(l*q*o + l*o*p + l*m*q + l*m*p + l*m*o + i*o*q + i*o*p + i*m*q + i*m*p + i*m*o + i*l*q + i*l*p + i*l*o))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
+        h: (j*e*r*l*(o*q + o*p + m*q + m*p + m*o))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
+        k: (j*e*r*o*l*(q + p))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
+        n: (j*e*r*o*q*l)/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
+    }
+
+    assert solve_lin_sys(eqs, domain) == sol
+
+def test_eqs_to_matrix():
+    domain, x1,x2 = ring("x1,x2", QQ)
+    eqs_coeff = [{x1: QQ(1), x2: QQ(1)}, {x1: QQ(2), x2: QQ(-1)}]
+    eqs_rhs = [QQ(-5), QQ(0)]
+    M = eqs_to_matrix(eqs_coeff, eqs_rhs, [x1, x2], QQ)
+    assert M.to_Matrix() == Matrix([[1, 1, 5], [2, -1, 0]])

rl/Lib/site-packages/sympy/polys/tests/test_specialpolys.py

@@ -0,0 +1,152 @@
+"""Tests for functions for generating interesting polynomials. """
+
+from sympy.core.add import Add
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.ntheory.generate import prime
+from sympy.polys.domains.integerring import ZZ
+from sympy.polys.polytools import Poly
+from sympy.utilities.iterables import permute_signs
+from sympy.testing.pytest import raises
+
+from sympy.polys.specialpolys import (
+    swinnerton_dyer_poly,
+    cyclotomic_poly,
+    symmetric_poly,
+    random_poly,
+    interpolating_poly,
+    fateman_poly_F_1,
+    dmp_fateman_poly_F_1,
+    fateman_poly_F_2,
+    dmp_fateman_poly_F_2,
+    fateman_poly_F_3,
+    dmp_fateman_poly_F_3,
+)
+
+from sympy.abc import x, y, z
+
+
+def test_swinnerton_dyer_poly():
+    raises(ValueError, lambda: swinnerton_dyer_poly(0, x))
+
+    assert swinnerton_dyer_poly(1, x, polys=True) == Poly(x**2 - 2)
+
+    assert swinnerton_dyer_poly(1, x) == x**2 - 2
+    assert swinnerton_dyer_poly(2, x) == x**4 - 10*x**2 + 1
+    assert swinnerton_dyer_poly(
+        3, x) == x**8 - 40*x**6 + 352*x**4 - 960*x**2 + 576
+    # we only need to check that the polys arg works but
+    # we may as well test that the roots are correct
+    p = [sqrt(prime(i)) for i in range(1, 5)]
+    assert str([i.n(3) for i in
+        swinnerton_dyer_poly(4, polys=True).all_roots()]
+        ) == str(sorted([Add(*i).n(3) for i in permute_signs(p)]))
+
+
+def test_cyclotomic_poly():
+    raises(ValueError, lambda: cyclotomic_poly(0, x))
+
+    assert cyclotomic_poly(1, x, polys=True) == Poly(x - 1)
+
+    assert cyclotomic_poly(1, x) == x - 1
+    assert cyclotomic_poly(2, x) == x + 1
+    assert cyclotomic_poly(3, x) == x**2 + x + 1
+    assert cyclotomic_poly(4, x) == x**2 + 1
+    assert cyclotomic_poly(5, x) == x**4 + x**3 + x**2 + x + 1
+    assert cyclotomic_poly(6, x) == x**2 - x + 1
+
+
+def test_symmetric_poly():
+    raises(ValueError, lambda: symmetric_poly(-1, x, y, z))
+    raises(ValueError, lambda: symmetric_poly(5, x, y, z))
+
+    assert symmetric_poly(1, x, y, z, polys=True) == Poly(x + y + z)
+    assert symmetric_poly(1, (x, y, z), polys=True) == Poly(x + y + z)
+
+    assert symmetric_poly(0, x, y, z) == 1
+    assert symmetric_poly(1, x, y, z) == x + y + z
+    assert symmetric_poly(2, x, y, z) == x*y + x*z + y*z
+    assert symmetric_poly(3, x, y, z) == x*y*z
+
+
+def test_random_poly():
+    poly = random_poly(x, 10, -100, 100, polys=False)
+
+    assert Poly(poly).degree() == 10
+    assert all(-100 <= coeff <= 100 for coeff in Poly(poly).coeffs()) is True
+
+    poly = random_poly(x, 10, -100, 100, polys=True)
+
+    assert poly.degree() == 10
+    assert all(-100 <= coeff <= 100 for coeff in poly.coeffs()) is True
+
+
+def test_interpolating_poly():
+    x0, x1, x2, x3, y0, y1, y2, y3 = symbols('x:4, y:4')
+
+    assert interpolating_poly(0, x) == 0
+    assert interpolating_poly(1, x) == y0
+
+    assert interpolating_poly(2, x) == \
+        y0*(x - x1)/(x0 - x1) + y1*(x - x0)/(x1 - x0)
+
+    assert interpolating_poly(3, x) == \
+        y0*(x - x1)*(x - x2)/((x0 - x1)*(x0 - x2)) + \
+        y1*(x - x0)*(x - x2)/((x1 - x0)*(x1 - x2)) + \
+        y2*(x - x0)*(x - x1)/((x2 - x0)*(x2 - x1))
+
+    assert interpolating_poly(4, x) == \
+        y0*(x - x1)*(x - x2)*(x - x3)/((x0 - x1)*(x0 - x2)*(x0 - x3)) + \
+        y1*(x - x0)*(x - x2)*(x - x3)/((x1 - x0)*(x1 - x2)*(x1 - x3)) + \
+        y2*(x - x0)*(x - x1)*(x - x3)/((x2 - x0)*(x2 - x1)*(x2 - x3)) + \
+        y3*(x - x0)*(x - x1)*(x - x2)/((x3 - x0)*(x3 - x1)*(x3 - x2))
+
+    raises(ValueError, lambda:
+        interpolating_poly(2, x, (x, 2), (1, 3)))
+    raises(ValueError, lambda:
+        interpolating_poly(2, x, (x + y, 2), (1, 3)))
+    raises(ValueError, lambda:
+        interpolating_poly(2, x + y, (x, 2), (1, 3)))
+    raises(ValueError, lambda:
+        interpolating_poly(2, 3, (4, 5), (6, 7)))
+    raises(ValueError, lambda:
+        interpolating_poly(2, 3, (4, 5), (6, 7, 8)))
+    assert interpolating_poly(0, x, (1, 2), (3, 4)) == 0
+    assert interpolating_poly(1, x, (1, 2), (3, 4)) == 3
+    assert interpolating_poly(2, x, (1, 2), (3, 4)) == x + 2
+
+
+def test_fateman_poly_F_1():
+    f, g, h = fateman_poly_F_1(1)
+    F, G, H = dmp_fateman_poly_F_1(1, ZZ)
+
+    assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H]
+
+    f, g, h = fateman_poly_F_1(3)
+    F, G, H = dmp_fateman_poly_F_1(3, ZZ)
+
+    assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H]
+
+
+def test_fateman_poly_F_2():
+    f, g, h = fateman_poly_F_2(1)
+    F, G, H = dmp_fateman_poly_F_2(1, ZZ)
+
+    assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H]
+
+    f, g, h = fateman_poly_F_2(3)
+    F, G, H = dmp_fateman_poly_F_2(3, ZZ)
+
+    assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H]
+
+
+def test_fateman_poly_F_3():
+    f, g, h = fateman_poly_F_3(1)
+    F, G, H = dmp_fateman_poly_F_3(1, ZZ)
+
+    assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H]
+
+    f, g, h = fateman_poly_F_3(3)
+    F, G, H = dmp_fateman_poly_F_3(3, ZZ)
+
+    assert [ t.rep.to_list() for t in [f, g, h] ] == [F, G, H]

rl/Lib/site-packages/sympy/polys/tests/test_sqfreetools.py

@@ -0,0 +1,160 @@
+"""Tests for square-free decomposition algorithms and related tools. """
+
+from sympy.polys.rings import ring
+from sympy.polys.domains import FF, ZZ, QQ
+from sympy.polys.specialpolys import f_polys
+
+from sympy.testing.pytest import raises
+from sympy.external.gmpy import MPQ
+
+f_0, f_1, f_2, f_3, f_4, f_5, f_6 = f_polys()
+
+def test_dup_sqf():
+    R, x = ring("x", ZZ)
+
+    assert R.dup_sqf_part(0) == 0
+    assert R.dup_sqf_p(0) is True
+
+    assert R.dup_sqf_part(7) == 1
+    assert R.dup_sqf_p(7) is True
+
+    assert R.dup_sqf_part(2*x + 2) == x + 1
+    assert R.dup_sqf_p(2*x + 2) is True
+
+    assert R.dup_sqf_part(x**3 + x + 1) == x**3 + x + 1
+    assert R.dup_sqf_p(x**3 + x + 1) is True
+
+    assert R.dup_sqf_part(-x**3 + x + 1) == x**3 - x - 1
+    assert R.dup_sqf_p(-x**3 + x + 1) is True
+
+    assert R.dup_sqf_part(2*x**3 + 3*x**2) == 2*x**2 + 3*x
+    assert R.dup_sqf_p(2*x**3 + 3*x**2) is False
+
+    assert R.dup_sqf_part(-2*x**3 + 3*x**2) == 2*x**2 - 3*x
+    assert R.dup_sqf_p(-2*x**3 + 3*x**2) is False
+
+    assert R.dup_sqf_list(0) == (0, [])
+    assert R.dup_sqf_list(1) == (1, [])
+
+    assert R.dup_sqf_list(x) == (1, [(x, 1)])
+    assert R.dup_sqf_list(2*x**2) == (2, [(x, 2)])
+    assert R.dup_sqf_list(3*x**3) == (3, [(x, 3)])
+
+    assert R.dup_sqf_list(-x**5 + x**4 + x - 1) == \
+        (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)])
+    assert R.dup_sqf_list(x**8 + 6*x**6 + 12*x**4 + 8*x**2) == \
+        ( 1, [(x, 2), (x**2 + 2, 3)])
+
+    assert R.dup_sqf_list(2*x**2 + 4*x + 2) == (2, [(x + 1, 2)])
+
+    R, x = ring("x", QQ)
+    assert R.dup_sqf_list(2*x**2 + 4*x + 2) == (2, [(x + 1, 2)])
+
+    R, x = ring("x", FF(2))
+    assert R.dup_sqf_list(x**2 + 1) == (1, [(x + 1, 2)])
+
+    R, x = ring("x", FF(3))
+    assert R.dup_sqf_list(x**10 + 2*x**7 + 2*x**4 + x) == \
+        (1, [(x, 1),
+             (x + 1, 3),
+             (x + 2, 6)])
+
+    R1, x = ring("x", ZZ)
+    R2, y = ring("y", FF(3))
+
+    f = x**3 + 1
+    g = y**3 + 1
+
+    assert R1.dup_sqf_part(f) == f
+    assert R2.dup_sqf_part(g) == y + 1
+
+    assert R1.dup_sqf_p(f) is True
+    assert R2.dup_sqf_p(g) is False
+
+    R, x, y = ring("x,y", ZZ)
+
+    A = x**4 - 3*x**2 + 6
+    D = x**6 - 5*x**4 + 5*x**2 + 4
+
+    f, g = D, R.dmp_sub(A, R.dmp_mul(R.dmp_diff(D, 1), y))
+    res = R.dmp_resultant(f, g)
+    h = (4*y**2 + 1).drop(x)
+
+    assert R.drop(x).dup_sqf_list(res) == (45796, [(h, 3)])
+
+    Rt, t = ring("t", ZZ)
+    R, x = ring("x", Rt)
+    assert R.dup_sqf_list_include(t**3*x**2) == [(t**3, 1), (x, 2)]
+
+
+def test_dmp_sqf():
+    R, x, y = ring("x,y", ZZ)
+    assert R.dmp_sqf_part(0) == 0
+    assert R.dmp_sqf_p(0) is True
+
+    assert R.dmp_sqf_part(7) == 1
+    assert R.dmp_sqf_p(7) is True
+
+    assert R.dmp_sqf_list(3) == (3, [])
+    assert R.dmp_sqf_list_include(3) == [(3, 1)]
+
+    R, x, y, z = ring("x,y,z", ZZ)
+    assert R.dmp_sqf_p(f_0) is True
+    assert R.dmp_sqf_p(f_0**2) is False
+    assert R.dmp_sqf_p(f_1) is True
+    assert R.dmp_sqf_p(f_1**2) is False
+    assert R.dmp_sqf_p(f_2) is True
+    assert R.dmp_sqf_p(f_2**2) is False
+    assert R.dmp_sqf_p(f_3) is True
+    assert R.dmp_sqf_p(f_3**2) is False
+    assert R.dmp_sqf_p(f_5) is False
+    assert R.dmp_sqf_p(f_5**2) is False
+
+    assert R.dmp_sqf_p(f_4) is True
+    assert R.dmp_sqf_part(f_4) == -f_4
+
+    assert R.dmp_sqf_part(f_5) == x + y - z
+
+    R, x, y, z, t = ring("x,y,z,t", ZZ)
+    assert R.dmp_sqf_p(f_6) is True
+    assert R.dmp_sqf_part(f_6) == f_6
+
+    R, x = ring("x", ZZ)
+    f = -x**5 + x**4 + x - 1
+
+    assert R.dmp_sqf_list(f) == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)])
+    assert R.dmp_sqf_list_include(f) == [(-x**3 - x**2 - x - 1, 1), (x - 1, 2)]
+
+    R, x, y = ring("x,y", ZZ)
+    f = -x**5 + x**4 + x - 1
+
+    assert R.dmp_sqf_list(f) == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)])
+    assert R.dmp_sqf_list_include(f) == [(-x**3 - x**2 - x - 1, 1), (x - 1, 2)]
+
+    f = -x**2 + 2*x - 1
+    assert R.dmp_sqf_list_include(f) == [(-1, 1), (x - 1, 2)]
+
+    f = (y**2 + 1)**2*(x**2 + 2*x + 2)
+    assert R.dmp_sqf_p(f) is False
+    assert R.dmp_sqf_list(f) == (1, [(x**2 + 2*x + 2, 1), (y**2 + 1, 2)])
+
+    R, x, y = ring("x,y", FF(2))
+    raises(NotImplementedError, lambda: R.dmp_sqf_list(y**2 + 1))
+
+
+def test_dup_gff_list():
+    R, x = ring("x", ZZ)
+
+    f = x**5 + 2*x**4 - x**3 - 2*x**2
+    assert R.dup_gff_list(f) == [(x, 1), (x + 2, 4)]
+
+    g = x**9 - 20*x**8 + 166*x**7 - 744*x**6 + 1965*x**5 - 3132*x**4 + 2948*x**3 - 1504*x**2 + 320*x
+    assert R.dup_gff_list(g) == [(x**2 - 5*x + 4, 1), (x**2 - 5*x + 4, 2), (x, 3)]
+
+    raises(ValueError, lambda: R.dup_gff_list(0))
+
+def test_issue_26178():
+    R, x, y, z = ring(['x', 'y', 'z'], QQ)
+    assert (x**2 - 2*y**2 + 1).sqf_list() == (MPQ(1,1), [(x**2 - 2*y**2 + 1, 1)])
+    assert (x**2 - 2*z**2 + 1).sqf_list() == (MPQ(1,1), [(x**2 - 2*z**2 + 1, 1)])
+    assert (y**2 - 2*z**2 + 1).sqf_list() == (MPQ(1,1), [(y**2 - 2*z**2 + 1, 1)])

rl/Lib/site-packages/sympy/polys/tests/test_subresultants_qq_zz.py

@@ -0,0 +1,347 @@
+from sympy.core.symbol import var
+from sympy.polys.polytools import (pquo, prem, sturm, subresultants)
+from sympy.matrices import Matrix
+from sympy.polys.subresultants_qq_zz import (sylvester, res, res_q, res_z, bezout,
+    subresultants_sylv,   modified_subresultants_sylv,
+    subresultants_bezout, modified_subresultants_bezout,
+    backward_eye,
+    sturm_pg, sturm_q, sturm_amv, euclid_pg, euclid_q,
+    euclid_amv, modified_subresultants_pg, subresultants_pg,
+    subresultants_amv_q, quo_z, rem_z, subresultants_amv,
+    modified_subresultants_amv, subresultants_rem,
+    subresultants_vv, subresultants_vv_2)
+
+
+def test_sylvester():
+    x = var('x')
+
+    assert sylvester(x**3 -7, 0, x) == sylvester(x**3 -7, 0, x, 1) == Matrix([[0]])
+    assert sylvester(0, x**3 -7, x) == sylvester(0, x**3 -7, x, 1) == Matrix([[0]])
+    assert sylvester(x**3 -7, 0, x, 2) == Matrix([[0]])
+    assert sylvester(0, x**3 -7, x, 2) == Matrix([[0]])
+
+    assert sylvester(x**3 -7, 7, x).det() == sylvester(x**3 -7, 7, x, 1).det() == 343
+    assert sylvester(7, x**3 -7, x).det() == sylvester(7, x**3 -7, x, 1).det() == 343
+    assert sylvester(x**3 -7, 7, x, 2).det() == -343
+    assert sylvester(7, x**3 -7, x, 2).det() == 343
+
+    assert sylvester(3, 7, x).det() == sylvester(3, 7, x, 1).det() == sylvester(3, 7, x, 2).det() == 1
+
+    assert sylvester(3, 0, x).det() == sylvester(3, 0, x, 1).det() == sylvester(3, 0, x, 2).det() == 1
+
+    assert sylvester(x - 3, x - 8, x) == sylvester(x - 3, x - 8, x, 1) == sylvester(x - 3, x - 8, x, 2) == Matrix([[1, -3], [1, -8]])
+
+    assert sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x) == sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x, 1) == Matrix([[1, 0, -7,  7,  0], [0, 1,  0, -7,  7], [3, 0, -7,  0,  0], [0, 3,  0, -7,  0], [0, 0,  3,  0, -7]])
+
+    assert sylvester(x**3 - 7*x + 7, 3*x**2 - 7, x, 2) == Matrix([
+[1, 0, -7,  7,  0,  0], [0, 3,  0, -7,  0,  0], [0, 1,  0, -7,  7,  0], [0, 0,  3,  0, -7,  0], [0, 0,  1,  0, -7,  7], [0, 0,  0,  3,  0, -7]])
+
+def test_subresultants_sylv():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_sylv(p, q, x) == subresultants(p, q, x)
+    assert subresultants_sylv(p, q, x)[-1] == res(p, q, x)
+    assert subresultants_sylv(p, q, x) != euclid_amv(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_sylv(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_sylv(p, q, x) == euclid_amv(p, q, x)
+
+def test_modified_subresultants_sylv():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert modified_subresultants_sylv(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))]
+    assert modified_subresultants_sylv(p, q, x)[-1] != res_q(p + x**8, q, x)
+    assert modified_subresultants_sylv(p, q, x) != sturm_amv(p, q, x)
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert modified_subresultants_sylv(p, q, x) == sturm_amv(p, q, x)
+    assert modified_subresultants_sylv(-p, q, x) != sturm_amv(-p, q, x)
+
+def test_res():
+    x = var('x')
+
+    assert res(3, 5, x) == 1
+
+def test_res_q():
+    x = var('x')
+
+    assert res_q(3, 5, x) == 1
+
+def test_res_z():
+    x = var('x')
+
+    assert res_z(3, 5, x) == 1
+    assert res(3, 5, x) == res_q(3, 5, x) == res_z(3, 5, x)
+
+def test_bezout():
+    x = var('x')
+
+    p = -2*x**5+7*x**3+9*x**2-3*x+1
+    q = -10*x**4+21*x**2+18*x-3
+    assert bezout(p, q, x, 'bz').det() == sylvester(p, q, x, 2).det()
+    assert bezout(p, q, x, 'bz').det() != sylvester(p, q, x, 1).det()
+    assert bezout(p, q, x, 'prs') == backward_eye(5) * bezout(p, q, x, 'bz') * backward_eye(5)
+
+def test_subresultants_bezout():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_bezout(p, q, x) == subresultants(p, q, x)
+    assert subresultants_bezout(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert subresultants_bezout(p, q, x) != euclid_amv(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_bezout(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_bezout(p, q, x) == euclid_amv(p, q, x)
+
+def test_modified_subresultants_bezout():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert modified_subresultants_bezout(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))]
+    assert modified_subresultants_bezout(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
+    assert modified_subresultants_bezout(p, q, x) != sturm_amv(p, q, x)
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert modified_subresultants_bezout(p, q, x) == sturm_amv(p, q, x)
+    assert modified_subresultants_bezout(-p, q, x) != sturm_amv(-p, q, x)
+
+def test_sturm_pg():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert sturm_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
+    sam_factors = [1, 1, -1, -1, 1, 1]
+    assert sturm_pg(p, q, x) == [i*j for i,j in zip(sam_factors, euclid_pg(p, q, x))]
+
+    p = -9*x**5 - 5*x**3 - 9
+    q = -45*x**4 - 15*x**2
+    assert sturm_pg(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det()
+    assert sturm_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
+    assert sturm_pg(-p, q, x)[-1] == sylvester(-p, q, x, 2).det()
+    assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x)
+
+def test_sturm_q():
+    x = var('x')
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert sturm_q(p, q, x) == sturm(p)
+    assert sturm_q(-p, -q, x) != sturm(-p)
+
+
+def test_sturm_amv():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
+    sam_factors = [1, 1, -1, -1, 1, 1]
+    assert sturm_amv(p, q, x) == [i*j for i,j in zip(sam_factors, euclid_amv(p, q, x))]
+
+    p = -9*x**5 - 5*x**3 - 9
+    q = -45*x**4 - 15*x**2
+    assert sturm_amv(p, q, x, 1)[-1] == sylvester(p, q, x, 1).det()
+    assert sturm_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
+    assert sturm_amv(-p, q, x)[-1] == sylvester(-p, q, x, 2).det()
+    assert sturm_pg(-p, q, x) == modified_subresultants_pg(-p, q, x)
+
+
+def test_euclid_pg():
+    x = var('x')
+
+    p = x**6+x**5-x**4-x**3+x**2-x+1
+    q = 6*x**5+5*x**4-4*x**3-3*x**2+2*x-1
+    assert euclid_pg(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert euclid_pg(p, q, x) == subresultants_pg(p, q, x)
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert euclid_pg(p, q, x)[-1] != sylvester(p, q, x, 2).det()
+    sam_factors = [1, 1, -1, -1, 1, 1]
+    assert euclid_pg(p, q, x) == [i*j for i,j in zip(sam_factors, sturm_pg(p, q, x))]
+
+
+def test_euclid_q():
+    x = var('x')
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert euclid_q(p, q, x)[-1] == -sturm(p)[-1]
+
+
+def test_euclid_amv():
+    x = var('x')
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert euclid_amv(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert euclid_amv(p, q, x) == subresultants_amv(p, q, x)
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert euclid_amv(p, q, x)[-1] != sylvester(p, q, x, 2).det()
+    sam_factors = [1, 1, -1, -1, 1, 1]
+    assert euclid_amv(p, q, x) == [i*j for i,j in zip(sam_factors, sturm_amv(p, q, x))]
+
+
+def test_modified_subresultants_pg():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert modified_subresultants_pg(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_pg(p, q, x))]
+    assert modified_subresultants_pg(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
+    assert modified_subresultants_pg(p, q, x) != sturm_pg(p, q, x)
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert modified_subresultants_pg(p, q, x) == sturm_pg(p, q, x)
+    assert modified_subresultants_pg(-p, q, x) != sturm_pg(-p, q, x)
+
+
+def test_subresultants_pg():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_pg(p, q, x) == subresultants(p, q, x)
+    assert subresultants_pg(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert subresultants_pg(p, q, x) != euclid_pg(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_pg(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_pg(p, q, x) == euclid_pg(p, q, x)
+
+
+def test_subresultants_amv_q():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_amv_q(p, q, x) == subresultants(p, q, x)
+    assert subresultants_amv_q(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert subresultants_amv_q(p, q, x) != euclid_amv(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_amv_q(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_amv(p, q, x) == euclid_amv(p, q, x)
+
+
+def test_rem_z():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert rem_z(p, -q, x) != prem(p, -q, x)
+
+def test_quo_z():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert quo_z(p, -q, x) != pquo(p, -q, x)
+
+    y = var('y')
+    q = 3*x**6 + 5*y**4 - 4*x**2 - 9*x + 21
+    assert quo_z(p, -q, x) == pquo(p, -q, x)
+
+def test_subresultants_amv():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_amv(p, q, x) == subresultants(p, q, x)
+    assert subresultants_amv(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert subresultants_amv(p, q, x) != euclid_amv(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_amv(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_amv(p, q, x) == euclid_amv(p, q, x)
+
+
+def test_modified_subresultants_amv():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert modified_subresultants_amv(p, q, x) == [i*j for i, j in zip(amv_factors, subresultants_amv(p, q, x))]
+    assert modified_subresultants_amv(p, q, x)[-1] != sylvester(p + x**8, q, x).det()
+    assert modified_subresultants_amv(p, q, x) != sturm_amv(p, q, x)
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert modified_subresultants_amv(p, q, x) == sturm_amv(p, q, x)
+    assert modified_subresultants_amv(-p, q, x) != sturm_amv(-p, q, x)
+
+
+def test_subresultants_rem():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_rem(p, q, x) == subresultants(p, q, x)
+    assert subresultants_rem(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert subresultants_rem(p, q, x) != euclid_amv(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_rem(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_rem(p, q, x) == euclid_amv(p, q, x)
+
+
+def test_subresultants_vv():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_vv(p, q, x) == subresultants(p, q, x)
+    assert subresultants_vv(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert subresultants_vv(p, q, x) != euclid_amv(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_vv(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_vv(p, q, x) == euclid_amv(p, q, x)
+
+
+def test_subresultants_vv_2():
+    x = var('x')
+
+    p = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
+    q = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
+    assert subresultants_vv_2(p, q, x) == subresultants(p, q, x)
+    assert subresultants_vv_2(p, q, x)[-1] == sylvester(p, q, x).det()
+    assert subresultants_vv_2(p, q, x) != euclid_amv(p, q, x)
+    amv_factors = [1, 1, -1, 1, -1, 1]
+    assert subresultants_vv_2(p, q, x) == [i*j for i, j in zip(amv_factors, modified_subresultants_amv(p, q, x))]
+
+    p = x**3 - 7*x + 7
+    q = 3*x**2 - 7
+    assert subresultants_vv_2(p, q, x) == euclid_amv(p, q, x)