I am done

rl/Lib/site-packages/sympy/polys/__init__.py

@@ -0,0 +1,130 @@
+"""Polynomial manipulation algorithms and algebraic objects. """
+
+__all__ = [
+    'Poly', 'PurePoly', 'poly_from_expr', 'parallel_poly_from_expr', 'degree',
+    'total_degree', 'degree_list', 'LC', 'LM', 'LT', 'pdiv', 'prem', 'pquo',
+    'pexquo', 'div', 'rem', 'quo', 'exquo', 'half_gcdex', 'gcdex', 'invert',
+    'subresultants', 'resultant', 'discriminant', 'cofactors', 'gcd_list',
+    'gcd', 'lcm_list', 'lcm', 'terms_gcd', 'trunc', 'monic', 'content',
+    'primitive', 'compose', 'decompose', 'sturm', 'gff_list', 'gff',
+    'sqf_norm', 'sqf_part', 'sqf_list', 'sqf', 'factor_list', 'factor',
+    'intervals', 'refine_root', 'count_roots', 'all_roots', 'real_roots',
+    'nroots', 'ground_roots', 'nth_power_roots_poly', 'cancel', 'reduced',
+    'groebner', 'is_zero_dimensional', 'GroebnerBasis', 'poly',
+
+    'symmetrize', 'horner', 'interpolate', 'rational_interpolate', 'viete',
+
+    'together',
+
+    'BasePolynomialError', 'ExactQuotientFailed', 'PolynomialDivisionFailed',
+    'OperationNotSupported', 'HeuristicGCDFailed', 'HomomorphismFailed',
+    'IsomorphismFailed', 'ExtraneousFactors', 'EvaluationFailed',
+    'RefinementFailed', 'CoercionFailed', 'NotInvertible', 'NotReversible',
+    'NotAlgebraic', 'DomainError', 'PolynomialError', 'UnificationFailed',
+    'GeneratorsError', 'GeneratorsNeeded', 'ComputationFailed',
+    'UnivariatePolynomialError', 'MultivariatePolynomialError',
+    'PolificationFailed', 'OptionError', 'FlagError',
+
+    'minpoly', 'minimal_polynomial', 'primitive_element', 'field_isomorphism',
+    'to_number_field', 'isolate', 'round_two', 'prime_decomp',
+    'prime_valuation', 'galois_group',
+
+    'itermonomials', 'Monomial',
+
+    'lex', 'grlex', 'grevlex', 'ilex', 'igrlex', 'igrevlex',
+
+    'CRootOf', 'rootof', 'RootOf', 'ComplexRootOf', 'RootSum',
+
+    'roots',
+
+    'Domain', 'FiniteField', 'IntegerRing', 'RationalField', 'RealField',
+    'ComplexField', 'PythonFiniteField', 'GMPYFiniteField',
+    'PythonIntegerRing', 'GMPYIntegerRing', 'PythonRational',
+    'GMPYRationalField', 'AlgebraicField', 'PolynomialRing', 'FractionField',
+    'ExpressionDomain', 'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy',
+    'QQ_python', 'QQ_gmpy', 'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR',
+    'CC', 'EX', 'EXRAW',
+
+    'construct_domain',
+
+    'swinnerton_dyer_poly', 'cyclotomic_poly', 'symmetric_poly',
+    'random_poly', 'interpolating_poly',
+
+    'jacobi_poly', 'chebyshevt_poly', 'chebyshevu_poly', 'hermite_poly',
+    'hermite_prob_poly', 'legendre_poly', 'laguerre_poly',
+
+    'bernoulli_poly', 'bernoulli_c_poly', 'genocchi_poly', 'euler_poly',
+    'andre_poly',
+
+    'apart', 'apart_list', 'assemble_partfrac_list',
+
+    'Options',
+
+    'ring', 'xring', 'vring', 'sring',
+
+    'field', 'xfield', 'vfield', 'sfield'
+]
+
+from .polytools import (Poly, PurePoly, poly_from_expr,
+        parallel_poly_from_expr, degree, total_degree, degree_list, LC, LM,
+        LT, pdiv, prem, pquo, pexquo, div, rem, quo, exquo, half_gcdex, gcdex,
+        invert, subresultants, resultant, discriminant, cofactors, gcd_list,
+        gcd, lcm_list, lcm, terms_gcd, trunc, monic, content, primitive,
+        compose, decompose, sturm, gff_list, gff, sqf_norm, sqf_part,
+        sqf_list, sqf, factor_list, factor, intervals, refine_root,
+        count_roots, all_roots, real_roots, nroots, ground_roots,
+        nth_power_roots_poly, cancel, reduced, groebner, is_zero_dimensional,
+        GroebnerBasis, poly)
+
+from .polyfuncs import (symmetrize, horner, interpolate,
+        rational_interpolate, viete)
+
+from .rationaltools import together
+
+from .polyerrors import (BasePolynomialError, ExactQuotientFailed,
+        PolynomialDivisionFailed, OperationNotSupported, HeuristicGCDFailed,
+        HomomorphismFailed, IsomorphismFailed, ExtraneousFactors,
+        EvaluationFailed, RefinementFailed, CoercionFailed, NotInvertible,
+        NotReversible, NotAlgebraic, DomainError, PolynomialError,
+        UnificationFailed, GeneratorsError, GeneratorsNeeded,
+        ComputationFailed, UnivariatePolynomialError,
+        MultivariatePolynomialError, PolificationFailed, OptionError,
+        FlagError)
+
+from .numberfields import (minpoly, minimal_polynomial, primitive_element,
+        field_isomorphism, to_number_field, isolate, round_two, prime_decomp,
+        prime_valuation, galois_group)
+
+from .monomials import itermonomials, Monomial
+
+from .orderings import lex, grlex, grevlex, ilex, igrlex, igrevlex
+
+from .rootoftools import CRootOf, rootof, RootOf, ComplexRootOf, RootSum
+
+from .polyroots import roots
+
+from .domains import (Domain, FiniteField, IntegerRing, RationalField,
+        RealField, ComplexField, PythonFiniteField, GMPYFiniteField,
+        PythonIntegerRing, GMPYIntegerRing, PythonRational, GMPYRationalField,
+        AlgebraicField, PolynomialRing, FractionField, ExpressionDomain,
+        FF_python, FF_gmpy, ZZ_python, ZZ_gmpy, QQ_python, QQ_gmpy, GF, FF,
+        ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX, EXRAW)
+
+from .constructor import construct_domain
+
+from .specialpolys import (swinnerton_dyer_poly, cyclotomic_poly,
+        symmetric_poly, random_poly, interpolating_poly)
+
+from .orthopolys import (jacobi_poly, chebyshevt_poly, chebyshevu_poly,
+        hermite_poly, hermite_prob_poly, legendre_poly, laguerre_poly)
+
+from .appellseqs import (bernoulli_poly, bernoulli_c_poly, genocchi_poly,
+        euler_poly, andre_poly)
+
+from .partfrac import apart, apart_list, assemble_partfrac_list
+
+from .polyoptions import Options
+
+from .rings import ring, xring, vring, sring
+
+from .fields import field, xfield, vfield, sfield

rl/Lib/site-packages/sympy/polys/agca/__init__.py

@@ -0,0 +1,5 @@
+"""Module for algebraic geometry and commutative algebra."""
+
+from .homomorphisms import homomorphism
+
+__all__ = ['homomorphism']

rl/Lib/site-packages/sympy/polys/agca/extensions.py

@@ -0,0 +1,356 @@
+"""Finite extensions of ring domains."""
+
+from sympy.polys.domains.domain import Domain
+from sympy.polys.domains.domainelement import DomainElement
+from sympy.polys.polyerrors import (CoercionFailed, NotInvertible,
+        GeneratorsError)
+from sympy.polys.polytools import Poly
+from sympy.printing.defaults import DefaultPrinting
+
+
+class ExtensionElement(DomainElement, DefaultPrinting):
+    """
+    Element of a finite extension.
+
+    A class of univariate polynomials modulo the ``modulus``
+    of the extension ``ext``. It is represented by the
+    unique polynomial ``rep`` of lowest degree. Both
+    ``rep`` and the representation ``mod`` of ``modulus``
+    are of class DMP.
+
+    """
+    __slots__ = ('rep', 'ext')
+
+    def __init__(self, rep, ext):
+        self.rep = rep
+        self.ext = ext
+
+    def parent(f):
+        return f.ext
+
+    def as_expr(f):
+        return f.ext.to_sympy(f)
+
+    def __bool__(f):
+        return bool(f.rep)
+
+    def __pos__(f):
+        return f
+
+    def __neg__(f):
+        return ExtElem(-f.rep, f.ext)
+
+    def _get_rep(f, g):
+        if isinstance(g, ExtElem):
+            if g.ext == f.ext:
+                return g.rep
+            else:
+                return None
+        else:
+            try:
+                g = f.ext.convert(g)
+                return g.rep
+            except CoercionFailed:
+                return None
+
+    def __add__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem(f.rep + rep, f.ext)
+        else:
+            return NotImplemented
+
+    __radd__ = __add__
+
+    def __sub__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem(f.rep - rep, f.ext)
+        else:
+            return NotImplemented
+
+    def __rsub__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem(rep - f.rep, f.ext)
+        else:
+            return NotImplemented
+
+    def __mul__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem((f.rep * rep) % f.ext.mod, f.ext)
+        else:
+            return NotImplemented
+
+    __rmul__ = __mul__
+
+    def _divcheck(f):
+        """Raise if division is not implemented for this divisor"""
+        if not f:
+            raise NotInvertible('Zero divisor')
+        elif f.ext.is_Field:
+            return True
+        elif f.rep.is_ground and f.ext.domain.is_unit(f.rep.LC()):
+            return True
+        else:
+            # Some cases like (2*x + 2)/2 over ZZ will fail here. It is
+            # unclear how to implement division in general if the ground
+            # domain is not a field so for now it was decided to restrict the
+            # implementation to division by invertible constants.
+            msg = (f"Can not invert {f} in {f.ext}. "
+                    "Only division by invertible constants is implemented.")
+            raise NotImplementedError(msg)
+
+    def inverse(f):
+        """Multiplicative inverse.
+
+        Raises
+        ======
+
+        NotInvertible
+            If the element is a zero divisor.
+
+        """
+        f._divcheck()
+
+        if f.ext.is_Field:
+            invrep = f.rep.invert(f.ext.mod)
+        else:
+            R = f.ext.ring
+            invrep = R.exquo(R.one, f.rep)
+
+        return ExtElem(invrep, f.ext)
+
+    def __truediv__(f, g):
+        rep = f._get_rep(g)
+        if rep is None:
+            return NotImplemented
+        g = ExtElem(rep, f.ext)
+
+        try:
+            ginv = g.inverse()
+        except NotInvertible:
+            raise ZeroDivisionError(f"{f} / {g}")
+
+        return f * ginv
+
+    __floordiv__ = __truediv__
+
+    def __rtruediv__(f, g):
+        try:
+            g = f.ext.convert(g)
+        except CoercionFailed:
+            return NotImplemented
+        return g / f
+
+    __rfloordiv__ = __rtruediv__
+
+    def __mod__(f, g):
+        rep = f._get_rep(g)
+        if rep is None:
+            return NotImplemented
+        g = ExtElem(rep, f.ext)
+
+        try:
+            g._divcheck()
+        except NotInvertible:
+            raise ZeroDivisionError(f"{f} % {g}")
+
+        # Division where defined is always exact so there is no remainder
+        return f.ext.zero
+
+    def __rmod__(f, g):
+        try:
+            g = f.ext.convert(g)
+        except CoercionFailed:
+            return NotImplemented
+        return g % f
+
+    def __pow__(f, n):
+        if not isinstance(n, int):
+            raise TypeError("exponent of type 'int' expected")
+        if n < 0:
+            try:
+                f, n = f.inverse(), -n
+            except NotImplementedError:
+                raise ValueError("negative powers are not defined")
+
+        b = f.rep
+        m = f.ext.mod
+        r = f.ext.one.rep
+        while n > 0:
+            if n % 2:
+                r = (r*b) % m
+            b = (b*b) % m
+            n //= 2
+
+        return ExtElem(r, f.ext)
+
+    def __eq__(f, g):
+        if isinstance(g, ExtElem):
+            return f.rep == g.rep and f.ext == g.ext
+        else:
+            return NotImplemented
+
+    def __ne__(f, g):
+        return not f == g
+
+    def __hash__(f):
+        return hash((f.rep, f.ext))
+
+    def __str__(f):
+        from sympy.printing.str import sstr
+        return sstr(f.as_expr())
+
+    __repr__ = __str__
+
+    @property
+    def is_ground(f):
+        return f.rep.is_ground
+
+    def to_ground(f):
+        [c] = f.rep.to_list()
+        return c
+
+ExtElem = ExtensionElement
+
+
+class MonogenicFiniteExtension(Domain):
+    r"""
+    Finite extension generated by an integral element.
+
+    The generator is defined by a monic univariate
+    polynomial derived from the argument ``mod``.
+
+    A shorter alias is ``FiniteExtension``.
+
+    Examples
+    ========
+
+    Quadratic integer ring $\mathbb{Z}[\sqrt2]$:
+
+    >>> from sympy import Symbol, Poly
+    >>> from sympy.polys.agca.extensions import FiniteExtension
+    >>> x = Symbol('x')
+    >>> R = FiniteExtension(Poly(x**2 - 2)); R
+    ZZ[x]/(x**2 - 2)
+    >>> R.rank
+    2
+    >>> R(1 + x)*(3 - 2*x)
+    x - 1
+
+    Finite field $GF(5^3)$ defined by the primitive
+    polynomial $x^3 + x^2 + 2$ (over $\mathbb{Z}_5$).
+
+    >>> F = FiniteExtension(Poly(x**3 + x**2 + 2, modulus=5)); F
+    GF(5)[x]/(x**3 + x**2 + 2)
+    >>> F.basis
+    (1, x, x**2)
+    >>> F(x + 3)/(x**2 + 2)
+    -2*x**2 + x + 2
+
+    Function field of an elliptic curve:
+
+    >>> t = Symbol('t')
+    >>> FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
+    ZZ(x)[t]/(t**2 - x**3 - x + 1)
+
+    """
+    is_FiniteExtension = True
+
+    dtype = ExtensionElement
+
+    def __init__(self, mod):
+        if not (isinstance(mod, Poly) and mod.is_univariate):
+            raise TypeError("modulus must be a univariate Poly")
+
+        # Using auto=True (default) potentially changes the ground domain to a
+        # field whereas auto=False raises if division is not exact.  We'll let
+        # the caller decide whether or not they want to put the ground domain
+        # over a field. In most uses mod is already monic.
+        mod = mod.monic(auto=False)
+
+        self.rank = mod.degree()
+        self.modulus = mod
+        self.mod = mod.rep  # DMP representation
+
+        self.domain = dom = mod.domain
+        self.ring = dom.old_poly_ring(*mod.gens)
+
+        self.zero = self.convert(self.ring.zero)
+        self.one = self.convert(self.ring.one)
+
+        gen = self.ring.gens[0]
+        self.symbol = self.ring.symbols[0]
+        self.generator = self.convert(gen)
+        self.basis = tuple(self.convert(gen**i) for i in range(self.rank))
+
+        # XXX: It might be necessary to check mod.is_irreducible here
+        self.is_Field = self.domain.is_Field
+
+    def new(self, arg):
+        rep = self.ring.convert(arg)
+        return ExtElem(rep % self.mod, self)
+
+    def __eq__(self, other):
+        if not isinstance(other, FiniteExtension):
+            return False
+        return self.modulus == other.modulus
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.modulus))
+
+    def __str__(self):
+        return "%s/(%s)" % (self.ring, self.modulus.as_expr())
+
+    __repr__ = __str__
+
+    @property
+    def has_CharacteristicZero(self):
+        return self.domain.has_CharacteristicZero
+
+    def characteristic(self):
+        return self.domain.characteristic()
+
+    def convert(self, f, base=None):
+        rep = self.ring.convert(f, base)
+        return ExtElem(rep % self.mod, self)
+
+    def convert_from(self, f, base):
+        rep = self.ring.convert(f, base)
+        return ExtElem(rep % self.mod, self)
+
+    def to_sympy(self, f):
+        return self.ring.to_sympy(f.rep)
+
+    def from_sympy(self, f):
+        return self.convert(f)
+
+    def set_domain(self, K):
+        mod = self.modulus.set_domain(K)
+        return self.__class__(mod)
+
+    def drop(self, *symbols):
+        if self.symbol in symbols:
+            raise GeneratorsError('Can not drop generator from FiniteExtension')
+        K = self.domain.drop(*symbols)
+        return self.set_domain(K)
+
+    def quo(self, f, g):
+        return self.exquo(f, g)
+
+    def exquo(self, f, g):
+        rep = self.ring.exquo(f.rep, g.rep)
+        return ExtElem(rep % self.mod, self)
+
+    def is_negative(self, a):
+        return False
+
+    def is_unit(self, a):
+        if self.is_Field:
+            return bool(a)
+        elif a.is_ground:
+            return self.domain.is_unit(a.to_ground())
+
+FiniteExtension = MonogenicFiniteExtension

rl/Lib/site-packages/sympy/polys/agca/homomorphisms.py

@@ -0,0 +1,691 @@
+"""
+Computations with homomorphisms of modules and rings.
+
+This module implements classes for representing homomorphisms of rings and
+their modules. Instead of instantiating the classes directly, you should use
+the function ``homomorphism(from, to, matrix)`` to create homomorphism objects.
+"""
+
+
+from sympy.polys.agca.modules import (Module, FreeModule, QuotientModule,
+    SubModule, SubQuotientModule)
+from sympy.polys.polyerrors import CoercionFailed
+
+# The main computational task for module homomorphisms is kernels.
+# For this reason, the concrete classes are organised by domain module type.
+
+
+class ModuleHomomorphism:
+    """
+    Abstract base class for module homomoprhisms. Do not instantiate.
+
+    Instead, use the ``homomorphism`` function:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> F = QQ.old_poly_ring(x).free_module(2)
+    >>> homomorphism(F, F, [[1, 0], [0, 1]])
+    Matrix([
+    [1, 0], : QQ[x]**2 -> QQ[x]**2
+    [0, 1]])
+
+    Attributes:
+
+    - ring - the ring over which we are considering modules
+    - domain - the domain module
+    - codomain - the codomain module
+    - _ker - cached kernel
+    - _img - cached image
+
+    Non-implemented methods:
+
+    - _kernel
+    - _image
+    - _restrict_domain
+    - _restrict_codomain
+    - _quotient_domain
+    - _quotient_codomain
+    - _apply
+    - _mul_scalar
+    - _compose
+    - _add
+    """
+
+    def __init__(self, domain, codomain):
+        if not isinstance(domain, Module):
+            raise TypeError('Source must be a module, got %s' % domain)
+        if not isinstance(codomain, Module):
+            raise TypeError('Target must be a module, got %s' % codomain)
+        if domain.ring != codomain.ring:
+            raise ValueError('Source and codomain must be over same ring, '
+                             'got %s != %s' % (domain, codomain))
+        self.domain = domain
+        self.codomain = codomain
+        self.ring = domain.ring
+        self._ker = None
+        self._img = None
+
+    def kernel(self):
+        r"""
+        Compute the kernel of ``self``.
+
+        That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute
+        `ker(\phi) = \{x \in M | \phi(x) = 0\}`.  This is a submodule of `M`.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> homomorphism(F, F, [[1, 0], [x, 0]]).kernel()
+        <[x, -1]>
+        """
+        if self._ker is None:
+            self._ker = self._kernel()
+        return self._ker
+
+    def image(self):
+        r"""
+        Compute the image of ``self``.
+
+        That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute
+        `im(\phi) = \{\phi(x) | x \in M \}`.  This is a submodule of `N`.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> homomorphism(F, F, [[1, 0], [x, 0]]).image() == F.submodule([1, 0])
+        True
+        """
+        if self._img is None:
+            self._img = self._image()
+        return self._img
+
+    def _kernel(self):
+        """Compute the kernel of ``self``."""
+        raise NotImplementedError
+
+    def _image(self):
+        """Compute the image of ``self``."""
+        raise NotImplementedError
+
+    def _restrict_domain(self, sm):
+        """Implementation of domain restriction."""
+        raise NotImplementedError
+
+    def _restrict_codomain(self, sm):
+        """Implementation of codomain restriction."""
+        raise NotImplementedError
+
+    def _quotient_domain(self, sm):
+        """Implementation of domain quotient."""
+        raise NotImplementedError
+
+    def _quotient_codomain(self, sm):
+        """Implementation of codomain quotient."""
+        raise NotImplementedError
+
+    def restrict_domain(self, sm):
+        """
+        Return ``self``, with the domain restricted to ``sm``.
+
+        Here ``sm`` has to be a submodule of ``self.domain``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.restrict_domain(F.submodule([1, 0]))
+        Matrix([
+        [1, x], : <[1, 0]> -> QQ[x]**2
+        [0, 0]])
+
+        This is the same as just composing on the right with the submodule
+        inclusion:
+
+        >>> h * F.submodule([1, 0]).inclusion_hom()
+        Matrix([
+        [1, x], : <[1, 0]> -> QQ[x]**2
+        [0, 0]])
+        """
+        if not self.domain.is_submodule(sm):
+            raise ValueError('sm must be a submodule of %s, got %s'
+                             % (self.domain, sm))
+        if sm == self.domain:
+            return self
+        return self._restrict_domain(sm)
+
+    def restrict_codomain(self, sm):
+        """
+        Return ``self``, with codomain restricted to to ``sm``.
+
+        Here ``sm`` has to be a submodule of ``self.codomain`` containing the
+        image.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.restrict_codomain(F.submodule([1, 0]))
+        Matrix([
+        [1, x], : QQ[x]**2 -> <[1, 0]>
+        [0, 0]])
+        """
+        if not sm.is_submodule(self.image()):
+            raise ValueError('the image %s must contain sm, got %s'
+                             % (self.image(), sm))
+        if sm == self.codomain:
+            return self
+        return self._restrict_codomain(sm)
+
+    def quotient_domain(self, sm):
+        """
+        Return ``self`` with domain replaced by ``domain/sm``.
+
+        Here ``sm`` must be a submodule of ``self.kernel()``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.quotient_domain(F.submodule([-x, 1]))
+        Matrix([
+        [1, x], : QQ[x]**2/<[-x, 1]> -> QQ[x]**2
+        [0, 0]])
+        """
+        if not self.kernel().is_submodule(sm):
+            raise ValueError('kernel %s must contain sm, got %s' %
+                             (self.kernel(), sm))
+        if sm.is_zero():
+            return self
+        return self._quotient_domain(sm)
+
+    def quotient_codomain(self, sm):
+        """
+        Return ``self`` with codomain replaced by ``codomain/sm``.
+
+        Here ``sm`` must be a submodule of ``self.codomain``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.quotient_codomain(F.submodule([1, 1]))
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2/<[1, 1]>
+        [0, 0]])
+
+        This is the same as composing with the quotient map on the left:
+
+        >>> (F/[(1, 1)]).quotient_hom() * h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2/<[1, 1]>
+        [0, 0]])
+        """
+        if not self.codomain.is_submodule(sm):
+            raise ValueError('sm must be a submodule of codomain %s, got %s'
+                             % (self.codomain, sm))
+        if sm.is_zero():
+            return self
+        return self._quotient_codomain(sm)
+
+    def _apply(self, elem):
+        """Apply ``self`` to ``elem``."""
+        raise NotImplementedError
+
+    def __call__(self, elem):
+        return self.codomain.convert(self._apply(self.domain.convert(elem)))
+
+    def _compose(self, oth):
+        """
+        Compose ``self`` with ``oth``, that is, return the homomorphism
+        obtained by first applying then ``self``, then ``oth``.
+
+        (This method is private since in this syntax, it is non-obvious which
+        homomorphism is executed first.)
+        """
+        raise NotImplementedError
+
+    def _mul_scalar(self, c):
+        """Scalar multiplication. ``c`` is guaranteed in self.ring."""
+        raise NotImplementedError
+
+    def _add(self, oth):
+        """
+        Homomorphism addition.
+        ``oth`` is guaranteed to be a homomorphism with same domain/codomain.
+        """
+        raise NotImplementedError
+
+    def _check_hom(self, oth):
+        """Helper to check that oth is a homomorphism with same domain/codomain."""
+        if not isinstance(oth, ModuleHomomorphism):
+            return False
+        return oth.domain == self.domain and oth.codomain == self.codomain
+
+    def __mul__(self, oth):
+        if isinstance(oth, ModuleHomomorphism) and self.domain == oth.codomain:
+            return oth._compose(self)
+        try:
+            return self._mul_scalar(self.ring.convert(oth))
+        except CoercionFailed:
+            return NotImplemented
+
+    # NOTE: _compose will never be called from rmul
+    __rmul__ = __mul__
+
+    def __truediv__(self, oth):
+        try:
+            return self._mul_scalar(1/self.ring.convert(oth))
+        except CoercionFailed:
+            return NotImplemented
+
+    def __add__(self, oth):
+        if self._check_hom(oth):
+            return self._add(oth)
+        return NotImplemented
+
+    def __sub__(self, oth):
+        if self._check_hom(oth):
+            return self._add(oth._mul_scalar(self.ring.convert(-1)))
+        return NotImplemented
+
+    def is_injective(self):
+        """
+        Return True if ``self`` is injective.
+
+        That is, check if the elements of the domain are mapped to the same
+        codomain element.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h.is_injective()
+        False
+        >>> h.quotient_domain(h.kernel()).is_injective()
+        True
+        """
+        return self.kernel().is_zero()
+
+    def is_surjective(self):
+        """
+        Return True if ``self`` is surjective.
+
+        That is, check if every element of the codomain has at least one
+        preimage.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h.is_surjective()
+        False
+        >>> h.restrict_codomain(h.image()).is_surjective()
+        True
+        """
+        return self.image() == self.codomain
+
+    def is_isomorphism(self):
+        """
+        Return True if ``self`` is an isomorphism.
+
+        That is, check if every element of the codomain has precisely one
+        preimage. Equivalently, ``self`` is both injective and surjective.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h = h.restrict_codomain(h.image())
+        >>> h.is_isomorphism()
+        False
+        >>> h.quotient_domain(h.kernel()).is_isomorphism()
+        True
+        """
+        return self.is_injective() and self.is_surjective()
+
+    def is_zero(self):
+        """
+        Return True if ``self`` is a zero morphism.
+
+        That is, check if every element of the domain is mapped to zero
+        under self.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h.is_zero()
+        False
+        >>> h.restrict_domain(F.submodule()).is_zero()
+        True
+        >>> h.quotient_codomain(h.image()).is_zero()
+        True
+        """
+        return self.image().is_zero()
+
+    def __eq__(self, oth):
+        try:
+            return (self - oth).is_zero()
+        except TypeError:
+            return False
+
+    def __ne__(self, oth):
+        return not (self == oth)
+
+
+class MatrixHomomorphism(ModuleHomomorphism):
+    r"""
+    Helper class for all homomoprhisms which are expressed via a matrix.
+
+    That is, for such homomorphisms ``domain`` is contained in a module
+    generated by finitely many elements `e_1, \ldots, e_n`, so that the
+    homomorphism is determined uniquely by its action on the `e_i`. It
+    can thus be represented as a vector of elements of the codomain module,
+    or potentially a supermodule of the codomain module
+    (and hence conventionally as a matrix, if there is a similar interpretation
+    for elements of the codomain module).
+
+    Note that this class does *not* assume that the `e_i` freely generate a
+    submodule, nor that ``domain`` is even all of this submodule. It exists
+    only to unify the interface.
+
+    Do not instantiate.
+
+    Attributes:
+
+    - matrix - the list of images determining the homomorphism.
+    NOTE: the elements of matrix belong to either self.codomain or
+          self.codomain.container
+
+    Still non-implemented methods:
+
+    - kernel
+    - _apply
+    """
+
+    def __init__(self, domain, codomain, matrix):
+        ModuleHomomorphism.__init__(self, domain, codomain)
+        if len(matrix) != domain.rank:
+            raise ValueError('Need to provide %s elements, got %s'
+                             % (domain.rank, len(matrix)))
+
+        converter = self.codomain.convert
+        if isinstance(self.codomain, (SubModule, SubQuotientModule)):
+            converter = self.codomain.container.convert
+        self.matrix = tuple(converter(x) for x in matrix)
+
+    def _sympy_matrix(self):
+        """Helper function which returns a SymPy matrix ``self.matrix``."""
+        from sympy.matrices import Matrix
+        c = lambda x: x
+        if isinstance(self.codomain, (QuotientModule, SubQuotientModule)):
+            c = lambda x: x.data
+        return Matrix([[self.ring.to_sympy(y) for y in c(x)] for x in self.matrix]).T
+
+    def __repr__(self):
+        lines = repr(self._sympy_matrix()).split('\n')
+        t = " : %s -> %s" % (self.domain, self.codomain)
+        s = ' '*len(t)
+        n = len(lines)
+        for i in range(n // 2):
+            lines[i] += s
+        lines[n // 2] += t
+        for i in range(n//2 + 1, n):
+            lines[i] += s
+        return '\n'.join(lines)
+
+    def _restrict_domain(self, sm):
+        """Implementation of domain restriction."""
+        return SubModuleHomomorphism(sm, self.codomain, self.matrix)
+
+    def _restrict_codomain(self, sm):
+        """Implementation of codomain restriction."""
+        return self.__class__(self.domain, sm, self.matrix)
+
+    def _quotient_domain(self, sm):
+        """Implementation of domain quotient."""
+        return self.__class__(self.domain/sm, self.codomain, self.matrix)
+
+    def _quotient_codomain(self, sm):
+        """Implementation of codomain quotient."""
+        Q = self.codomain/sm
+        converter = Q.convert
+        if isinstance(self.codomain, SubModule):
+            converter = Q.container.convert
+        return self.__class__(self.domain, self.codomain/sm,
+            [converter(x) for x in self.matrix])
+
+    def _add(self, oth):
+        return self.__class__(self.domain, self.codomain,
+                              [x + y for x, y in zip(self.matrix, oth.matrix)])
+
+    def _mul_scalar(self, c):
+        return self.__class__(self.domain, self.codomain, [c*x for x in self.matrix])
+
+    def _compose(self, oth):
+        return self.__class__(self.domain, oth.codomain, [oth(x) for x in self.matrix])
+
+
+class FreeModuleHomomorphism(MatrixHomomorphism):
+    """
+    Concrete class for homomorphisms with domain a free module or a quotient
+    thereof.
+
+    Do not instantiate; the constructor does not check that your data is well
+    defined. Use the ``homomorphism`` function instead:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> F = QQ.old_poly_ring(x).free_module(2)
+    >>> homomorphism(F, F, [[1, 0], [0, 1]])
+    Matrix([
+    [1, 0], : QQ[x]**2 -> QQ[x]**2
+    [0, 1]])
+    """
+
+    def _apply(self, elem):
+        if isinstance(self.domain, QuotientModule):
+            elem = elem.data
+        return sum(x * e for x, e in zip(elem, self.matrix))
+
+    def _image(self):
+        return self.codomain.submodule(*self.matrix)
+
+    def _kernel(self):
+        # The domain is either a free module or a quotient thereof.
+        # It does not matter if it is a quotient, because that won't increase
+        # the kernel.
+        # Our generators {e_i} are sent to the matrix entries {b_i}.
+        # The kernel is essentially the syzygy module of these {b_i}.
+        syz = self.image().syzygy_module()
+        return self.domain.submodule(*syz.gens)
+
+
+class SubModuleHomomorphism(MatrixHomomorphism):
+    """
+    Concrete class for homomorphism with domain a submodule of a free module
+    or a quotient thereof.
+
+    Do not instantiate; the constructor does not check that your data is well
+    defined. Use the ``homomorphism`` function instead:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> M = QQ.old_poly_ring(x).free_module(2)*x
+    >>> homomorphism(M, M, [[1, 0], [0, 1]])
+    Matrix([
+    [1, 0], : <[x, 0], [0, x]> -> <[x, 0], [0, x]>
+    [0, 1]])
+    """
+
+    def _apply(self, elem):
+        if isinstance(self.domain, SubQuotientModule):
+            elem = elem.data
+        return sum(x * e for x, e in zip(elem, self.matrix))
+
+    def _image(self):
+        return self.codomain.submodule(*[self(x) for x in self.domain.gens])
+
+    def _kernel(self):
+        syz = self.image().syzygy_module()
+        return self.domain.submodule(
+            *[sum(xi*gi for xi, gi in zip(s, self.domain.gens))
+              for s in syz.gens])
+
+
+def homomorphism(domain, codomain, matrix):
+    r"""
+    Create a homomorphism object.
+
+    This function tries to build a homomorphism from ``domain`` to ``codomain``
+    via the matrix ``matrix``.
+
+    Examples
+    ========
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> R = QQ.old_poly_ring(x)
+    >>> T = R.free_module(2)
+
+    If ``domain`` is a free module generated by `e_1, \ldots, e_n`, then
+    ``matrix`` should be an n-element iterable `(b_1, \ldots, b_n)` where
+    the `b_i` are elements of ``codomain``. The constructed homomorphism is the
+    unique homomorphism sending `e_i` to `b_i`.
+
+    >>> F = R.free_module(2)
+    >>> h = homomorphism(F, T, [[1, x], [x**2, 0]])
+    >>> h
+    Matrix([
+    [1, x**2], : QQ[x]**2 -> QQ[x]**2
+    [x,    0]])
+    >>> h([1, 0])
+    [1, x]
+    >>> h([0, 1])
+    [x**2, 0]
+    >>> h([1, 1])
+    [x**2 + 1, x]
+
+    If ``domain`` is a submodule of a free module, them ``matrix`` determines
+    a homomoprhism from the containing free module to ``codomain``, and the
+    homomorphism returned is obtained by restriction to ``domain``.
+
+    >>> S = F.submodule([1, 0], [0, x])
+    >>> homomorphism(S, T, [[1, x], [x**2, 0]])
+    Matrix([
+    [1, x**2], : <[1, 0], [0, x]> -> QQ[x]**2
+    [x,    0]])
+
+    If ``domain`` is a (sub)quotient `N/K`, then ``matrix`` determines a
+    homomorphism from `N` to ``codomain``. If the kernel contains `K`, this
+    homomorphism descends to ``domain`` and is returned; otherwise an exception
+    is raised.
+
+    >>> homomorphism(S/[(1, 0)], T, [0, [x**2, 0]])
+    Matrix([
+    [0, x**2], : <[1, 0] + <[1, 0]>, [0, x] + <[1, 0]>, [1, 0] + <[1, 0]>> -> QQ[x]**2
+    [0,    0]])
+    >>> homomorphism(S/[(0, x)], T, [0, [x**2, 0]])
+    Traceback (most recent call last):
+    ...
+    ValueError: kernel <[1, 0], [0, 0]> must contain sm, got <[0,x]>
+
+    """
+    def freepres(module):
+        """
+        Return a tuple ``(F, S, Q, c)`` where ``F`` is a free module, ``S`` is a
+        submodule of ``F``, and ``Q`` a submodule of ``S``, such that
+        ``module = S/Q``, and ``c`` is a conversion function.
+        """
+        if isinstance(module, FreeModule):
+            return module, module, module.submodule(), lambda x: module.convert(x)
+        if isinstance(module, QuotientModule):
+            return (module.base, module.base, module.killed_module,
+                    lambda x: module.convert(x).data)
+        if isinstance(module, SubQuotientModule):
+            return (module.base.container, module.base, module.killed_module,
+                    lambda x: module.container.convert(x).data)
+        # an ordinary submodule
+        return (module.container, module, module.submodule(),
+                lambda x: module.container.convert(x))
+
+    SF, SS, SQ, _ = freepres(domain)
+    TF, TS, TQ, c = freepres(codomain)
+    # NOTE this is probably a bit inefficient (redundant checks)
+    return FreeModuleHomomorphism(SF, TF, [c(x) for x in matrix]
+         ).restrict_domain(SS).restrict_codomain(TS
+         ).quotient_codomain(TQ).quotient_domain(SQ)

rl/Lib/site-packages/sympy/polys/agca/ideals.py

@@ -0,0 +1,395 @@
+"""Computations with ideals of polynomial rings."""
+
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.polys.polyutils import IntegerPowerable
+
+
+class Ideal(IntegerPowerable):
+    """
+    Abstract base class for ideals.
+
+    Do not instantiate - use explicit constructors in the ring class instead:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> QQ.old_poly_ring(x).ideal(x+1)
+    <x + 1>
+
+    Attributes
+
+    - ring - the ring this ideal belongs to
+
+    Non-implemented methods:
+
+    - _contains_elem
+    - _contains_ideal
+    - _quotient
+    - _intersect
+    - _union
+    - _product
+    - is_whole_ring
+    - is_zero
+    - is_prime, is_maximal, is_primary, is_radical
+    - is_principal
+    - height, depth
+    - radical
+
+    Methods that likely should be overridden in subclasses:
+
+    - reduce_element
+    """
+
+    def _contains_elem(self, x):
+        """Implementation of element containment."""
+        raise NotImplementedError
+
+    def _contains_ideal(self, I):
+        """Implementation of ideal containment."""
+        raise NotImplementedError
+
+    def _quotient(self, J):
+        """Implementation of ideal quotient."""
+        raise NotImplementedError
+
+    def _intersect(self, J):
+        """Implementation of ideal intersection."""
+        raise NotImplementedError
+
+    def is_whole_ring(self):
+        """Return True if ``self`` is the whole ring."""
+        raise NotImplementedError
+
+    def is_zero(self):
+        """Return True if ``self`` is the zero ideal."""
+        raise NotImplementedError
+
+    def _equals(self, J):
+        """Implementation of ideal equality."""
+        return self._contains_ideal(J) and J._contains_ideal(self)
+
+    def is_prime(self):
+        """Return True if ``self`` is a prime ideal."""
+        raise NotImplementedError
+
+    def is_maximal(self):
+        """Return True if ``self`` is a maximal ideal."""
+        raise NotImplementedError
+
+    def is_radical(self):
+        """Return True if ``self`` is a radical ideal."""
+        raise NotImplementedError
+
+    def is_primary(self):
+        """Return True if ``self`` is a primary ideal."""
+        raise NotImplementedError
+
+    def is_principal(self):
+        """Return True if ``self`` is a principal ideal."""
+        raise NotImplementedError
+
+    def radical(self):
+        """Compute the radical of ``self``."""
+        raise NotImplementedError
+
+    def depth(self):
+        """Compute the depth of ``self``."""
+        raise NotImplementedError
+
+    def height(self):
+        """Compute the height of ``self``."""
+        raise NotImplementedError
+
+    # TODO more
+
+    # non-implemented methods end here
+
+    def __init__(self, ring):
+        self.ring = ring
+
+    def _check_ideal(self, J):
+        """Helper to check ``J`` is an ideal of our ring."""
+        if not isinstance(J, Ideal) or J.ring != self.ring:
+            raise ValueError(
+                'J must be an ideal of %s, got %s' % (self.ring, J))
+
+    def contains(self, elem):
+        """
+        Return True if ``elem`` is an element of this ideal.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).ideal(x+1, x-1).contains(3)
+        True
+        >>> QQ.old_poly_ring(x).ideal(x**2, x**3).contains(x)
+        False
+        """
+        return self._contains_elem(self.ring.convert(elem))
+
+    def subset(self, other):
+        """
+        Returns True if ``other`` is is a subset of ``self``.
+
+        Here ``other`` may be an ideal.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> I = QQ.old_poly_ring(x).ideal(x+1)
+        >>> I.subset([x**2 - 1, x**2 + 2*x + 1])
+        True
+        >>> I.subset([x**2 + 1, x + 1])
+        False
+        >>> I.subset(QQ.old_poly_ring(x).ideal(x**2 - 1))
+        True
+        """
+        if isinstance(other, Ideal):
+            return self._contains_ideal(other)
+        return all(self._contains_elem(x) for x in other)
+
+    def quotient(self, J, **opts):
+        r"""
+        Compute the ideal quotient of ``self`` by ``J``.
+
+        That is, if ``self`` is the ideal `I`, compute the set
+        `I : J = \{x \in R | xJ \subset I \}`.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import QQ
+        >>> R = QQ.old_poly_ring(x, y)
+        >>> R.ideal(x*y).quotient(R.ideal(x))
+        <y>
+        """
+        self._check_ideal(J)
+        return self._quotient(J, **opts)
+
+    def intersect(self, J):
+        """
+        Compute the intersection of self with ideal J.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import QQ
+        >>> R = QQ.old_poly_ring(x, y)
+        >>> R.ideal(x).intersect(R.ideal(y))
+        <x*y>
+        """
+        self._check_ideal(J)
+        return self._intersect(J)
+
+    def saturate(self, J):
+        r"""
+        Compute the ideal saturation of ``self`` by ``J``.
+
+        That is, if ``self`` is the ideal `I`, compute the set
+        `I : J^\infty = \{x \in R | xJ^n \subset I \text{ for some } n\}`.
+        """
+        raise NotImplementedError
+        # Note this can be implemented using repeated quotient
+
+    def union(self, J):
+        """
+        Compute the ideal generated by the union of ``self`` and ``J``.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).ideal(x**2 - 1).union(QQ.old_poly_ring(x).ideal((x+1)**2)) == QQ.old_poly_ring(x).ideal(x+1)
+        True
+        """
+        self._check_ideal(J)
+        return self._union(J)
+
+    def product(self, J):
+        r"""
+        Compute the ideal product of ``self`` and ``J``.
+
+        That is, compute the ideal generated by products `xy`, for `x` an element
+        of ``self`` and `y \in J`.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x, y).ideal(x).product(QQ.old_poly_ring(x, y).ideal(y))
+        <x*y>
+        """
+        self._check_ideal(J)
+        return self._product(J)
+
+    def reduce_element(self, x):
+        """
+        Reduce the element ``x`` of our ring modulo the ideal ``self``.
+
+        Here "reduce" has no specific meaning: it could return a unique normal
+        form, simplify the expression a bit, or just do nothing.
+        """
+        return x
+
+    def __add__(self, e):
+        if not isinstance(e, Ideal):
+            R = self.ring.quotient_ring(self)
+            if isinstance(e, R.dtype):
+                return e
+            if isinstance(e, R.ring.dtype):
+                return R(e)
+            return R.convert(e)
+        self._check_ideal(e)
+        return self.union(e)
+
+    __radd__ = __add__
+
+    def __mul__(self, e):
+        if not isinstance(e, Ideal):
+            try:
+                e = self.ring.ideal(e)
+            except CoercionFailed:
+                return NotImplemented
+        self._check_ideal(e)
+        return self.product(e)
+
+    __rmul__ = __mul__
+
+    def _zeroth_power(self):
+        return self.ring.ideal(1)
+
+    def _first_power(self):
+        # Raising to any power but 1 returns a new instance. So we mult by 1
+        # here so that the first power is no exception.
+        return self * 1
+
+    def __eq__(self, e):
+        if not isinstance(e, Ideal) or e.ring != self.ring:
+            return False
+        return self._equals(e)
+
+    def __ne__(self, e):
+        return not (self == e)
+
+
+class ModuleImplementedIdeal(Ideal):
+    """
+    Ideal implementation relying on the modules code.
+
+    Attributes:
+
+    - _module - the underlying module
+    """
+
+    def __init__(self, ring, module):
+        Ideal.__init__(self, ring)
+        self._module = module
+
+    def _contains_elem(self, x):
+        return self._module.contains([x])
+
+    def _contains_ideal(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self._module.is_submodule(J._module)
+
+    def _intersect(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self.__class__(self.ring, self._module.intersect(J._module))
+
+    def _quotient(self, J, **opts):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self._module.module_quotient(J._module, **opts)
+
+    def _union(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self.__class__(self.ring, self._module.union(J._module))
+
+    @property
+    def gens(self):
+        """
+        Return generators for ``self``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x, y
+        >>> list(QQ.old_poly_ring(x, y).ideal(x, y, x**2 + y).gens)
+        [DMP_Python([[1], []], QQ), DMP_Python([[1, 0]], QQ), DMP_Python([[1], [], [1, 0]], QQ)]
+        """
+        return (x[0] for x in self._module.gens)
+
+    def is_zero(self):
+        """
+        Return True if ``self`` is the zero ideal.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).ideal(x).is_zero()
+        False
+        >>> QQ.old_poly_ring(x).ideal().is_zero()
+        True
+        """
+        return self._module.is_zero()
+
+    def is_whole_ring(self):
+        """
+        Return True if ``self`` is the whole ring, i.e. one generator is a unit.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ, ilex
+        >>> QQ.old_poly_ring(x).ideal(x).is_whole_ring()
+        False
+        >>> QQ.old_poly_ring(x).ideal(3).is_whole_ring()
+        True
+        >>> QQ.old_poly_ring(x, order=ilex).ideal(2 + x).is_whole_ring()
+        True
+        """
+        return self._module.is_full_module()
+
+    def __repr__(self):
+        from sympy.printing.str import sstr
+        gens = [self.ring.to_sympy(x) for [x] in self._module.gens]
+        return '<' + ','.join(sstr(g) for g in gens) + '>'
+
+    # NOTE this is the only method using the fact that the module is a SubModule
+    def _product(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self.__class__(self.ring, self._module.submodule(
+            *[[x*y] for [x] in self._module.gens for [y] in J._module.gens]))
+
+    def in_terms_of_generators(self, e):
+        """
+        Express ``e`` in terms of the generators of ``self``.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> I = QQ.old_poly_ring(x).ideal(x**2 + 1, x)
+        >>> I.in_terms_of_generators(1)  # doctest: +SKIP
+        [DMP_Python([1], QQ), DMP_Python([-1, 0], QQ)]
+        """
+        return self._module.in_terms_of_generators([e])
+
+    def reduce_element(self, x, **options):
+        return self._module.reduce_element([x], **options)[0]

rl/Lib/site-packages/sympy/polys/agca/tests/test_extensions.py

@@ -0,0 +1,196 @@
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.polys import QQ, ZZ
+from sympy.polys.polytools import Poly
+from sympy.polys.polyerrors import NotInvertible
+from sympy.polys.agca.extensions import FiniteExtension
+from sympy.polys.domainmatrix import DomainMatrix
+
+from sympy.testing.pytest import raises
+
+from sympy.abc import x, y, t
+
+
+def test_FiniteExtension():
+    # Gaussian integers
+    A = FiniteExtension(Poly(x**2 + 1, x))
+    assert A.rank == 2
+    assert str(A) == 'ZZ[x]/(x**2 + 1)'
+    i = A.generator
+    assert i.parent() is A
+
+    assert i*i == A(-1)
+    raises(TypeError, lambda: i*())
+
+    assert A.basis == (A.one, i)
+    assert A(1) == A.one
+    assert i**2 == A(-1)
+    assert i**2 != -1  # no coercion
+    assert (2 + i)*(1 - i) == 3 - i
+    assert (1 + i)**8 == A(16)
+    assert A(1).inverse() == A(1)
+    raises(NotImplementedError, lambda: A(2).inverse())
+
+    # Finite field of order 27
+    F = FiniteExtension(Poly(x**3 - x + 1, x, modulus=3))
+    assert F.rank == 3
+    a = F.generator  # also generates the cyclic group F - {0}
+    assert F.basis == (F(1), a, a**2)
+    assert a**27 == a
+    assert a**26 == F(1)
+    assert a**13 == F(-1)
+    assert a**9 == a + 1
+    assert a**3 == a - 1
+    assert a**6 == a**2 + a + 1
+    assert F(x**2 + x).inverse() == 1 - a
+    assert F(x + 2)**(-1) == F(x + 2).inverse()
+    assert a**19 * a**(-19) == F(1)
+    assert (a - 1) / (2*a**2 - 1) == a**2 + 1
+    assert (a - 1) // (2*a**2 - 1) == a**2 + 1
+    assert 2/(a**2 + 1) == a**2 - a + 1
+    assert (a**2 + 1)/2 == -a**2 - 1
+    raises(NotInvertible, lambda: F(0).inverse())
+
+    # Function field of an elliptic curve
+    K = FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
+    assert K.rank == 2
+    assert str(K) == 'ZZ(x)[t]/(t**2 - x**3 - x + 1)'
+    y = K.generator
+    c = 1/(x**3 - x**2 + x - 1)
+    assert ((y + x)*(y - x)).inverse() == K(c)
+    assert (y + x)*(y - x)*c == K(1)  # explicit inverse of y + x
+
+
+def test_FiniteExtension_eq_hash():
+    # Test eq and hash
+    p1 = Poly(x**2 - 2, x, domain=ZZ)
+    p2 = Poly(x**2 - 2, x, domain=QQ)
+    K1 = FiniteExtension(p1)
+    K2 = FiniteExtension(p2)
+    assert K1 == FiniteExtension(Poly(x**2 - 2))
+    assert K2 != FiniteExtension(Poly(x**2 - 2))
+    assert len({K1, K2, FiniteExtension(p1)}) == 2
+
+
+def test_FiniteExtension_mod():
+    # Test mod
+    K = FiniteExtension(Poly(x**3 + 1, x, domain=QQ))
+    xf = K(x)
+    assert (xf**2 - 1) % 1 == K.zero
+    assert 1 % (xf**2 - 1) == K.zero
+    assert (xf**2 - 1) / (xf - 1) == xf + 1
+    assert (xf**2 - 1) // (xf - 1) == xf + 1
+    assert (xf**2 - 1) % (xf - 1) == K.zero
+    raises(ZeroDivisionError, lambda: (xf**2 - 1) % 0)
+    raises(TypeError, lambda: xf % [])
+    raises(TypeError, lambda: [] % xf)
+
+    # Test mod over ring
+    K = FiniteExtension(Poly(x**3 + 1, x, domain=ZZ))
+    xf = K(x)
+    assert (xf**2 - 1) % 1 == K.zero
+    raises(NotImplementedError, lambda: (xf**2 - 1) % (xf - 1))
+
+
+def test_FiniteExtension_from_sympy():
+    # Test to_sympy/from_sympy
+    K = FiniteExtension(Poly(x**3 + 1, x, domain=ZZ))
+    xf = K(x)
+    assert K.from_sympy(x) == xf
+    assert K.to_sympy(xf) == x
+
+
+def test_FiniteExtension_set_domain():
+    KZ = FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))
+    KQ = FiniteExtension(Poly(x**2 + 1, x, domain='QQ'))
+    assert KZ.set_domain(QQ) == KQ
+
+
+def test_FiniteExtension_exquo():
+    # Test exquo
+    K = FiniteExtension(Poly(x**4 + 1))
+    xf = K(x)
+    assert K.exquo(xf**2 - 1, xf - 1) == xf + 1
+
+
+def test_FiniteExtension_convert():
+    # Test from_MonogenicFiniteExtension
+    K1 = FiniteExtension(Poly(x**2 + 1))
+    K2 = QQ[x]
+    x1, x2 = K1(x), K2(x)
+    assert K1.convert(x2) == x1
+    assert K2.convert(x1) == x2
+
+    K = FiniteExtension(Poly(x**2 - 1, domain=QQ))
+    assert K.convert_from(QQ(1, 2), QQ) == K.one/2
+
+
+def test_FiniteExtension_division_ring():
+    # Test division in FiniteExtension over a ring
+    KQ = FiniteExtension(Poly(x**2 - 1, x, domain=QQ))
+    KZ = FiniteExtension(Poly(x**2 - 1, x, domain=ZZ))
+    KQt = FiniteExtension(Poly(x**2 - 1, x, domain=QQ[t]))
+    KQtf = FiniteExtension(Poly(x**2 - 1, x, domain=QQ.frac_field(t)))
+    assert KQ.is_Field is True
+    assert KZ.is_Field is False
+    assert KQt.is_Field is False
+    assert KQtf.is_Field is True
+    for K in KQ, KZ, KQt, KQtf:
+        xK = K.convert(x)
+        assert xK / K.one == xK
+        assert xK // K.one == xK
+        assert xK % K.one == K.zero
+        raises(ZeroDivisionError, lambda: xK / K.zero)
+        raises(ZeroDivisionError, lambda: xK // K.zero)
+        raises(ZeroDivisionError, lambda: xK % K.zero)
+        if K.is_Field:
+            assert xK / xK == K.one
+            assert xK // xK == K.one
+            assert xK % xK == K.zero
+        else:
+            raises(NotImplementedError, lambda: xK / xK)
+            raises(NotImplementedError, lambda: xK // xK)
+            raises(NotImplementedError, lambda: xK % xK)
+
+
+def test_FiniteExtension_Poly():
+    K = FiniteExtension(Poly(x**2 - 2))
+    p = Poly(x, y, domain=K)
+    assert p.domain == K
+    assert p.as_expr() == x
+    assert (p**2).as_expr() == 2
+
+    K = FiniteExtension(Poly(x**2 - 2, x, domain=QQ))
+    K2 = FiniteExtension(Poly(t**2 - 2, t, domain=K))
+    assert str(K2) == 'QQ[x]/(x**2 - 2)[t]/(t**2 - 2)'
+
+    eK = K2.convert(x + t)
+    assert K2.to_sympy(eK) == x + t
+    assert K2.to_sympy(eK ** 2) == 4 + 2*x*t
+    p = Poly(x + t, y, domain=K2)
+    assert p**2 == Poly(4 + 2*x*t, y, domain=K2)
+
+
+def test_FiniteExtension_sincos_jacobian():
+    # Use FiniteExtensino to compute the Jacobian of a matrix involving sin
+    # and cos of different symbols.
+    r, p, t = symbols('rho, phi, theta')
+    elements = [
+        [sin(p)*cos(t), r*cos(p)*cos(t), -r*sin(p)*sin(t)],
+        [sin(p)*sin(t), r*cos(p)*sin(t),  r*sin(p)*cos(t)],
+        [       cos(p),       -r*sin(p),                0],
+    ]
+
+    def make_extension(K):
+        K = FiniteExtension(Poly(sin(p)**2+cos(p)**2-1, sin(p), domain=K[cos(p)]))
+        K = FiniteExtension(Poly(sin(t)**2+cos(t)**2-1, sin(t), domain=K[cos(t)]))
+        return K
+
+    Ksc1 = make_extension(ZZ[r])
+    Ksc2 = make_extension(ZZ)[r]
+
+    for K in [Ksc1, Ksc2]:
+        elements_K = [[K.convert(e) for e in row] for row in elements]
+        J = DomainMatrix(elements_K, (3, 3), K)
+        det = J.charpoly()[-1] * (-K.one)**3
+        assert det == K.convert(r**2*sin(p))

rl/Lib/site-packages/sympy/polys/agca/tests/test_homomorphisms.py

@@ -0,0 +1,113 @@
+"""Tests for homomorphisms."""
+
+from sympy.core.singleton import S
+from sympy.polys.domains.rationalfield import QQ
+from sympy.abc import x, y
+from sympy.polys.agca import homomorphism
+from sympy.testing.pytest import raises
+
+
+def test_printing():
+    R = QQ.old_poly_ring(x)
+
+    assert str(homomorphism(R.free_module(1), R.free_module(1), [0])) == \
+        'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1'
+    assert str(homomorphism(R.free_module(2), R.free_module(2), [0, 0])) == \
+        'Matrix([                       \n[0, 0], : QQ[x]**2 -> QQ[x]**2\n[0, 0]])                       '
+    assert str(homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])) == \
+        'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1/<[x]>'
+    assert str(R.free_module(0).identity_hom()) == 'Matrix(0, 0, []) : QQ[x]**0 -> QQ[x]**0'
+
+def test_operations():
+    F = QQ.old_poly_ring(x).free_module(2)
+    G = QQ.old_poly_ring(x).free_module(3)
+    f = F.identity_hom()
+    g = homomorphism(F, F, [0, [1, x]])
+    h = homomorphism(F, F, [[1, 0], 0])
+    i = homomorphism(F, G, [[1, 0, 0], [0, 1, 0]])
+
+    assert f == f
+    assert f != g
+    assert f != i
+    assert (f != F.identity_hom()) is False
+    assert 2*f == f*2 == homomorphism(F, F, [[2, 0], [0, 2]])
+    assert f/2 == homomorphism(F, F, [[S.Half, 0], [0, S.Half]])
+    assert f + g == homomorphism(F, F, [[1, 0], [1, x + 1]])
+    assert f - g == homomorphism(F, F, [[1, 0], [-1, 1 - x]])
+    assert f*g == g == g*f
+    assert h*g == homomorphism(F, F, [0, [1, 0]])
+    assert g*h == homomorphism(F, F, [0, 0])
+    assert i*f == i
+    assert f([1, 2]) == [1, 2]
+    assert g([1, 2]) == [2, 2*x]
+
+    assert i.restrict_domain(F.submodule([x, x]))([x, x]) == i([x, x])
+    h1 = h.quotient_domain(F.submodule([0, 1]))
+    assert h1([1, 0]) == h([1, 0])
+    assert h1.restrict_domain(h1.domain.submodule([x, 0]))([x, 0]) == h([x, 0])
+
+    raises(TypeError, lambda: f/g)
+    raises(TypeError, lambda: f + 1)
+    raises(TypeError, lambda: f + i)
+    raises(TypeError, lambda: f - 1)
+    raises(TypeError, lambda: f*i)
+
+
+def test_creation():
+    F = QQ.old_poly_ring(x).free_module(3)
+    G = QQ.old_poly_ring(x).free_module(2)
+    SM = F.submodule([1, 1, 1])
+    Q = F / SM
+    SQ = Q.submodule([1, 0, 0])
+
+    matrix = [[1, 0], [0, 1], [-1, -1]]
+    h = homomorphism(F, G, matrix)
+    h2 = homomorphism(Q, G, matrix)
+    assert h.quotient_domain(SM) == h2
+    raises(ValueError, lambda: h.quotient_domain(F.submodule([1, 0, 0])))
+    assert h2.restrict_domain(SQ) == homomorphism(SQ, G, matrix)
+    raises(ValueError, lambda: h.restrict_domain(G))
+    raises(ValueError, lambda: h.restrict_codomain(G.submodule([1, 0])))
+    raises(ValueError, lambda: h.quotient_codomain(F))
+
+    im = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
+    for M in [F, SM, Q, SQ]:
+        assert M.identity_hom() == homomorphism(M, M, im)
+    assert SM.inclusion_hom() == homomorphism(SM, F, im)
+    assert SQ.inclusion_hom() == homomorphism(SQ, Q, im)
+    assert Q.quotient_hom() == homomorphism(F, Q, im)
+    assert SQ.quotient_hom() == homomorphism(SQ.base, SQ, im)
+
+    class conv:
+        def convert(x, y=None):
+            return x
+
+    class dummy:
+        container = conv()
+
+        def submodule(*args):
+            return None
+    raises(TypeError, lambda: homomorphism(dummy(), G, matrix))
+    raises(TypeError, lambda: homomorphism(F, dummy(), matrix))
+    raises(
+        ValueError, lambda: homomorphism(QQ.old_poly_ring(x, y).free_module(3), G, matrix))
+    raises(ValueError, lambda: homomorphism(F, G, [0, 0]))
+
+
+def test_properties():
+    R = QQ.old_poly_ring(x, y)
+    F = R.free_module(2)
+    h = homomorphism(F, F, [[x, 0], [y, 0]])
+    assert h.kernel() == F.submodule([-y, x])
+    assert h.image() == F.submodule([x, 0], [y, 0])
+    assert not h.is_injective()
+    assert not h.is_surjective()
+    assert h.restrict_codomain(h.image()).is_surjective()
+    assert h.restrict_domain(F.submodule([1, 0])).is_injective()
+    assert h.quotient_domain(
+        h.kernel()).restrict_codomain(h.image()).is_isomorphism()
+
+    R2 = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
+    F = R2.free_module(2)
+    h = homomorphism(F, F, [[x, 0], [y, y + 1]])
+    assert h.is_isomorphism()

rl/Lib/site-packages/sympy/polys/agca/tests/test_ideals.py

@@ -0,0 +1,131 @@
+"""Test ideals.py code."""
+
+from sympy.polys import QQ, ilex
+from sympy.abc import x, y, z
+from sympy.testing.pytest import raises
+
+
+def test_ideal_operations():
+    R = QQ.old_poly_ring(x, y)
+    I = R.ideal(x)
+    J = R.ideal(y)
+    S = R.ideal(x*y)
+    T = R.ideal(x, y)
+
+    assert not (I == J)
+    assert I == I
+
+    assert I.union(J) == T
+    assert I + J == T
+    assert I + T == T
+
+    assert not I.subset(T)
+    assert T.subset(I)
+
+    assert I.product(J) == S
+    assert I*J == S
+    assert x*J == S
+    assert I*y == S
+    assert R.convert(x)*J == S
+    assert I*R.convert(y) == S
+
+    assert not I.is_zero()
+    assert not J.is_whole_ring()
+
+    assert R.ideal(x**2 + 1, x).is_whole_ring()
+    assert R.ideal() == R.ideal(0)
+    assert R.ideal().is_zero()
+
+    assert T.contains(x*y)
+    assert T.subset([x, y])
+
+    assert T.in_terms_of_generators(x) == [R(1), R(0)]
+
+    assert T**0 == R.ideal(1)
+    assert T**1 == T
+    assert T**2 == R.ideal(x**2, y**2, x*y)
+    assert I**5 == R.ideal(x**5)
+
+
+def test_exceptions():
+    I = QQ.old_poly_ring(x).ideal(x)
+    J = QQ.old_poly_ring(y).ideal(1)
+    raises(ValueError, lambda: I.union(x))
+    raises(ValueError, lambda: I + J)
+    raises(ValueError, lambda: I * J)
+    raises(ValueError, lambda: I.union(J))
+    assert (I == J) is False
+    assert I != J
+
+
+def test_nontriv_global():
+    R = QQ.old_poly_ring(x, y, z)
+
+    def contains(I, f):
+        return R.ideal(*I).contains(f)
+
+    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_nontriv_local():
+    R = QQ.old_poly_ring(x, y, z, order=ilex)
+
+    def contains(I, f):
+        return R.ideal(*I).contains(f)
+
+    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_intersection():
+    R = QQ.old_poly_ring(x, y, z)
+    # SCA, example 1.8.11
+    assert R.ideal(x, y).intersect(R.ideal(y**2, z)) == R.ideal(y**2, y*z, x*z)
+
+    assert R.ideal(x, y).intersect(R.ideal()).is_zero()
+
+    R = QQ.old_poly_ring(x, y, z, order="ilex")
+    assert R.ideal(x, y).intersect(R.ideal(y**2 + y**2*z, z + z*x**3*y)) == \
+        R.ideal(y**2, y*z, x*z)
+
+
+def test_quotient():
+    # SCA, example 1.8.13
+    R = QQ.old_poly_ring(x, y, z)
+    assert R.ideal(x, y).quotient(R.ideal(y**2, z)) == R.ideal(x, y)
+
+
+def test_reduction():
+    from sympy.polys.distributedmodules import sdm_nf_buchberger_reduced
+    R = QQ.old_poly_ring(x, y)
+    I = R.ideal(x**5, y)
+    e = R.convert(x**3 + y**2)
+    assert I.reduce_element(e) == e
+    assert I.reduce_element(e, NF=sdm_nf_buchberger_reduced) == R.convert(x**3)

rl/Lib/site-packages/sympy/polys/agca/tests/test_modules.py

@@ -0,0 +1,408 @@
+"""Test modules.py code."""
+
+from sympy.polys.agca.modules import FreeModule, ModuleOrder, FreeModulePolyRing
+from sympy.polys import CoercionFailed, QQ, lex, grlex, ilex, ZZ
+from sympy.abc import x, y, z
+from sympy.testing.pytest import raises
+from sympy.core.numbers import Rational
+
+
+def test_FreeModuleElement():
+    M = QQ.old_poly_ring(x).free_module(3)
+    e = M.convert([1, x, x**2])
+    f = [QQ.old_poly_ring(x).convert(1), QQ.old_poly_ring(x).convert(x), QQ.old_poly_ring(x).convert(x**2)]
+    assert list(e) == f
+    assert f[0] == e[0]
+    assert f[1] == e[1]
+    assert f[2] == e[2]
+    raises(IndexError, lambda: e[3])
+
+    g = M.convert([x, 0, 0])
+    assert e + g == M.convert([x + 1, x, x**2])
+    assert f + g == M.convert([x + 1, x, x**2])
+    assert -e == M.convert([-1, -x, -x**2])
+    assert e - g == M.convert([1 - x, x, x**2])
+    assert e != g
+
+    assert M.convert([x, x, x]) / QQ.old_poly_ring(x).convert(x) == [1, 1, 1]
+    R = QQ.old_poly_ring(x, order="ilex")
+    assert R.free_module(1).convert([x]) / R.convert(x) == [1]
+
+
+def test_FreeModule():
+    M1 = FreeModule(QQ.old_poly_ring(x), 2)
+    assert M1 == FreeModule(QQ.old_poly_ring(x), 2)
+    assert M1 != FreeModule(QQ.old_poly_ring(y), 2)
+    assert M1 != FreeModule(QQ.old_poly_ring(x), 3)
+    M2 = FreeModule(QQ.old_poly_ring(x, order="ilex"), 2)
+
+    assert [x, 1] in M1
+    assert [x] not in M1
+    assert [2, y] not in M1
+    assert [1/(x + 1), 2] not in M1
+
+    e = M1.convert([x, x**2 + 1])
+    X = QQ.old_poly_ring(x).convert(x)
+    assert e == [X, X**2 + 1]
+    assert e == [x, x**2 + 1]
+    assert 2*e == [2*x, 2*x**2 + 2]
+    assert e*2 == [2*x, 2*x**2 + 2]
+    assert e/2 == [x/2, (x**2 + 1)/2]
+    assert x*e == [x**2, x**3 + x]
+    assert e*x == [x**2, x**3 + x]
+    assert X*e == [x**2, x**3 + x]
+    assert e*X == [x**2, x**3 + x]
+
+    assert [x, 1] in M2
+    assert [x] not in M2
+    assert [2, y] not in M2
+    assert [1/(x + 1), 2] in M2
+
+    e = M2.convert([x, x**2 + 1])
+    X = QQ.old_poly_ring(x, order="ilex").convert(x)
+    assert e == [X, X**2 + 1]
+    assert e == [x, x**2 + 1]
+    assert 2*e == [2*x, 2*x**2 + 2]
+    assert e*2 == [2*x, 2*x**2 + 2]
+    assert e/2 == [x/2, (x**2 + 1)/2]
+    assert x*e == [x**2, x**3 + x]
+    assert e*x == [x**2, x**3 + x]
+    assert e/(1 + x) == [x/(1 + x), (x**2 + 1)/(1 + x)]
+    assert X*e == [x**2, x**3 + x]
+    assert e*X == [x**2, x**3 + x]
+
+    M3 = FreeModule(QQ.old_poly_ring(x, y), 2)
+    assert M3.convert(e) == M3.convert([x, x**2 + 1])
+
+    assert not M3.is_submodule(0)
+    assert not M3.is_zero()
+
+    raises(NotImplementedError, lambda: ZZ.old_poly_ring(x).free_module(2))
+    raises(NotImplementedError, lambda: FreeModulePolyRing(ZZ, 2))
+    raises(CoercionFailed, lambda: M1.convert(QQ.old_poly_ring(x).free_module(3)
+           .convert([1, 2, 3])))
+    raises(CoercionFailed, lambda: M3.convert(1))
+
+
+def test_ModuleOrder():
+    o1 = ModuleOrder(lex, grlex, False)
+    o2 = ModuleOrder(ilex, lex, False)
+
+    assert o1 == ModuleOrder(lex, grlex, False)
+    assert (o1 != ModuleOrder(lex, grlex, False)) is False
+    assert o1 != o2
+
+    assert o1((1, 2, 3)) == (1, (5, (2, 3)))
+    assert o2((1, 2, 3)) == (-1, (2, 3))
+
+
+def test_SubModulePolyRing_global():
+    R = QQ.old_poly_ring(x, y)
+    F = R.free_module(3)
+    Fd = F.submodule([1, 0, 0], [1, 2, 0], [1, 2, 3])
+    M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
+
+    assert F == Fd
+    assert Fd == F
+    assert F != M
+    assert M != F
+    assert Fd != M
+    assert M != Fd
+    assert Fd == F.submodule(*F.basis())
+
+    assert Fd.is_full_module()
+    assert not M.is_full_module()
+    assert not Fd.is_zero()
+    assert not M.is_zero()
+    assert Fd.submodule().is_zero()
+
+    assert M.contains([x**2 + y**2 + x, 1 + y, 1])
+    assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
+    assert M.contains([y**2, 1 - x*y, -x])
+
+    assert not F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
+    assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
+    assert not M.is_submodule(0)
+
+    m = F.convert([x**2 + y**2, 1, 0])
+    n = M.convert(m)
+    assert m.module is F
+    assert n.module is M
+
+    raises(ValueError, lambda: M.submodule([1, 0, 0]))
+    raises(TypeError, lambda: M.union(1))
+    raises(ValueError, lambda: M.union(R.free_module(1).submodule([x])))
+
+    assert F.submodule([x, x, x]) != F.submodule([x, x, x], order="ilex")
+
+
+def test_SubModulePolyRing_local():
+    R = QQ.old_poly_ring(x, y, order=ilex)
+    F = R.free_module(3)
+    Fd = F.submodule([1 + x, 0, 0], [1 + y, 2 + 2*y, 0], [1, 2, 3])
+    M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
+
+    assert F == Fd
+    assert Fd == F
+    assert F != M
+    assert M != F
+    assert Fd != M
+    assert M != Fd
+    assert Fd == F.submodule(*F.basis())
+
+    assert Fd.is_full_module()
+    assert not M.is_full_module()
+    assert not Fd.is_zero()
+    assert not M.is_zero()
+    assert Fd.submodule().is_zero()
+
+    assert M.contains([x**2 + y**2 + x, 1 + y, 1])
+    assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
+    assert M.contains([y**2, 1 - x*y, -x])
+
+    assert F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
+    assert F.submodule(
+        [1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1 + x*y])) == F
+
+    raises(ValueError, lambda: M.submodule([1, 0, 0]))
+
+
+def test_SubModulePolyRing_nontriv_global():
+    R = QQ.old_poly_ring(x, y, z)
+    F = R.free_module(1)
+
+    def contains(I, f):
+        return F.submodule(*[[g] for g in I]).contains([f])
+
+    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_SubModulePolyRing_nontriv_local():
+    R = QQ.old_poly_ring(x, y, z, order=ilex)
+    F = R.free_module(1)
+
+    def contains(I, f):
+        return F.submodule(*[[g] for g in I]).contains([f])
+
+    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_syzygy():
+    R = QQ.old_poly_ring(x, y, z)
+    M = R.free_module(1).submodule([x*y], [y*z], [x*z])
+    S = R.free_module(3).submodule([0, x, -y], [z, -x, 0])
+    assert M.syzygy_module() == S
+
+    M2 = M / ([x*y*z],)
+    S2 = R.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
+    assert M2.syzygy_module() == S2
+
+    F = R.free_module(3)
+    assert F.submodule(*F.basis()).syzygy_module() == F.submodule()
+
+    R2 = QQ.old_poly_ring(x, y, z) / [x*y*z]
+    M3 = R2.free_module(1).submodule([x*y], [y*z], [x*z])
+    S3 = R2.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
+    assert M3.syzygy_module() == S3
+
+
+def test_in_terms_of_generators():
+    R = QQ.old_poly_ring(x, order="ilex")
+    M = R.free_module(2).submodule([2*x, 0], [1, 2])
+    assert M.in_terms_of_generators(
+        [x, x]) == [R.convert(Rational(1, 4)), R.convert(x/2)]
+    raises(ValueError, lambda: M.in_terms_of_generators([1, 0]))
+
+    M = R.free_module(2) / ([x, 0], [1, 1])
+    SM = M.submodule([1, x])
+    assert SM.in_terms_of_generators([2, 0]) == [R.convert(-2/(x - 1))]
+
+    R = QQ.old_poly_ring(x, y) / [x**2 - y**2]
+    M = R.free_module(2)
+    SM = M.submodule([x, 0], [0, y])
+    assert SM.in_terms_of_generators(
+        [x**2, x**2]) == [R.convert(x), R.convert(y)]
+
+
+def test_QuotientModuleElement():
+    R = QQ.old_poly_ring(x)
+    F = R.free_module(3)
+    N = F.submodule([1, x, x**2])
+    M = F/N
+    e = M.convert([x**2, 2, 0])
+
+    assert M.convert([x + 1, x**2 + x, x**3 + x**2]) == 0
+    assert e == [x**2, 2, 0] + N == F.convert([x**2, 2, 0]) + N == \
+        M.convert(F.convert([x**2, 2, 0]))
+
+    assert M.convert([x**2 + 1, 2*x + 2, x**2]) == e + [0, x, 0] == \
+        e + M.convert([0, x, 0]) == e + F.convert([0, x, 0])
+    assert M.convert([x**2 + 1, 2, x**2]) == e - [0, x, 0] == \
+        e - M.convert([0, x, 0]) == e - F.convert([0, x, 0])
+    assert M.convert([0, 2, 0]) == M.convert([x**2, 4, 0]) - e == \
+        [x**2, 4, 0] - e == F.convert([x**2, 4, 0]) - e
+    assert M.convert([x**3 + x**2, 2*x + 2, 0]) == (1 + x)*e == \
+        R.convert(1 + x)*e == e*(1 + x) == e*R.convert(1 + x)
+    assert -e == [-x**2, -2, 0]
+
+    f = [x, x, 0] + N
+    assert M.convert([1, 1, 0]) == f / x == f / R.convert(x)
+
+    M2 = F/[(2, 2*x, 2*x**2), (0, 0, 1)]
+    G = R.free_module(2)
+    M3 = G/[[1, x]]
+    M4 = F.submodule([1, x, x**2], [1, 0, 0]) / N
+    raises(CoercionFailed, lambda: M.convert(G.convert([1, x])))
+    raises(CoercionFailed, lambda: M.convert(M3.convert([1, x])))
+    raises(CoercionFailed, lambda: M.convert(M2.convert([1, x, x])))
+    assert M2.convert(M.convert([2, x, x**2])) == [2, x, 0]
+    assert M.convert(M4.convert([2, 0, 0])) == [2, 0, 0]
+
+
+def test_QuotientModule():
+    R = QQ.old_poly_ring(x)
+    F = R.free_module(3)
+    N = F.submodule([1, x, x**2])
+    M = F/N
+
+    assert M != F
+    assert M != N
+    assert M == F / [(1, x, x**2)]
+    assert not M.is_zero()
+    assert (F / F.basis()).is_zero()
+
+    SQ = F.submodule([1, x, x**2], [2, 0, 0]) / N
+    assert SQ == M.submodule([2, x, x**2])
+    assert SQ != M.submodule([2, 1, 0])
+    assert SQ != M
+    assert M.is_submodule(SQ)
+    assert not SQ.is_full_module()
+
+    raises(ValueError, lambda: N/F)
+    raises(ValueError, lambda: F.submodule([2, 0, 0]) / N)
+    raises(ValueError, lambda: R.free_module(2)/F)
+    raises(CoercionFailed, lambda: F.convert(M.convert([1, x, x**2])))
+
+    M1 = F / [[1, 1, 1]]
+    M2 = M1.submodule([1, 0, 0], [0, 1, 0])
+    assert M1 == M2
+
+
+def test_ModulesQuotientRing():
+    R = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
+    M1 = R.free_module(2)
+    assert M1 == R.free_module(2)
+    assert M1 != QQ.old_poly_ring(x).free_module(2)
+    assert M1 != R.free_module(3)
+
+    assert [x, 1] in M1
+    assert [x] not in M1
+    assert [1/(R.convert(x) + 1), 2] in M1
+    assert [1, 2/(1 + y)] in M1
+    assert [1, 2/y] not in M1
+
+    assert M1.convert([x**2, y]) == [-1, y]
+
+    F = R.free_module(3)
+    Fd = F.submodule([x**2, 0, 0], [1, 2, 0], [1, 2, 3])
+    M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
+
+    assert F == Fd
+    assert Fd == F
+    assert F != M
+    assert M != F
+    assert Fd != M
+    assert M != Fd
+    assert Fd == F.submodule(*F.basis())
+
+    assert Fd.is_full_module()
+    assert not M.is_full_module()
+    assert not Fd.is_zero()
+    assert not M.is_zero()
+    assert Fd.submodule().is_zero()
+
+    assert M.contains([x**2 + y**2 + x, -x**2 + y, 1])
+    assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
+    assert M.contains([y**2, 1 - x*y, -x])
+
+    assert F.submodule([x, 0, 0]) == F.submodule([1, 0, 0])
+    assert not F.submodule([y, 0, 0]) == F.submodule([1, 0, 0])
+    assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
+    assert not M.is_submodule(0)
+
+
+def test_module_mul():
+    R = QQ.old_poly_ring(x)
+    M = R.free_module(2)
+    S1 = M.submodule([x, 0], [0, x])
+    S2 = M.submodule([x**2, 0], [0, x**2])
+    I = R.ideal(x)
+
+    assert I*M == M*I == S1 == x*M == M*x
+    assert I*S1 == S2 == x*S1
+
+
+def test_intersection():
+    # SCA, example 2.8.5
+    F = QQ.old_poly_ring(x, y).free_module(2)
+    M1 = F.submodule([x, y], [y, 1])
+    M2 = F.submodule([0, y - 1], [x, 1], [y, x])
+    I = F.submodule([x, y], [y**2 - y, y - 1], [x*y + y, x + 1])
+    I1, rel1, rel2 = M1.intersect(M2, relations=True)
+    assert I1 == M2.intersect(M1) == I
+    for i, g in enumerate(I1.gens):
+        assert g == sum(c*x for c, x in zip(rel1[i], M1.gens)) \
+                 == sum(d*y for d, y in zip(rel2[i], M2.gens))
+
+    assert F.submodule([x, y]).intersect(F.submodule([y, x])).is_zero()
+
+
+def test_quotient():
+    # SCA, example 2.8.6
+    R = QQ.old_poly_ring(x, y, z)
+    F = R.free_module(2)
+    assert F.submodule([x*y, x*z], [y*z, x*y]).module_quotient(
+        F.submodule([y, z], [z, y])) == QQ.old_poly_ring(x, y, z).ideal(x**2*y**2 - x*y*z**2)
+    assert F.submodule([x, y]).module_quotient(F.submodule()).is_whole_ring()
+
+    M = F.submodule([x**2, x**2], [y**2, y**2])
+    N = F.submodule([x + y, x + y])
+    q, rel = M.module_quotient(N, relations=True)
+    assert q == R.ideal(y**2, x - y)
+    for i, g in enumerate(q.gens):
+        assert g*N.gens[0] == sum(c*x for c, x in zip(rel[i], M.gens))
+
+
+def test_groebner_extendend():
+    M = QQ.old_poly_ring(x, y, z).free_module(3).submodule([x + 1, y, 1], [x*y, z, z**2])
+    G, R = M._groebner_vec(extended=True)
+    for i, g in enumerate(G):
+        assert g == sum(c*gen for c, gen in zip(R[i], M.gens))

rl/Lib/site-packages/sympy/polys/appellseqs.py

@@ -0,0 +1,269 @@
+r"""
+Efficient functions for generating Appell sequences.
+
+An Appell sequence is a zero-indexed sequence of polynomials `p_i(x)`
+satisfying `p_{i+1}'(x)=(i+1)p_i(x)` for all `i`. This definition leads
+to the following iterative algorithm:
+
+.. math :: p_0(x) = c_0,\ p_i(x) = i \int_0^x p_{i-1}(t)\,dt + c_i
+
+The constant coefficients `c_i` are usually determined from the
+just-evaluated integral and `i`.
+
+Appell sequences satisfy the following identity from umbral calculus:
+
+.. math :: p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) y^{n-k}
+
+References
+==========
+
+.. [1] https://en.wikipedia.org/wiki/Appell_sequence
+.. [2] Peter Luschny, "An introduction to the Bernoulli function",
+       https://arxiv.org/abs/2009.06743
+"""
+from sympy.polys.densearith import dup_mul_ground, dup_sub_ground, dup_quo_ground
+from sympy.polys.densetools import dup_eval, dup_integrate
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.polytools import named_poly
+from sympy.utilities import public
+
+def dup_bernoulli(n, K):
+    """Low-level implementation of Bernoulli polynomials."""
+    if n < 1:
+        return [K.one]
+    p = [K.one, K(-1,2)]
+    for i in range(2, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K(1,2), K) * K(1<<(i-1), (1<<i)-1), K)
+    return p
+
+@public
+def bernoulli_poly(n, x=None, polys=False):
+    r"""Generates the Bernoulli polynomial `\operatorname{B}_n(x)`.
+
+    `\operatorname{B}_n(x)` is the unique polynomial satisfying
+
+    .. math :: \int_{x}^{x+1} \operatorname{B}_n(t) \,dt = x^n.
+
+    Based on this, we have for nonnegative integer `s` and integer
+    `a` and `b`
+
+    .. math :: \sum_{k=a}^{b} k^s = \frac{\operatorname{B}_{s+1}(b+1) -
+            \operatorname{B}_{s+1}(a)}{s+1}
+
+    which is related to Jakob Bernoulli's original motivation for introducing
+    the Bernoulli numbers, the values of these polynomials at `x = 1`.
+
+    Examples
+    ========
+
+    >>> from sympy import summation
+    >>> from sympy.abc import x
+    >>> from sympy.polys import bernoulli_poly
+    >>> bernoulli_poly(5, x)
+    x**5 - 5*x**4/2 + 5*x**3/3 - x/6
+
+    >>> def psum(p, a, b):
+    ...     return (bernoulli_poly(p+1,b+1) - bernoulli_poly(p+1,a)) / (p+1)
+    >>> psum(4, -6, 27)
+    3144337
+    >>> summation(x**4, (x, -6, 27))
+    3144337
+
+    >>> psum(1, 1, x).factor()
+    x*(x + 1)/2
+    >>> psum(2, 1, x).factor()
+    x*(x + 1)*(2*x + 1)/6
+    >>> psum(3, 1, x).factor()
+    x**2*(x + 1)**2/4
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.bernoulli
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Bernoulli_polynomials
+    """
+    return named_poly(n, dup_bernoulli, QQ, "Bernoulli polynomial", (x,), polys)
+
+def dup_bernoulli_c(n, K):
+    """Low-level implementation of central Bernoulli polynomials."""
+    p = [K.one]
+    for i in range(1, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K.one, K) * K((1<<(i-1))-1, (1<<i)-1), K)
+    return p
+
+@public
+def bernoulli_c_poly(n, x=None, polys=False):
+    r"""Generates the central Bernoulli polynomial `\operatorname{B}_n^c(x)`.
+
+    These are scaled and shifted versions of the plain Bernoulli polynomials,
+    done in such a way that `\operatorname{B}_n^c(x)` is an even or odd function
+    for even or odd `n` respectively:
+
+    .. math :: \operatorname{B}_n^c(x) = 2^n \operatorname{B}_n
+            \left(\frac{x+1}{2}\right)
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+    """
+    return named_poly(n, dup_bernoulli_c, QQ, "central Bernoulli polynomial", (x,), polys)
+
+def dup_genocchi(n, K):
+    """Low-level implementation of Genocchi polynomials."""
+    if n < 1:
+        return [K.zero]
+    p = [-K.one]
+    for i in range(2, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K.one, K) // K(2), K)
+    return p
+
+@public
+def genocchi_poly(n, x=None, polys=False):
+    r"""Generates the Genocchi polynomial `\operatorname{G}_n(x)`.
+
+    `\operatorname{G}_n(x)` is twice the difference between the plain and
+    central Bernoulli polynomials, so has degree `n-1`:
+
+    .. math :: \operatorname{G}_n(x) = 2 (\operatorname{B}_n(x) -
+            \operatorname{B}_n^c(x))
+
+    The factor of 2 in the definition endows `\operatorname{G}_n(x)` with
+    integer coefficients.
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial plus one.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.genocchi
+    """
+    return named_poly(n, dup_genocchi, ZZ, "Genocchi polynomial", (x,), polys)
+
+def dup_euler(n, K):
+    """Low-level implementation of Euler polynomials."""
+    return dup_quo_ground(dup_genocchi(n+1, ZZ), K(-n-1), K)
+
+@public
+def euler_poly(n, x=None, polys=False):
+    r"""Generates the Euler polynomial `\operatorname{E}_n(x)`.
+
+    These are scaled and reindexed versions of the Genocchi polynomials:
+
+    .. math :: \operatorname{E}_n(x) = -\frac{\operatorname{G}_{n+1}(x)}{n+1}
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.euler
+    """
+    return named_poly(n, dup_euler, QQ, "Euler polynomial", (x,), polys)
+
+def dup_andre(n, K):
+    """Low-level implementation of Andre polynomials."""
+    p = [K.one]
+    for i in range(1, n+1):
+        p = dup_integrate(dup_mul_ground(p, K(i), K), 1, K)
+        if i % 2 == 0:
+            p = dup_sub_ground(p, dup_eval(p, K.one, K), K)
+    return p
+
+@public
+def andre_poly(n, x=None, polys=False):
+    r"""Generates the Andre polynomial `\mathcal{A}_n(x)`.
+
+    This is the Appell sequence where the constant coefficients form the sequence
+    of Euler numbers ``euler(n)``. As such they have integer coefficients
+    and parities matching the parity of `n`.
+
+    Luschny calls these the *Swiss-knife polynomials* because their values
+    at 0 and 1 can be simply transformed into both the Bernoulli and Euler
+    numbers. Here they are called the Andre polynomials because
+    `|\mathcal{A}_n(n\bmod 2)|` for `n \ge 0` generates what Luschny calls
+    the *Andre numbers*, A000111 in the OEIS.
+
+    Examples
+    ========
+
+    >>> from sympy import bernoulli, euler, genocchi
+    >>> from sympy.abc import x
+    >>> from sympy.polys import andre_poly
+    >>> andre_poly(9, x)
+    x**9 - 36*x**7 + 630*x**5 - 5124*x**3 + 12465*x
+
+    >>> [andre_poly(n, 0) for n in range(11)]
+    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]
+    >>> [euler(n) for n in range(11)]
+    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521]
+    >>> [andre_poly(n-1, 1) * n / (4**n - 2**n) for n in range(1, 11)]
+    [1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
+    >>> [bernoulli(n) for n in range(1, 11)]
+    [1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
+    >>> [-andre_poly(n-1, -1) * n / (-2)**(n-1) for n in range(1, 11)]
+    [-1, -1, 0, 1, 0, -3, 0, 17, 0, -155]
+    >>> [genocchi(n) for n in range(1, 11)]
+    [-1, -1, 0, 1, 0, -3, 0, 17, 0, -155]
+
+    >>> [abs(andre_poly(n, n%2)) for n in range(11)]
+    [1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521]
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of the polynomial.
+    x : optional
+    polys : bool, optional
+        If True, return a Poly, otherwise (default) return an expression.
+
+    See Also
+    ========
+
+    sympy.functions.combinatorial.numbers.andre
+
+    References
+    ==========
+
+    .. [1] Peter Luschny, "An introduction to the Bernoulli function",
+           https://arxiv.org/abs/2009.06743
+    """
+    return named_poly(n, dup_andre, ZZ, "Andre polynomial", (x,), polys)

rl/Lib/site-packages/sympy/polys/benchmarks/bench_galoispolys.py

@@ -0,0 +1,66 @@
+"""Benchmarks for polynomials over Galois fields. """
+
+
+from sympy.polys.galoistools import gf_from_dict, gf_factor_sqf
+from sympy.polys.domains import ZZ
+from sympy.core.numbers import pi
+from sympy.ntheory.generate import nextprime
+
+
+def gathen_poly(n, p, K):
+    return gf_from_dict({n: K.one, 1: K.one, 0: K.one}, p, K)
+
+
+def shoup_poly(n, p, K):
+    f = [K.one] * (n + 1)
+    for i in range(1, n + 1):
+        f[i] = (f[i - 1]**2 + K.one) % p
+    return f
+
+
+def genprime(n, K):
+    return K(nextprime(int((2**n * pi).evalf())))
+
+p_10 = genprime(10, ZZ)
+f_10 = gathen_poly(10, p_10, ZZ)
+
+p_20 = genprime(20, ZZ)
+f_20 = gathen_poly(20, p_20, ZZ)
+
+
+def timeit_gathen_poly_f10_zassenhaus():
+    gf_factor_sqf(f_10, p_10, ZZ, method='zassenhaus')
+
+
+def timeit_gathen_poly_f10_shoup():
+    gf_factor_sqf(f_10, p_10, ZZ, method='shoup')
+
+
+def timeit_gathen_poly_f20_zassenhaus():
+    gf_factor_sqf(f_20, p_20, ZZ, method='zassenhaus')
+
+
+def timeit_gathen_poly_f20_shoup():
+    gf_factor_sqf(f_20, p_20, ZZ, method='shoup')
+
+P_08 = genprime(8, ZZ)
+F_10 = shoup_poly(10, P_08, ZZ)
+
+P_18 = genprime(18, ZZ)
+F_20 = shoup_poly(20, P_18, ZZ)
+
+
+def timeit_shoup_poly_F10_zassenhaus():
+    gf_factor_sqf(F_10, P_08, ZZ, method='zassenhaus')
+
+
+def timeit_shoup_poly_F10_shoup():
+    gf_factor_sqf(F_10, P_08, ZZ, method='shoup')
+
+
+def timeit_shoup_poly_F20_zassenhaus():
+    gf_factor_sqf(F_20, P_18, ZZ, method='zassenhaus')
+
+
+def timeit_shoup_poly_F20_shoup():
+    gf_factor_sqf(F_20, P_18, ZZ, method='shoup')

rl/Lib/site-packages/sympy/polys/benchmarks/bench_groebnertools.py

@@ -0,0 +1,25 @@
+"""Benchmark of the Groebner bases algorithms. """
+
+
+from sympy.polys.rings import ring
+from sympy.polys.domains import QQ
+from sympy.polys.groebnertools import groebner
+
+R, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = ring("x1:13", QQ)
+
+V = R.gens
+E = [(x1, x2), (x2, x3), (x1, x4), (x1, x6), (x1, x12), (x2, x5), (x2, x7), (x3, x8),
+     (x3, x10), (x4, x11), (x4, x9), (x5, x6), (x6, x7), (x7, x8), (x8, x9), (x9, x10),
+     (x10, x11), (x11, x12), (x5, x12), (x5, x9), (x6, x10), (x7, x11), (x8, x12)]
+
+F3 = [ x**3 - 1 for x in V ]
+Fg = [ x**2 + x*y + y**2 for x, y in E ]
+
+F_1 = F3 + Fg
+F_2 = F3 + Fg + [x3**2 + x3*x4 + x4**2]
+
+def time_vertex_color_12_vertices_23_edges():
+    assert groebner(F_1, R) != [1]
+
+def time_vertex_color_12_vertices_24_edges():
+    assert groebner(F_2, R) == [1]

rl/Lib/site-packages/sympy/polys/benchmarks/bench_solvers.py

@@ -0,0 +1,543 @@
+from sympy.polys.rings import ring
+from sympy.polys.fields import field
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.solvers import solve_lin_sys
+
+# Expected times on 3.4 GHz i7:
+
+# In [1]: %timeit time_solve_lin_sys_189x49()
+# 1 loops, best of 3: 864 ms per loop
+# In [2]: %timeit time_solve_lin_sys_165x165()
+# 1 loops, best of 3: 1.83 s per loop
+# In [3]: %timeit time_solve_lin_sys_10x8()
+# 1 loops, best of 3: 2.31 s per loop
+
+# Benchmark R_165: shows how fast are arithmetics in QQ.
+
+R_165, uk_0, uk_1, uk_2, uk_3, uk_4, uk_5, uk_6, uk_7, uk_8, uk_9, uk_10, uk_11, uk_12, uk_13, uk_14, uk_15, uk_16, uk_17, uk_18, uk_19, uk_20, uk_21, uk_22, uk_23, uk_24, uk_25, uk_26, uk_27, uk_28, uk_29, uk_30, uk_31, uk_32, uk_33, uk_34, uk_35, uk_36, uk_37, uk_38, uk_39, uk_40, uk_41, uk_42, uk_43, uk_44, uk_45, uk_46, uk_47, uk_48, uk_49, uk_50, uk_51, uk_52, uk_53, uk_54, uk_55, uk_56, uk_57, uk_58, uk_59, uk_60, uk_61, uk_62, uk_63, uk_64, uk_65, uk_66, uk_67, uk_68, uk_69, uk_70, uk_71, uk_72, uk_73, uk_74, uk_75, uk_76, uk_77, uk_78, uk_79, uk_80, uk_81, uk_82, uk_83, uk_84, uk_85, uk_86, uk_87, uk_88, uk_89, uk_90, uk_91, uk_92, uk_93, uk_94, uk_95, uk_96, uk_97, uk_98, uk_99, uk_100, uk_101, uk_102, uk_103, uk_104, uk_105, uk_106, uk_107, uk_108, uk_109, uk_110, uk_111, uk_112, uk_113, uk_114, uk_115, uk_116, uk_117, uk_118, uk_119, uk_120, uk_121, uk_122, uk_123, uk_124, uk_125, uk_126, uk_127, uk_128, uk_129, uk_130, uk_131, uk_132, uk_133, uk_134, uk_135, uk_136, uk_137, uk_138, uk_139, uk_140, uk_141, uk_142, uk_143, uk_144, uk_145, uk_146, uk_147, uk_148, uk_149, uk_150, uk_151, uk_152, uk_153, uk_154, uk_155, uk_156, uk_157, uk_158, uk_159, uk_160, uk_161, uk_162, uk_163, uk_164 = ring("uk_:165", QQ)
+
+def eqs_165x165():
+    return [
+        uk_0 + 50719*uk_1 + 2789545*uk_10 + 411400*uk_100 + 1683000*uk_101 + 166375*uk_103 + 680625*uk_104 + 2784375*uk_106 + 729*uk_109 + 456471*uk_11 + 4131*uk_110 + 11016*uk_111 + 4455*uk_112 + 18225*uk_113 + 23409*uk_115 + 62424*uk_116 + 25245*uk_117 + 103275*uk_118 + 2586669*uk_12 + 166464*uk_120 + 67320*uk_121 + 275400*uk_122 + 27225*uk_124 + 111375*uk_125 + 455625*uk_127 + 6897784*uk_13 + 132651*uk_130 + 353736*uk_131 + 143055*uk_132 + 585225*uk_133 + 943296*uk_135 + 381480*uk_136 + 1560600*uk_137 + 154275*uk_139 + 2789545*uk_14 + 631125*uk_140 + 2581875*uk_142 + 2515456*uk_145 + 1017280*uk_146 + 4161600*uk_147 + 411400*uk_149 + 11411775*uk_15 + 1683000*uk_150 + 6885000*uk_152 + 166375*uk_155 + 680625*uk_156 + 2784375*uk_158 + 11390625*uk_161 + 3025*uk_17 + 495*uk_18 + 2805*uk_19 + 55*uk_2 + 7480*uk_20 + 3025*uk_21 + 12375*uk_22 + 81*uk_24 + 459*uk_25 + 1224*uk_26 + 495*uk_27 + 2025*uk_28 + 9*uk_3 + 2601*uk_30 + 6936*uk_31 + 2805*uk_32 + 11475*uk_33 + 18496*uk_35 + 7480*uk_36 + 30600*uk_37 + 3025*uk_39 + 51*uk_4 + 12375*uk_40 + 50625*uk_42 + 130470415844959*uk_45 + 141482932855*uk_46 + 23151752649*uk_47 + 131193265011*uk_48 + 349848706696*uk_49 + 136*uk_5 + 141482932855*uk_50 + 578793816225*uk_51 + 153424975*uk_53 + 25105905*uk_54 + 142266795*uk_55 + 379378120*uk_56 + 153424975*uk_57 + 627647625*uk_58 + 55*uk_6 + 4108239*uk_60 + 23280021*uk_61 + 62080056*uk_62 + 25105905*uk_63 + 102705975*uk_64 + 131920119*uk_66 + 351786984*uk_67 + 142266795*uk_68 + 582000525*uk_69 + 225*uk_7 + 938098624*uk_71 + 379378120*uk_72 + 1552001400*uk_73 + 153424975*uk_75 + 627647625*uk_76 + 2567649375*uk_78 + 166375*uk_81 + 27225*uk_82 + 154275*uk_83 + 411400*uk_84 + 166375*uk_85 + 680625*uk_86 + 4455*uk_88 + 25245*uk_89 + 2572416961*uk_9 + 67320*uk_90 + 27225*uk_91 + 111375*uk_92 + 143055*uk_94 + 381480*uk_95 + 154275*uk_96 + 631125*uk_97 + 1017280*uk_99,
+        uk_0 + 50719*uk_1 + 2789545*uk_10 + 413820*uk_100 + 1633500*uk_101 + 65340*uk_102 + 178695*uk_103 + 705375*uk_104 + 28215*uk_105 + 2784375*uk_106 + 111375*uk_107 + 4455*uk_108 + 97336*uk_109 + 2333074*uk_11 + 19044*uk_110 + 279312*uk_111 + 120612*uk_112 + 476100*uk_113 + 19044*uk_114 + 3726*uk_115 + 54648*uk_116 + 23598*uk_117 + 93150*uk_118 + 3726*uk_119 + 456471*uk_12 + 801504*uk_120 + 346104*uk_121 + 1366200*uk_122 + 54648*uk_123 + 149454*uk_124 + 589950*uk_125 + 23598*uk_126 + 2328750*uk_127 + 93150*uk_128 + 3726*uk_129 + 6694908*uk_13 + 729*uk_130 + 10692*uk_131 + 4617*uk_132 + 18225*uk_133 + 729*uk_134 + 156816*uk_135 + 67716*uk_136 + 267300*uk_137 + 10692*uk_138 + 29241*uk_139 + 2890983*uk_14 + 115425*uk_140 + 4617*uk_141 + 455625*uk_142 + 18225*uk_143 + 729*uk_144 + 2299968*uk_145 + 993168*uk_146 + 3920400*uk_147 + 156816*uk_148 + 428868*uk_149 + 11411775*uk_15 + 1692900*uk_150 + 67716*uk_151 + 6682500*uk_152 + 267300*uk_153 + 10692*uk_154 + 185193*uk_155 + 731025*uk_156 + 29241*uk_157 + 2885625*uk_158 + 115425*uk_159 + 456471*uk_16 + 4617*uk_160 + 11390625*uk_161 + 455625*uk_162 + 18225*uk_163 + 729*uk_164 + 3025*uk_17 + 2530*uk_18 + 495*uk_19 + 55*uk_2 + 7260*uk_20 + 3135*uk_21 + 12375*uk_22 + 495*uk_23 + 2116*uk_24 + 414*uk_25 + 6072*uk_26 + 2622*uk_27 + 10350*uk_28 + 414*uk_29 + 46*uk_3 + 81*uk_30 + 1188*uk_31 + 513*uk_32 + 2025*uk_33 + 81*uk_34 + 17424*uk_35 + 7524*uk_36 + 29700*uk_37 + 1188*uk_38 + 3249*uk_39 + 9*uk_4 + 12825*uk_40 + 513*uk_41 + 50625*uk_42 + 2025*uk_43 + 81*uk_44 + 130470415844959*uk_45 + 141482932855*uk_46 + 118331180206*uk_47 + 23151752649*uk_48 + 339559038852*uk_49 + 132*uk_5 + 146627766777*uk_50 + 578793816225*uk_51 + 23151752649*uk_52 + 153424975*uk_53 + 128319070*uk_54 + 25105905*uk_55 + 368219940*uk_56 + 159004065*uk_57 + 627647625*uk_58 + 25105905*uk_59 + 57*uk_6 + 107321404*uk_60 + 20997666*uk_61 + 307965768*uk_62 + 132985218*uk_63 + 524941650*uk_64 + 20997666*uk_65 + 4108239*uk_66 + 60254172*uk_67 + 26018847*uk_68 + 102705975*uk_69 + 225*uk_7 + 4108239*uk_70 + 883727856*uk_71 + 381609756*uk_72 + 1506354300*uk_73 + 60254172*uk_74 + 164786031*uk_75 + 650471175*uk_76 + 26018847*uk_77 + 2567649375*uk_78 + 102705975*uk_79 + 9*uk_8 + 4108239*uk_80 + 166375*uk_81 + 139150*uk_82 + 27225*uk_83 + 399300*uk_84 + 172425*uk_85 + 680625*uk_86 + 27225*uk_87 + 116380*uk_88 + 22770*uk_89 + 2572416961*uk_9 + 333960*uk_90 + 144210*uk_91 + 569250*uk_92 + 22770*uk_93 + 4455*uk_94 + 65340*uk_95 + 28215*uk_96 + 111375*uk_97 + 4455*uk_98 + 958320*uk_99,
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+        uk_0 + 47353*uk_1 + 2983239*uk_10 + 105336*uk_100 + 109368*uk_101 + 79128*uk_102 + 2751903*uk_103 + 2857239*uk_104 + 2067219*uk_105 + 2966607*uk_106 + 2146347*uk_107 + 1552887*uk_108 + 1685159*uk_109 + 5635007*uk_11 + 2223277*uk_110 + 113288*uk_111 + 2959649*uk_112 + 3072937*uk_113 + 2223277*uk_114 + 2933231*uk_115 + 149464*uk_116 + 3904747*uk_117 + 4054211*uk_118 + 2933231*uk_119 + 7434421*uk_12 + 7616*uk_120 + 198968*uk_121 + 206584*uk_122 + 149464*uk_123 + 5198039*uk_124 + 5397007*uk_125 + 3904747*uk_126 + 5603591*uk_127 + 4054211*uk_128 + 2933231*uk_129 + 378824*uk_13 + 3869893*uk_130 + 197192*uk_131 + 5151641*uk_132 + 5348833*uk_133 + 3869893*uk_134 + 10048*uk_135 + 262504*uk_136 + 272552*uk_137 + 197192*uk_138 + 6857917*uk_139 + 9896777*uk_14 + 7120421*uk_140 + 5151641*uk_141 + 7392973*uk_142 + 5348833*uk_143 + 3869893*uk_144 + 512*uk_145 + 13376*uk_146 + 13888*uk_147 + 10048*uk_148 + 349448*uk_149 + 10275601*uk_15 + 362824*uk_150 + 262504*uk_151 + 376712*uk_152 + 272552*uk_153 + 197192*uk_154 + 9129329*uk_155 + 9478777*uk_156 + 6857917*uk_157 + 9841601*uk_158 + 7120421*uk_159 + 7434421*uk_16 + 5151641*uk_160 + 10218313*uk_161 + 7392973*uk_162 + 5348833*uk_163 + 3869893*uk_164 + 3969*uk_17 + 7497*uk_18 + 9891*uk_19 + 63*uk_2 + 504*uk_20 + 13167*uk_21 + 13671*uk_22 + 9891*uk_23 + 14161*uk_24 + 18683*uk_25 + 952*uk_26 + 24871*uk_27 + 25823*uk_28 + 18683*uk_29 + 119*uk_3 + 24649*uk_30 + 1256*uk_31 + 32813*uk_32 + 34069*uk_33 + 24649*uk_34 + 64*uk_35 + 1672*uk_36 + 1736*uk_37 + 1256*uk_38 + 43681*uk_39 + 157*uk_4 + 45353*uk_40 + 32813*uk_41 + 47089*uk_42 + 34069*uk_43 + 24649*uk_44 + 106179944855977*uk_45 + 141265316367*uk_46 + 266834486471*uk_47 + 352042137613*uk_48 + 17938452872*uk_49 + 8*uk_5 + 468642081281*uk_50 + 486580534153*uk_51 + 352042137613*uk_52 + 187944057*uk_53 + 355005441*uk_54 + 468368523*uk_55 + 23865912*uk_56 + 623496951*uk_57 + 647362863*uk_58 + 468368523*uk_59 + 209*uk_6 + 670565833*uk_60 + 884696099*uk_61 + 45080056*uk_62 + 1177716463*uk_63 + 1222796519*uk_64 + 884696099*uk_65 + 1167204097*uk_66 + 59475368*uk_67 + 1553793989*uk_68 + 1613269357*uk_69 + 217*uk_7 + 1167204097*uk_70 + 3030592*uk_71 + 79174216*uk_72 + 82204808*uk_73 + 59475368*uk_74 + 2068426393*uk_75 + 2147600609*uk_76 + 1553793989*uk_77 + 2229805417*uk_78 + 1613269357*uk_79 + 157*uk_8 + 1167204097*uk_80 + 250047*uk_81 + 472311*uk_82 + 623133*uk_83 + 31752*uk_84 + 829521*uk_85 + 861273*uk_86 + 623133*uk_87 + 892143*uk_88 + 1177029*uk_89 + 2242306609*uk_9 + 59976*uk_90 + 1566873*uk_91 + 1626849*uk_92 + 1177029*uk_93 + 1552887*uk_94 + 79128*uk_95 + 2067219*uk_96 + 2146347*uk_97 + 1552887*uk_98 + 4032*uk_99,
+        uk_0 + 47353*uk_1 + 2983239*uk_10 + 106344*uk_100 + 109368*uk_101 + 59976*uk_102 + 2804823*uk_103 + 2884581*uk_104 + 1581867*uk_105 + 2966607*uk_106 + 1626849*uk_107 + 892143*uk_108 + 704969*uk_109 + 4214417*uk_11 + 942599*uk_110 + 63368*uk_111 + 1671331*uk_112 + 1718857*uk_113 + 942599*uk_114 + 1260329*uk_115 + 84728*uk_116 + 2234701*uk_117 + 2298247*uk_118 + 1260329*uk_119 + 5635007*uk_12 + 5696*uk_120 + 150232*uk_121 + 154504*uk_122 + 84728*uk_123 + 3962369*uk_124 + 4075043*uk_125 + 2234701*uk_126 + 4190921*uk_127 + 2298247*uk_128 + 1260329*uk_129 + 378824*uk_13 + 1685159*uk_130 + 113288*uk_131 + 2987971*uk_132 + 3072937*uk_133 + 1685159*uk_134 + 7616*uk_135 + 200872*uk_136 + 206584*uk_137 + 113288*uk_138 + 5297999*uk_139 + 9991483*uk_14 + 5448653*uk_140 + 2987971*uk_141 + 5603591*uk_142 + 3072937*uk_143 + 1685159*uk_144 + 512*uk_145 + 13504*uk_146 + 13888*uk_147 + 7616*uk_148 + 356168*uk_149 + 10275601*uk_15 + 366296*uk_150 + 200872*uk_151 + 376712*uk_152 + 206584*uk_153 + 113288*uk_154 + 9393931*uk_155 + 9661057*uk_156 + 5297999*uk_157 + 9935779*uk_158 + 5448653*uk_159 + 5635007*uk_16 + 2987971*uk_160 + 10218313*uk_161 + 5603591*uk_162 + 3072937*uk_163 + 1685159*uk_164 + 3969*uk_17 + 5607*uk_18 + 7497*uk_19 + 63*uk_2 + 504*uk_20 + 13293*uk_21 + 13671*uk_22 + 7497*uk_23 + 7921*uk_24 + 10591*uk_25 + 712*uk_26 + 18779*uk_27 + 19313*uk_28 + 10591*uk_29 + 89*uk_3 + 14161*uk_30 + 952*uk_31 + 25109*uk_32 + 25823*uk_33 + 14161*uk_34 + 64*uk_35 + 1688*uk_36 + 1736*uk_37 + 952*uk_38 + 44521*uk_39 + 119*uk_4 + 45787*uk_40 + 25109*uk_41 + 47089*uk_42 + 25823*uk_43 + 14161*uk_44 + 106179944855977*uk_45 + 141265316367*uk_46 + 199565288201*uk_47 + 266834486471*uk_48 + 17938452872*uk_49 + 8*uk_5 + 473126694499*uk_50 + 486580534153*uk_51 + 266834486471*uk_52 + 187944057*uk_53 + 265508271*uk_54 + 355005441*uk_55 + 23865912*uk_56 + 629463429*uk_57 + 647362863*uk_58 + 355005441*uk_59 + 211*uk_6 + 375083113*uk_60 + 501515623*uk_61 + 33715336*uk_62 + 889241987*uk_63 + 914528489*uk_64 + 501515623*uk_65 + 670565833*uk_66 + 45080056*uk_67 + 1188986477*uk_68 + 1222796519*uk_69 + 217*uk_7 + 670565833*uk_70 + 3030592*uk_71 + 79931864*uk_72 + 82204808*uk_73 + 45080056*uk_74 + 2108202913*uk_75 + 2168151811*uk_76 + 1188986477*uk_77 + 2229805417*uk_78 + 1222796519*uk_79 + 119*uk_8 + 670565833*uk_80 + 250047*uk_81 + 353241*uk_82 + 472311*uk_83 + 31752*uk_84 + 837459*uk_85 + 861273*uk_86 + 472311*uk_87 + 499023*uk_88 + 667233*uk_89 + 2242306609*uk_9 + 44856*uk_90 + 1183077*uk_91 + 1216719*uk_92 + 667233*uk_93 + 892143*uk_94 + 59976*uk_95 + 1581867*uk_96 + 1626849*uk_97 + 892143*uk_98 + 4032*uk_99,
+        uk_0 + 47353*uk_1 + 2983239*uk_10 + 107352*uk_100 + 109368*uk_101 + 44856*uk_102 + 2858247*uk_103 + 2911923*uk_104 + 1194291*uk_105 + 2966607*uk_106 + 1216719*uk_107 + 499023*uk_108 + 300763*uk_109 + 3172651*uk_11 + 399521*uk_110 + 35912*uk_111 + 956157*uk_112 + 974113*uk_113 + 399521*uk_114 + 530707*uk_115 + 47704*uk_116 + 1270119*uk_117 + 1293971*uk_118 + 530707*uk_119 + 4214417*uk_12 + 4288*uk_120 + 114168*uk_121 + 116312*uk_122 + 47704*uk_123 + 3039723*uk_124 + 3096807*uk_125 + 1270119*uk_126 + 3154963*uk_127 + 1293971*uk_128 + 530707*uk_129 + 378824*uk_13 + 704969*uk_130 + 63368*uk_131 + 1687173*uk_132 + 1718857*uk_133 + 704969*uk_134 + 5696*uk_135 + 151656*uk_136 + 154504*uk_137 + 63368*uk_138 + 4037841*uk_139 + 10086189*uk_14 + 4113669*uk_140 + 1687173*uk_141 + 4190921*uk_142 + 1718857*uk_143 + 704969*uk_144 + 512*uk_145 + 13632*uk_146 + 13888*uk_147 + 5696*uk_148 + 362952*uk_149 + 10275601*uk_15 + 369768*uk_150 + 151656*uk_151 + 376712*uk_152 + 154504*uk_153 + 63368*uk_154 + 9663597*uk_155 + 9845073*uk_156 + 4037841*uk_157 + 10029957*uk_158 + 4113669*uk_159 + 4214417*uk_16 + 1687173*uk_160 + 10218313*uk_161 + 4190921*uk_162 + 1718857*uk_163 + 704969*uk_164 + 3969*uk_17 + 4221*uk_18 + 5607*uk_19 + 63*uk_2 + 504*uk_20 + 13419*uk_21 + 13671*uk_22 + 5607*uk_23 + 4489*uk_24 + 5963*uk_25 + 536*uk_26 + 14271*uk_27 + 14539*uk_28 + 5963*uk_29 + 67*uk_3 + 7921*uk_30 + 712*uk_31 + 18957*uk_32 + 19313*uk_33 + 7921*uk_34 + 64*uk_35 + 1704*uk_36 + 1736*uk_37 + 712*uk_38 + 45369*uk_39 + 89*uk_4 + 46221*uk_40 + 18957*uk_41 + 47089*uk_42 + 19313*uk_43 + 7921*uk_44 + 106179944855977*uk_45 + 141265316367*uk_46 + 150234542803*uk_47 + 199565288201*uk_48 + 17938452872*uk_49 + 8*uk_5 + 477611307717*uk_50 + 486580534153*uk_51 + 199565288201*uk_52 + 187944057*uk_53 + 199877013*uk_54 + 265508271*uk_55 + 23865912*uk_56 + 635429907*uk_57 + 647362863*uk_58 + 265508271*uk_59 + 213*uk_6 + 212567617*uk_60 + 282365939*uk_61 + 25381208*uk_62 + 675774663*uk_63 + 688465267*uk_64 + 282365939*uk_65 + 375083113*uk_66 + 33715336*uk_67 + 897670821*uk_68 + 914528489*uk_69 + 217*uk_7 + 375083113*uk_70 + 3030592*uk_71 + 80689512*uk_72 + 82204808*uk_73 + 33715336*uk_74 + 2148358257*uk_75 + 2188703013*uk_76 + 897670821*uk_77 + 2229805417*uk_78 + 914528489*uk_79 + 89*uk_8 + 375083113*uk_80 + 250047*uk_81 + 265923*uk_82 + 353241*uk_83 + 31752*uk_84 + 845397*uk_85 + 861273*uk_86 + 353241*uk_87 + 282807*uk_88 + 375669*uk_89 + 2242306609*uk_9 + 33768*uk_90 + 899073*uk_91 + 915957*uk_92 + 375669*uk_93 + 499023*uk_94 + 44856*uk_95 + 1194291*uk_96 + 1216719*uk_97 + 499023*uk_98 + 4032*uk_99,
+        uk_0 + 47353*uk_1 + 2983239*uk_10 + 108360*uk_100 + 109368*uk_101 + 33768*uk_102 + 2912175*uk_103 + 2939265*uk_104 + 907515*uk_105 + 2966607*uk_106 + 915957*uk_107 + 282807*uk_108 + 148877*uk_109 + 2509709*uk_11 + 188203*uk_110 + 22472*uk_111 + 603935*uk_112 + 609553*uk_113 + 188203*uk_114 + 237917*uk_115 + 28408*uk_116 + 763465*uk_117 + 770567*uk_118 + 237917*uk_119 + 3172651*uk_12 + 3392*uk_120 + 91160*uk_121 + 92008*uk_122 + 28408*uk_123 + 2449925*uk_124 + 2472715*uk_125 + 763465*uk_126 + 2495717*uk_127 + 770567*uk_128 + 237917*uk_129 + 378824*uk_13 + 300763*uk_130 + 35912*uk_131 + 965135*uk_132 + 974113*uk_133 + 300763*uk_134 + 4288*uk_135 + 115240*uk_136 + 116312*uk_137 + 35912*uk_138 + 3097075*uk_139 + 10180895*uk_14 + 3125885*uk_140 + 965135*uk_141 + 3154963*uk_142 + 974113*uk_143 + 300763*uk_144 + 512*uk_145 + 13760*uk_146 + 13888*uk_147 + 4288*uk_148 + 369800*uk_149 + 10275601*uk_15 + 373240*uk_150 + 115240*uk_151 + 376712*uk_152 + 116312*uk_153 + 35912*uk_154 + 9938375*uk_155 + 10030825*uk_156 + 3097075*uk_157 + 10124135*uk_158 + 3125885*uk_159 + 3172651*uk_16 + 965135*uk_160 + 10218313*uk_161 + 3154963*uk_162 + 974113*uk_163 + 300763*uk_164 + 3969*uk_17 + 3339*uk_18 + 4221*uk_19 + 63*uk_2 + 504*uk_20 + 13545*uk_21 + 13671*uk_22 + 4221*uk_23 + 2809*uk_24 + 3551*uk_25 + 424*uk_26 + 11395*uk_27 + 11501*uk_28 + 3551*uk_29 + 53*uk_3 + 4489*uk_30 + 536*uk_31 + 14405*uk_32 + 14539*uk_33 + 4489*uk_34 + 64*uk_35 + 1720*uk_36 + 1736*uk_37 + 536*uk_38 + 46225*uk_39 + 67*uk_4 + 46655*uk_40 + 14405*uk_41 + 47089*uk_42 + 14539*uk_43 + 4489*uk_44 + 106179944855977*uk_45 + 141265316367*uk_46 + 118842250277*uk_47 + 150234542803*uk_48 + 17938452872*uk_49 + 8*uk_5 + 482095920935*uk_50 + 486580534153*uk_51 + 150234542803*uk_52 + 187944057*uk_53 + 158111667*uk_54 + 199877013*uk_55 + 23865912*uk_56 + 641396385*uk_57 + 647362863*uk_58 + 199877013*uk_59 + 215*uk_6 + 133014577*uk_60 + 168150503*uk_61 + 20077672*uk_62 + 539587435*uk_63 + 544606853*uk_64 + 168150503*uk_65 + 212567617*uk_66 + 25381208*uk_67 + 682119965*uk_68 + 688465267*uk_69 + 217*uk_7 + 212567617*uk_70 + 3030592*uk_71 + 81447160*uk_72 + 82204808*uk_73 + 25381208*uk_74 + 2188892425*uk_75 + 2209254215*uk_76 + 682119965*uk_77 + 2229805417*uk_78 + 688465267*uk_79 + 67*uk_8 + 212567617*uk_80 + 250047*uk_81 + 210357*uk_82 + 265923*uk_83 + 31752*uk_84 + 853335*uk_85 + 861273*uk_86 + 265923*uk_87 + 176967*uk_88 + 223713*uk_89 + 2242306609*uk_9 + 26712*uk_90 + 717885*uk_91 + 724563*uk_92 + 223713*uk_93 + 282807*uk_94 + 33768*uk_95 + 907515*uk_96 + 915957*uk_97 + 282807*uk_98 + 4032*uk_99,
+        uk_0 + 47353*uk_1 + 2983239*uk_10 + 109368*uk_100 + 109368*uk_101 + 26712*uk_102 + 2966607*uk_103 + 2966607*uk_104 + 724563*uk_105 + 2966607*uk_106 + 724563*uk_107 + 176967*uk_108 + 103823*uk_109 + 2225591*uk_11 + 117077*uk_110 + 17672*uk_111 + 479353*uk_112 + 479353*uk_113 + 117077*uk_114 + 132023*uk_115 + 19928*uk_116 + 540547*uk_117 + 540547*uk_118 + 132023*uk_119 + 2509709*uk_12 + 3008*uk_120 + 81592*uk_121 + 81592*uk_122 + 19928*uk_123 + 2213183*uk_124 + 2213183*uk_125 + 540547*uk_126 + 2213183*uk_127 + 540547*uk_128 + 132023*uk_129 + 378824*uk_13 + 148877*uk_130 + 22472*uk_131 + 609553*uk_132 + 609553*uk_133 + 148877*uk_134 + 3392*uk_135 + 92008*uk_136 + 92008*uk_137 + 22472*uk_138 + 2495717*uk_139 + 10275601*uk_14 + 2495717*uk_140 + 609553*uk_141 + 2495717*uk_142 + 609553*uk_143 + 148877*uk_144 + 512*uk_145 + 13888*uk_146 + 13888*uk_147 + 3392*uk_148 + 376712*uk_149 + 10275601*uk_15 + 376712*uk_150 + 92008*uk_151 + 376712*uk_152 + 92008*uk_153 + 22472*uk_154 + 10218313*uk_155 + 10218313*uk_156 + 2495717*uk_157 + 10218313*uk_158 + 2495717*uk_159 + 2509709*uk_16 + 609553*uk_160 + 10218313*uk_161 + 2495717*uk_162 + 609553*uk_163 + 148877*uk_164 + 3969*uk_17 + 2961*uk_18 + 3339*uk_19 + 63*uk_2 + 504*uk_20 + 13671*uk_21 + 13671*uk_22 + 3339*uk_23 + 2209*uk_24 + 2491*uk_25 + 376*uk_26 + 10199*uk_27 + 10199*uk_28 + 2491*uk_29 + 47*uk_3 + 2809*uk_30 + 424*uk_31 + 11501*uk_32 + 11501*uk_33 + 2809*uk_34 + 64*uk_35 + 1736*uk_36 + 1736*uk_37 + 424*uk_38 + 47089*uk_39 + 53*uk_4 + 47089*uk_40 + 11501*uk_41 + 47089*uk_42 + 11501*uk_43 + 2809*uk_44 + 106179944855977*uk_45 + 141265316367*uk_46 + 105388410623*uk_47 + 118842250277*uk_48 + 17938452872*uk_49 + 8*uk_5 + 486580534153*uk_50 + 486580534153*uk_51 + 118842250277*uk_52 + 187944057*uk_53 + 140212233*uk_54 + 158111667*uk_55 + 23865912*uk_56 + 647362863*uk_57 + 647362863*uk_58 + 158111667*uk_59 + 217*uk_6 + 104602777*uk_60 + 117956323*uk_61 + 17804728*uk_62 + 482953247*uk_63 + 482953247*uk_64 + 117956323*uk_65 + 133014577*uk_66 + 20077672*uk_67 + 544606853*uk_68 + 544606853*uk_69 + 217*uk_7 + 133014577*uk_70 + 3030592*uk_71 + 82204808*uk_72 + 82204808*uk_73 + 20077672*uk_74 + 2229805417*uk_75 + 2229805417*uk_76 + 544606853*uk_77 + 2229805417*uk_78 + 544606853*uk_79 + 53*uk_8 + 133014577*uk_80 + 250047*uk_81 + 186543*uk_82 + 210357*uk_83 + 31752*uk_84 + 861273*uk_85 + 861273*uk_86 + 210357*uk_87 + 139167*uk_88 + 156933*uk_89 + 2242306609*uk_9 + 23688*uk_90 + 642537*uk_91 + 642537*uk_92 + 156933*uk_93 + 176967*uk_94 + 26712*uk_95 + 724563*uk_96 + 724563*uk_97 + 176967*uk_98 + 4032*uk_99,
+    ]
+
+def sol_165x165():
+    return {
+        uk_0: -QQ(295441,1683)*uk_2 - QQ(175799,1683)*uk_7 + QQ(2401696807,1)*uk_9 - QQ(9606787228,1683)*uk_10 + QQ(9606787228,1683)*uk_15 - QQ(29030443,1683)*uk_17 - QQ(5965893,187)*uk_22 + QQ(262901,99)*uk_42 + QQ(235539209256104,1)*uk_45 - QQ(232597130667529,1683)*uk_46 + QQ(1364372733998209,1683)*uk_51 - QQ(1133600892904,1683)*uk_53 - QQ(172922170104,187)*uk_58 + QQ(249776467928,99)*uk_78 - QQ(2401889209,1683)*uk_81 - QQ(636292759,187)*uk_86 - QQ(1034157281,187)*uk_106 + QQ(10558824289,1683)*uk_161,
+        uk_1: QQ(4,1683)*uk_2 - QQ(4,1683)*uk_7 - QQ(98072,1)*uk_9 + QQ(96847,1683)*uk_10 - QQ(568087,1683)*uk_15 + QQ(472,1683)*uk_17 + QQ(72,187)*uk_22 - QQ(104,99)*uk_42 - QQ(7216420377,1)*uk_45 - QQ(108808244,1683)*uk_46 - QQ(46106641036,1683)*uk_51 + QQ(17259541,1683)*uk_53 + QQ(1095291,187)*uk_58 - QQ(9936587,99)*uk_78 + QQ(41836,1683)*uk_81 + QQ(10036,187)*uk_86 + QQ(10124,187)*uk_106 - QQ(8,1)*uk_149 - QQ(586156,1683)*uk_161,
+        uk_3: -QQ(295441,1683)*uk_18 - QQ(175799,1683)*uk_28 + QQ(2401696807,1)*uk_47 - QQ(9606787228,1683)*uk_54 + QQ(9606787228,1683)*uk_64 - QQ(29030443,1683)*uk_82 - QQ(5965893,187)*uk_92 + QQ(262901,99)*uk_127 + QQ(8,1)*uk_149,
+        uk_4: -QQ(295441,1683)*uk_19 + QQ(1602583,3366)*uk_29 - QQ(175799,1683)*uk_33 - QQ(45670,99)*uk_34 - QQ(76006,187)*uk_38 + QQ(295441,1683)*uk_41 - QQ(45670,99)*uk_44 + QQ(2401696807,1)*uk_48 - QQ(9606787228,1683)*uk_55 + QQ(74452601017,3366)*uk_65 + QQ(9606787228,1683)*uk_69 - QQ(2401696807,99)*uk_70 - QQ(4803393614,187)*uk_74 + QQ(9606787228,1683)*uk_77 - QQ(2401696807,99)*uk_80 - QQ(29030443,1683)*uk_83 + QQ(11596905,374)*uk_93 - QQ(5965893,187)*uk_97 - QQ(769658,33)*uk_98 - QQ(17335370,1683)*uk_102 + QQ(29030443,1683)*uk_105 - QQ(769658,33)*uk_108 + QQ(77314807,3366)*uk_114 + QQ(750229,198)*uk_119 + QQ(72457964,1683)*uk_123 + QQ(11596905,374)*uk_126 + QQ(31304645,306)*uk_128 + QQ(750229,198)*uk_129 - QQ(3191393,99)*uk_134 - QQ(647642,9)*uk_138 - QQ(769658,33)*uk_141 + QQ(262901,99)*uk_142 - QQ(10478626,99)*uk_143 - QQ(3191393,99)*uk_144 - QQ(20480616,187)*uk_148 - QQ(17335370,1683)*uk_151 - QQ(174199750,1683)*uk_153 - QQ(647642,9)*uk_154 + QQ(29030443,1683)*uk_157 + QQ(5965893,187)*uk_159 - QQ(769658,33)*uk_160 - QQ(10478626,99)*uk_163 - QQ(3191393,99)*uk_164,
+        uk_5: -QQ(295441,1683)*uk_20 - QQ(175799,1683)*uk_37 + QQ(2401696807,1)*uk_49 - QQ(9606787228,1683)*uk_56 + QQ(9606787228,1683)*uk_73 - QQ(29030443,1683)*uk_84 - QQ(5965893,187)*uk_101 + QQ(262901,99)*uk_152,
+        uk_6: -QQ(295441,1683)*uk_21 - QQ(175799,1683)*uk_40 + QQ(2401696807,1)*uk_50 - QQ(9606787228,1683)*uk_57 + QQ(9606787228,1683)*uk_76 - QQ(29030443,1683)*uk_85 - QQ(5965893,187)*uk_104 + QQ(262901,99)*uk_158,
+        uk_8: -QQ(295441,1683)*uk_23 - QQ(1602583,3366)*uk_29 + QQ(45670,99)*uk_34 + QQ(76006,187)*uk_38 - QQ(295441,1683)*uk_41 - QQ(175799,1683)*uk_43 + QQ(45670,99)*uk_44 + QQ(2401696807,1)*uk_52 - QQ(9606787228,1683)*uk_59 - QQ(74452601017,3366)*uk_65 + QQ(2401696807,99)*uk_70 + QQ(4803393614,187)*uk_74 - QQ(9606787228,1683)*uk_77 + QQ(9606787228,1683)*uk_79 + QQ(2401696807,99)*uk_80 - QQ(29030443,1683)*uk_87 - QQ(11596905,374)*uk_93 + QQ(769658,33)*uk_98 + QQ(17335370,1683)*uk_102 - QQ(29030443,1683)*uk_105 - QQ(5965893,187)*uk_107 + QQ(769658,33)*uk_108 - QQ(77314807,3366)*uk_114 - QQ(750229,198)*uk_119 - QQ(72457964,1683)*uk_123 - QQ(11596905,374)*uk_126 - QQ(31304645,306)*uk_128 - QQ(750229,198)*uk_129 + QQ(3191393,99)*uk_134 + QQ(647642,9)*uk_138 + QQ(769658,33)*uk_141 + QQ(10478626,99)*uk_143 + QQ(3191393,99)*uk_144 + QQ(20480616,187)*uk_148 + QQ(17335370,1683)*uk_151 + QQ(174199750,1683)*uk_153 + QQ(647642,9)*uk_154 - QQ(29030443,1683)*uk_157 - QQ(5965893,187)*uk_159 + QQ(769658,33)*uk_160 + QQ(262901,99)*uk_162 + QQ(10478626,99)*uk_163 + QQ(3191393,99)*uk_164,
+        uk_11: QQ(4,1683)*uk_18 - QQ(4,1683)*uk_28 - QQ(98072,1)*uk_47 + QQ(96847,1683)*uk_54 - QQ(568087,1683)*uk_64 + QQ(472,1683)*uk_82 + QQ(72,187)*uk_92 - QQ(104,99)*uk_127,
+        uk_12: QQ(4,1683)*uk_19 - QQ(31,3366)*uk_29 - QQ(4,1683)*uk_33 + QQ(1,99)*uk_34 + QQ(2,187)*uk_38 - QQ(4,1683)*uk_41 + QQ(1,99)*uk_44 - QQ(98072,1)*uk_48 + QQ(96847,1683)*uk_55 - QQ(1437649,3366)*uk_65 - QQ(568087,1683)*uk_69 + QQ(52402,99)*uk_70 + QQ(120138,187)*uk_74 - QQ(96847,1683)*uk_77 + QQ(52402,99)*uk_80 + QQ(472,1683)*uk_83 - QQ(225,374)*uk_93 + QQ(72,187)*uk_97 + QQ(17,33)*uk_98 + QQ(590,1683)*uk_102 - QQ(472,1683)*uk_105 + QQ(17,33)*uk_108 - QQ(1519,3366)*uk_114 - QQ(13,198)*uk_119 - QQ(1388,1683)*uk_123 - QQ(225,374)*uk_126 - QQ(605,306)*uk_128 - QQ(13,198)*uk_129 + QQ(68,99)*uk_134 + QQ(14,9)*uk_138 + QQ(17,33)*uk_141 - QQ(104,99)*uk_142 + QQ(229,99)*uk_143 + QQ(68,99)*uk_144 + QQ(472,187)*uk_148 + QQ(590,1683)*uk_151 + QQ(4450,1683)*uk_153 + QQ(14,9)*uk_154 - QQ(472,1683)*uk_157 - QQ(72,187)*uk_159 + QQ(17,33)*uk_160 + QQ(229,99)*uk_163 + QQ(68,99)*uk_164,
+        uk_13: QQ(4,1683)*uk_20 - QQ(4,1683)*uk_37 - QQ(98072,1)*uk_49 + QQ(96847,1683)*uk_56 - QQ(568087,1683)*uk_73 + QQ(472,1683)*uk_84 + QQ(72,187)*uk_101 - QQ(104,99)*uk_152,
+        uk_14: QQ(4,1683)*uk_21 - QQ(4,1683)*uk_40 - QQ(98072,1)*uk_50 + QQ(96847,1683)*uk_57 - QQ(568087,1683)*uk_76 + QQ(472,1683)*uk_85 + QQ(72,187)*uk_104 - QQ(104,99)*uk_158,
+        uk_16: QQ(4,1683)*uk_23 + QQ(31,3366)*uk_29 - QQ(1,99)*uk_34 - QQ(2,187)*uk_38 + QQ(4,1683)*uk_41 - QQ(4,1683)*uk_43 - QQ(1,99)*uk_44 - QQ(98072,1)*uk_52 + QQ(96847,1683)*uk_59 + QQ(1437649,3366)*uk_65 - QQ(52402,99)*uk_70 - QQ(120138,187)*uk_74 + QQ(96847,1683)*uk_77 - QQ(568087,1683)*uk_79 - QQ(52402,99)*uk_80 + QQ(472,1683)*uk_87 + QQ(225,374)*uk_93 - QQ(17,33)*uk_98 - QQ(590,1683)*uk_102 + QQ(472,1683)*uk_105 + QQ(72,187)*uk_107 - QQ(17,33)*uk_108 + QQ(1519,3366)*uk_114 + QQ(13,198)*uk_119 + QQ(1388,1683)*uk_123 + QQ(225,374)*uk_126 + QQ(605,306)*uk_128 + QQ(13,198)*uk_129 - QQ(68,99)*uk_134 - QQ(14,9)*uk_138 - QQ(17,33)*uk_141 - QQ(229,99)*uk_143 - QQ(68,99)*uk_144 - QQ(472,187)*uk_148 - QQ(590,1683)*uk_151 - QQ(4450,1683)*uk_153 - QQ(14,9)*uk_154 + QQ(472,1683)*uk_157 + QQ(72,187)*uk_159 - QQ(17,33)*uk_160 - QQ(104,99)*uk_162 - QQ(229,99)*uk_163 - QQ(68,99)*uk_164,
+        uk_24: -QQ(295441,1683)*uk_88 - QQ(175799,1683)*uk_113,
+        uk_26: -QQ(295441,1683)*uk_90 - QQ(175799,1683)*uk_122, uk_25: -uk_29 - QQ(295441,1683)*uk_89 - QQ(295441,1683)*uk_93 - QQ(175799,1683)*uk_118 - QQ(175799,1683)*uk_128,
+        uk_27: -QQ(295441,1683)*uk_91 - QQ(175799,1683)*uk_125 - QQ(4,1)*uk_149,
+        uk_30: -uk_34 - uk_44 - QQ(295441,1683)*uk_94 - QQ(295441,1683)*uk_98 - QQ(295441,1683)*uk_108 - QQ(175799,1683)*uk_133 - QQ(175799,1683)*uk_143 - QQ(175799,1683)*uk_163,
+        uk_31: -uk_38 - QQ(295441,1683)*uk_95 - QQ(295441,1683)*uk_102 - QQ(175799,1683)*uk_137 - QQ(175799,1683)*uk_153,
+        uk_32: -uk_41 - QQ(295441,1683)*uk_96 - QQ(295441,1683)*uk_105 - QQ(175799,1683)*uk_140 + QQ(4,1)*uk_149 - QQ(175799,1683)*uk_159,
+        uk_35: -QQ(295441,1683)*uk_99 - QQ(175799,1683)*uk_147,
+        uk_36: -QQ(295441,1683)*uk_100 - QQ(2,1)*uk_149 - QQ(175799,1683)*uk_150,
+        uk_39: -QQ(295441,1683)*uk_103 - QQ(175799,1683)*uk_156,
+        uk_60: QQ(4,1683)*uk_88 - QQ(4,1683)*uk_113,
+        uk_61: -uk_65 + QQ(4,1683)*uk_89 + QQ(4,1683)*uk_93 - QQ(4,1683)*uk_118 - QQ(4,1683)*uk_128,
+        uk_62: QQ(4,1683)*uk_90 - QQ(4,1683)*uk_122,
+        uk_63: QQ(4,1683)*uk_91 - QQ(4,1683)*uk_125,
+        uk_66: -uk_70 - uk_80 + QQ(4,1683)*uk_94 + QQ(4,1683)*uk_98 + QQ(4,1683)*uk_108 - QQ(4,1683)*uk_133 - QQ(4,1683)*uk_143 - QQ(4,1683)*uk_163,
+        uk_67: -uk_74 + QQ(4,1683)*uk_95 + QQ(4,1683)*uk_102 - QQ(4,1683)*uk_137 - QQ(4,1683)*uk_153,
+        uk_68: -uk_77 + QQ(4,1683)*uk_96 + QQ(4,1683)*uk_105 - QQ(4,1683)*uk_140 - QQ(4,1683)*uk_159,
+        uk_71: QQ(4,1683)*uk_99 - QQ(4,1683)*uk_147,
+        uk_72: QQ(4,1683)*uk_100 - QQ(4,1683)*uk_150,
+        uk_75: QQ(4,1683)*uk_103 - QQ(4,1683)*uk_156,
+        uk_109: 0,
+        uk_110: -uk_114,
+        uk_111: 0,
+        uk_112: 0,
+        uk_115: -uk_119 - uk_129,
+        uk_116: -uk_123,
+        uk_117: -uk_126,
+        uk_120: 0,
+        uk_121: 0,
+        uk_124: 0,
+        uk_130: -uk_134 - uk_144 - uk_164,
+        uk_131: -uk_138 - uk_154,
+        uk_132: -uk_141 - uk_160,
+        uk_135: -uk_148,
+        uk_136: -uk_151,
+        uk_139: -uk_157,
+        uk_145: 0,
+        uk_146: 0,
+        uk_155: 0,
+    }
+
+def time_eqs_165x165():
+    if len(eqs_165x165()) != 165:
+        raise ValueError("length should be 165")
+
+def time_solve_lin_sys_165x165():
+    eqs = eqs_165x165()
+    sol = solve_lin_sys(eqs, R_165)
+    if sol != sol_165x165():
+        raise ValueError("Value should be equal")
+
+def time_verify_sol_165x165():
+    eqs = eqs_165x165()
+    sol = sol_165x165()
+    zeros = [ eq.compose(sol) for eq in eqs ]
+    if not all(zero == 0 for zero in zeros):
+        raise ValueError("All should be 0")
+
+def time_to_expr_eqs_165x165():
+    eqs = eqs_165x165()
+    assert [ R_165.from_expr(eq.as_expr()) for eq in eqs ] == eqs
+
+# Benchmark R_49: shows how fast are arithmetics in rational function fields.
+F_abc, a, b, c = field("a,b,c", ZZ)
+R_49, k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, k15, k16, k17, k18, k19, k20, k21, k22, k23, k24, k25, k26, k27, k28, k29, k30, k31, k32, k33, k34, k35, k36, k37, k38, k39, k40, k41, k42, k43, k44, k45, k46, k47, k48, k49 = ring("k1:50", F_abc)
+
+def eqs_189x49():
+    return [
+        -b*k8/a+c*k8/a,
+        -b*k11/a+c*k11/a,
+        -b*k10/a+c*k10/a+k2,
+        -k3-b*k9/a+c*k9/a,
+        -b*k14/a+c*k14/a,
+        -b*k15/a+c*k15/a,
+        -b*k18/a+c*k18/a-k2,
+        -b*k17/a+c*k17/a,
+        -b*k16/a+c*k16/a+k4,
+        -b*k13/a+c*k13/a-b*k21/a+c*k21/a+b*k5/a-c*k5/a,
+        b*k44/a-c*k44/a,
+        -b*k45/a+c*k45/a,
+        -b*k20/a+c*k20/a,
+        -b*k44/a+c*k44/a,
+        b*k46/a-c*k46/a,
+        b**2*k47/a**2-2*b*c*k47/a**2+c**2*k47/a**2,
+        k3,
+        -k4,
+        -b*k12/a+c*k12/a-a*k6/b+c*k6/b,
+        -b*k19/a+c*k19/a+a*k7/c-b*k7/c,
+        b*k45/a-c*k45/a,
+        -b*k46/a+c*k46/a,
+        -k48+c*k48/a+c*k48/b-c**2*k48/(a*b),
+        -k49+b*k49/a+b*k49/c-b**2*k49/(a*c),
+        a*k1/b-c*k1/b,
+        a*k4/b-c*k4/b,
+        a*k3/b-c*k3/b+k9,
+        -k10+a*k2/b-c*k2/b,
+        a*k7/b-c*k7/b,
+        -k9,
+        k11,
+        b*k12/a-c*k12/a+a*k6/b-c*k6/b,
+        a*k15/b-c*k15/b,
+        k10+a*k18/b-c*k18/b,
+        -k11+a*k17/b-c*k17/b,
+        a*k16/b-c*k16/b,
+        -a*k13/b+c*k13/b+a*k21/b-c*k21/b+a*k5/b-c*k5/b,
+        -a*k44/b+c*k44/b,
+        a*k45/b-c*k45/b,
+        a*k14/c-b*k14/c+a*k20/b-c*k20/b,
+        a*k44/b-c*k44/b,
+        -a*k46/b+c*k46/b,
+        -k47+c*k47/a+c*k47/b-c**2*k47/(a*b),
+        a*k19/b-c*k19/b,
+        -a*k45/b+c*k45/b,
+        a*k46/b-c*k46/b,
+        a**2*k48/b**2-2*a*c*k48/b**2+c**2*k48/b**2,
+        -k49+a*k49/b+a*k49/c-a**2*k49/(b*c),
+        k16,
+        -k17,
+        -a*k1/c+b*k1/c,
+        -k16-a*k4/c+b*k4/c,
+        -a*k3/c+b*k3/c,
+        k18-a*k2/c+b*k2/c,
+        b*k19/a-c*k19/a-a*k7/c+b*k7/c,
+        -a*k6/c+b*k6/c,
+        -a*k8/c+b*k8/c,
+        -a*k11/c+b*k11/c+k17,
+        -a*k10/c+b*k10/c-k18,
+        -a*k9/c+b*k9/c,
+        -a*k14/c+b*k14/c-a*k20/b+c*k20/b,
+        -a*k13/c+b*k13/c+a*k21/c-b*k21/c-a*k5/c+b*k5/c,
+        a*k44/c-b*k44/c,
+        -a*k45/c+b*k45/c,
+        -a*k44/c+b*k44/c,
+        a*k46/c-b*k46/c,
+        -k47+b*k47/a+b*k47/c-b**2*k47/(a*c),
+        -a*k12/c+b*k12/c,
+        a*k45/c-b*k45/c,
+        -a*k46/c+b*k46/c,
+        -k48+a*k48/b+a*k48/c-a**2*k48/(b*c),
+        a**2*k49/c**2-2*a*b*k49/c**2+b**2*k49/c**2,
+        k8,
+        k11,
+        -k15,
+        k10-k18,
+        -k17,
+        k9,
+        -k16,
+        -k29,
+        k14-k32,
+        -k21+k23-k31,
+        -k24-k30,
+        -k35,
+        k44,
+        -k45,
+        k36,
+        k13-k23+k39,
+        -k20+k38,
+        k25+k37,
+        b*k26/a-c*k26/a-k34+k42,
+        -2*k44,
+        k45,
+        k46,
+        b*k47/a-c*k47/a,
+        k41,
+        k44,
+        -k46,
+        -b*k47/a+c*k47/a,
+        k12+k24,
+        -k19-k25,
+        -a*k27/b+c*k27/b-k33,
+        k45,
+        -k46,
+        -a*k48/b+c*k48/b,
+        a*k28/c-b*k28/c+k40,
+        -k45,
+        k46,
+        a*k48/b-c*k48/b,
+        a*k49/c-b*k49/c,
+        -a*k49/c+b*k49/c,
+        -k1,
+        -k4,
+        -k3,
+        k15,
+        k18-k2,
+        k17,
+        k16,
+        k22,
+        k25-k7,
+        k24+k30,
+        k21+k23-k31,
+        k28,
+        -k44,
+        k45,
+        -k30-k6,
+        k20+k32,
+        k27+b*k33/a-c*k33/a,
+        k44,
+        -k46,
+        -b*k47/a+c*k47/a,
+        -k36,
+        k31-k39-k5,
+        -k32-k38,
+        k19-k37,
+        k26-a*k34/b+c*k34/b-k42,
+        k44,
+        -2*k45,
+        k46,
+        a*k48/b-c*k48/b,
+        a*k35/c-b*k35/c-k41,
+        -k44,
+        k46,
+        b*k47/a-c*k47/a,
+        -a*k49/c+b*k49/c,
+        -k40,
+        k45,
+        -k46,
+        -a*k48/b+c*k48/b,
+        a*k49/c-b*k49/c,
+        k1,
+        k4,
+        k3,
+        -k8,
+        -k11,
+        -k10+k2,
+        -k9,
+        k37+k7,
+        -k14-k38,
+        -k22,
+        -k25-k37,
+        -k24+k6,
+        -k13-k23+k39,
+        -k28+b*k40/a-c*k40/a,
+        k44,
+        -k45,
+        -k27,
+        -k44,
+        k46,
+        b*k47/a-c*k47/a,
+        k29,
+        k32+k38,
+        k31-k39+k5,
+        -k12+k30,
+        k35-a*k41/b+c*k41/b,
+        -k44,
+        k45,
+        -k26+k34+a*k42/c-b*k42/c,
+        k44,
+        k45,
+        -2*k46,
+        -b*k47/a+c*k47/a,
+        -a*k48/b+c*k48/b,
+        a*k49/c-b*k49/c,
+        k33,
+        -k45,
+        k46,
+        a*k48/b-c*k48/b,
+        -a*k49/c+b*k49/c,
+    ]
+
+def sol_189x49():
+    return {
+        k49: 0, k48: 0, k47: 0, k46: 0, k45: 0, k44: 0, k41: 0, k40: 0,
+        k38: 0, k37: 0, k36: 0, k35: 0, k33: 0, k32: 0, k30: 0, k29: 0,
+        k28: 0, k27: 0, k25: 0, k24: 0, k22: 0, k21: 0, k20: 0, k19: 0,
+        k18: 0, k17: 0, k16: 0, k15: 0, k14: 0, k13: 0, k12: 0, k11: 0,
+        k10: 0, k9:  0, k8:  0, k7:  0, k6:  0, k5:  0, k4:  0, k3:  0,
+        k2:  0, k1:  0,
+        k34: b/c*k42,
+        k31: k39,
+        k26: a/c*k42,
+        k23: k39,
+    }
+
+def time_eqs_189x49():
+    if len(eqs_189x49()) != 189:
+        raise ValueError("Length should be equal to 189")
+
+def time_solve_lin_sys_189x49():
+    eqs = eqs_189x49()
+    sol = solve_lin_sys(eqs, R_49)
+    if sol != sol_189x49():
+        raise ValueError("Values should be equal")
+
+def time_verify_sol_189x49():
+    eqs = eqs_189x49()
+    sol = sol_189x49()
+    zeros = [ eq.compose(sol) for eq in eqs ]
+    assert all(zero == 0 for zero in zeros)
+
+def time_to_expr_eqs_189x49():
+    eqs = eqs_189x49()
+    assert [ R_49.from_expr(eq.as_expr()) for eq in eqs ] == eqs
+
+# Benchmark R_8: shows how fast polynomial GCDs are computed.
+
+F_a5_5, a_11, a_12, a_13, a_14, a_21, a_22, a_23, a_24, a_31, a_32, a_33, a_34, a_41, a_42, a_43, a_44 = field("a_(1:5)(1:5)", ZZ)
+R_8, x0, x1, x2, x3, x4, x5, x6, x7 = ring("x:8", F_a5_5)
+
+def eqs_10x8():
+    return [
+        (a_33*a_34 + a_33*a_44 + a_43*a_44)*x3 + (a_33*a_34 + a_33*a_44 + a_43*a_44)*x4 + (a_12*a_34 + a_12*a_44 + a_22*a_34 + a_22*a_44)*x5 + (a_12*a_44 + a_22*a_44)*x6 + (a_12*a_33 + a_22*a_33)*x7 - a_12*a_33 - a_12*a_43 - a_22*a_33 - a_22*a_43,
+        (a_33 + a_34 + a_43 + a_44)*x3 + (a_33 + a_34 + a_43 + a_44)*x4 + (a_12 + a_22 + a_34 + a_44)*x5 + (a_12 + a_22 + a_44)*x6 + (a_12 + a_22 + a_33)*x7 - a_12 - a_22 - a_33 - a_43,
+        x3 + x4 + x5 + x6 + x7 - 1,
+        (a_12*a_33*a_34 + a_12*a_33*a_44 + a_12*a_43*a_44 + a_22*a_33*a_34 + a_22*a_33*a_44 + a_22*a_43*a_44)*x0 + (a_22*a_33*a_34 + a_22*a_33*a_44 + a_22*a_43*a_44)*x1 + (a_12*a_33*a_34 + a_12*a_33*a_44 + a_12*a_43*a_44 + a_22*a_33*a_34 + a_22*a_33*a_44 + a_22*a_43*a_44)*x2 + (a_11*a_33*a_34 + a_11*a_33*a_44 + a_11*a_43*a_44 + a_31*a_33*a_34 + a_31*a_33*a_44 + a_31*a_43*a_44)*x3 + (a_11*a_33*a_34 + a_11*a_33*a_44 + a_11*a_43*a_44 + a_21*a_33*a_34 + a_21*a_33*a_44 + a_21*a_43*a_44 + a_31*a_33*a_34 + a_31*a_33*a_44 + a_31*a_43*a_44)*x4 + (a_11*a_12*a_34 + a_11*a_12*a_44 + a_11*a_22*a_34 + a_11*a_22*a_44 + a_12*a_31*a_34 + a_12*a_31*a_44 + a_21*a_22*a_34 + a_21*a_22*a_44 + a_22*a_31*a_34 + a_22*a_31*a_44)*x5 + (a_11*a_12*a_44 + a_11*a_22*a_44 + a_12*a_31*a_44 + a_21*a_22*a_44 + a_22*a_31*a_44)*x6 + (a_11*a_12*a_33 + a_11*a_22*a_33 + a_12*a_31*a_33 + a_21*a_22*a_33 + a_22*a_31*a_33)*x7 - a_11*a_12*a_33 - a_11*a_12*a_43 - a_11*a_22*a_33 - a_11*a_22*a_43 - a_12*a_31*a_33 - a_12*a_31*a_43 - a_21*a_22*a_33 - a_21*a_22*a_43 - a_22*a_31*a_33 - a_22*a_31*a_43,
+        (a_12*a_33 + a_12*a_34 + a_12*a_43 + a_12*a_44 + a_22*a_33 + a_22*a_34 + a_22*a_43 + a_22*a_44 + a_33*a_34 + a_33*a_44 + a_43*a_44)*x0 + (a_22*a_33 + a_22*a_34 + a_22*a_43 + a_22*a_44 + a_33*a_34 + a_33*a_44 + a_43*a_44)*x1 + (a_12*a_33 + a_12*a_34 + a_12*a_43 + a_12*a_44 + a_22*a_33 + a_22*a_34 + a_22*a_43 + a_22*a_44 + a_33*a_34 + a_33*a_44 + a_43*a_44)*x2 + (a_11*a_33 + a_11*a_34 + a_11*a_43 + a_11*a_44 + a_31*a_33 + a_31*a_34 + a_31*a_43 + a_31*a_44 + a_33*a_34 + a_33*a_44 + a_43*a_44)*x3 + (a_11*a_33 + a_11*a_34 + a_11*a_43 + a_11*a_44 + a_21*a_33 + a_21*a_34 + a_21*a_43 + a_21*a_44 + a_31*a_33 + a_31*a_34 + a_31*a_43 + a_31*a_44 + a_33*a_34 + a_33*a_44 + a_43*a_44)*x4 + (a_11*a_12 + a_11*a_22 + a_11*a_34 + a_11*a_44 + a_12*a_31 + a_12*a_34 + a_12*a_44 + a_21*a_22 + a_21*a_34 + a_21*a_44 + a_22*a_31 + a_22*a_34 + a_22*a_44 + a_31*a_34 + a_31*a_44)*x5 + (a_11*a_12 + a_11*a_22 + a_11*a_44 + a_12*a_31 + a_12*a_44 + a_21*a_22 + a_21*a_44 + a_22*a_31 + a_22*a_44 + a_31*a_44)*x6 + (a_11*a_12 + a_11*a_22 + a_11*a_33 + a_12*a_31 + a_12*a_33 + a_21*a_22 + a_21*a_33 + a_22*a_31 + a_22*a_33 + a_31*a_33)*x7 - a_11*a_12 - a_11*a_22 - a_11*a_33 - a_11*a_43 - a_12*a_31 - a_12*a_33 - a_12*a_43 - a_21*a_22 - a_21*a_33 - a_21*a_43 - a_22*a_31 - a_22*a_33 - a_22*a_43 - a_31*a_33 - a_31*a_43,
+        (a_12 + a_22 + a_33 + a_34 + a_43 + a_44)*x0 + (a_22 + a_33 + a_34 + a_43 + a_44)*x1 + (a_12 + a_22 + a_33 + a_34 + a_43 + a_44)*x2 + (a_11 + a_31 + a_33 + a_34 + a_43 + a_44)*x3 + (a_11 + a_21 + a_31 + a_33 + a_34 + a_43 + a_44)*x4 + (a_11 + a_12 + a_21 + a_22 + a_31 + a_34 + a_44)*x5 + (a_11 + a_12 + a_21 + a_22 + a_31 + a_44)*x6 + (a_11 + a_12 + a_21 + a_22 + a_31 + a_33)*x7 - a_11 - a_12 - a_21 - a_22 - a_31 - a_33 - a_43,
+        x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 - 1,
+        (a_12*a_34 + a_12*a_44 + a_22*a_34 + a_22*a_44)*x2 + (a_31*a_34 + a_31*a_44)*x3 + (a_31*a_34 + a_31*a_44)*x4 + (a_12*a_31 + a_22*a_31)*x7 - a_12*a_31 - a_22*a_31,
+        (a_12 + a_22 + a_34 + a_44)*x2 + a_31*x3 + a_31*x4 + a_31*x7 - a_31,
+        x2,
+    ]
+
+def sol_10x8():
+    return {
+        x0: -a_21/a_12*x4,
+        x1: a_21/a_12*x4,
+        x2: 0,
+        x3: -x4,
+        x5: a_43/a_34,
+        x6: -a_43/a_34,
+        x7: 1,
+    }
+
+def time_eqs_10x8():
+    if len(eqs_10x8()) != 10:
+        raise ValueError("Value should be equal to 10")
+
+def time_solve_lin_sys_10x8():
+    eqs = eqs_10x8()
+    sol = solve_lin_sys(eqs, R_8)
+    if sol != sol_10x8():
+        raise ValueError("Values should be equal")
+
+def time_verify_sol_10x8():
+    eqs = eqs_10x8()
+    sol = sol_10x8()
+    zeros = [ eq.compose(sol) for eq in eqs ]
+    if not all(zero == 0 for zero in zeros):
+        raise ValueError("All values in zero should be 0")
+
+def time_to_expr_eqs_10x8():
+    eqs = eqs_10x8()
+    assert [ R_8.from_expr(eq.as_expr()) for eq in eqs ] == eqs

rl/Lib/site-packages/sympy/polys/constructor.py

@@ -0,0 +1,387 @@
+"""Tools for constructing domains for expressions. """
+from math import prod
+
+from sympy.core import sympify
+from sympy.core.evalf import pure_complex
+from sympy.core.sorting import ordered
+from sympy.polys.domains import ZZ, QQ, ZZ_I, QQ_I, EX
+from sympy.polys.domains.complexfield import ComplexField
+from sympy.polys.domains.realfield import RealField
+from sympy.polys.polyoptions import build_options
+from sympy.polys.polyutils import parallel_dict_from_basic
+from sympy.utilities import public
+
+
+def _construct_simple(coeffs, opt):
+    """Handle simple domains, e.g.: ZZ, QQ, RR and algebraic domains. """
+    rationals = floats = complexes = algebraics = False
+    float_numbers = []
+
+    if opt.extension is True:
+        is_algebraic = lambda coeff: coeff.is_number and coeff.is_algebraic
+    else:
+        is_algebraic = lambda coeff: False
+
+    for coeff in coeffs:
+        if coeff.is_Rational:
+            if not coeff.is_Integer:
+                rationals = True
+        elif coeff.is_Float:
+            if algebraics:
+                # there are both reals and algebraics -> EX
+                return False
+            else:
+                floats = True
+                float_numbers.append(coeff)
+        else:
+            is_complex = pure_complex(coeff)
+            if is_complex:
+                complexes = True
+                x, y = is_complex
+                if x.is_Rational and y.is_Rational:
+                    if not (x.is_Integer and y.is_Integer):
+                        rationals = True
+                    continue
+                else:
+                    floats = True
+                    if x.is_Float:
+                        float_numbers.append(x)
+                    if y.is_Float:
+                        float_numbers.append(y)
+            elif is_algebraic(coeff):
+                if floats:
+                    # there are both algebraics and reals -> EX
+                    return False
+                algebraics = True
+            else:
+                # this is a composite domain, e.g. ZZ[X], EX
+                return None
+
+    # Use the maximum precision of all coefficients for the RR or CC
+    # precision
+    max_prec = max(c._prec for c in float_numbers) if float_numbers else 53
+
+    if algebraics:
+        domain, result = _construct_algebraic(coeffs, opt)
+    else:
+        if floats and complexes:
+            domain = ComplexField(prec=max_prec)
+        elif floats:
+            domain = RealField(prec=max_prec)
+        elif rationals or opt.field:
+            domain = QQ_I if complexes else QQ
+        else:
+            domain = ZZ_I if complexes else ZZ
+
+        result = [domain.from_sympy(coeff) for coeff in coeffs]
+
+    return domain, result
+
+
+def _construct_algebraic(coeffs, opt):
+    """We know that coefficients are algebraic so construct the extension. """
+    from sympy.polys.numberfields import primitive_element
+
+    exts = set()
+
+    def build_trees(args):
+        trees = []
+        for a in args:
+            if a.is_Rational:
+                tree = ('Q', QQ.from_sympy(a))
+            elif a.is_Add:
+                tree = ('+', build_trees(a.args))
+            elif a.is_Mul:
+                tree = ('*', build_trees(a.args))
+            else:
+                tree = ('e', a)
+                exts.add(a)
+            trees.append(tree)
+        return trees
+
+    trees = build_trees(coeffs)
+    exts = list(ordered(exts))
+
+    g, span, H = primitive_element(exts, ex=True, polys=True)
+    root = sum(s*ext for s, ext in zip(span, exts))
+
+    domain, g = QQ.algebraic_field((g, root)), g.rep.to_list()
+
+    exts_dom = [domain.dtype.from_list(h, g, QQ) for h in H]
+    exts_map = dict(zip(exts, exts_dom))
+
+    def convert_tree(tree):
+        op, args = tree
+        if op == 'Q':
+            return domain.dtype.from_list([args], g, QQ)
+        elif op == '+':
+            return sum((convert_tree(a) for a in args), domain.zero)
+        elif op == '*':
+            return prod(convert_tree(a) for a in args)
+        elif op == 'e':
+            return exts_map[args]
+        else:
+            raise RuntimeError
+
+    result = [convert_tree(tree) for tree in trees]
+
+    return domain, result
+
+
+def _construct_composite(coeffs, opt):
+    """Handle composite domains, e.g.: ZZ[X], QQ[X], ZZ(X), QQ(X). """
+    numers, denoms = [], []
+
+    for coeff in coeffs:
+        numer, denom = coeff.as_numer_denom()
+
+        numers.append(numer)
+        denoms.append(denom)
+
+    polys, gens = parallel_dict_from_basic(numers + denoms)  # XXX: sorting
+    if not gens:
+        return None
+
+    if opt.composite is None:
+        if any(gen.is_number and gen.is_algebraic for gen in gens):
+            return None # generators are number-like so lets better use EX
+
+        all_symbols = set()
+
+        for gen in gens:
+            symbols = gen.free_symbols
+
+            if all_symbols & symbols:
+                return None # there could be algebraic relations between generators
+            else:
+                all_symbols |= symbols
+
+    n = len(gens)
+    k = len(polys)//2
+
+    numers = polys[:k]
+    denoms = polys[k:]
+
+    if opt.field:
+        fractions = True
+    else:
+        fractions, zeros = False, (0,)*n
+
+        for denom in denoms:
+            if len(denom) > 1 or zeros not in denom:
+                fractions = True
+                break
+
+    coeffs = set()
+
+    if not fractions:
+        for numer, denom in zip(numers, denoms):
+            denom = denom[zeros]
+
+            for monom, coeff in numer.items():
+                coeff /= denom
+                coeffs.add(coeff)
+                numer[monom] = coeff
+    else:
+        for numer, denom in zip(numers, denoms):
+            coeffs.update(list(numer.values()))
+            coeffs.update(list(denom.values()))
+
+    rationals = floats = complexes = False
+    float_numbers = []
+
+    for coeff in coeffs:
+        if coeff.is_Rational:
+            if not coeff.is_Integer:
+                rationals = True
+        elif coeff.is_Float:
+            floats = True
+            float_numbers.append(coeff)
+        else:
+            is_complex = pure_complex(coeff)
+            if is_complex is not None:
+                complexes = True
+                x, y = is_complex
+                if x.is_Rational and y.is_Rational:
+                    if not (x.is_Integer and y.is_Integer):
+                        rationals = True
+                else:
+                    floats = True
+                    if x.is_Float:
+                        float_numbers.append(x)
+                    if y.is_Float:
+                        float_numbers.append(y)
+
+    max_prec = max(c._prec for c in float_numbers) if float_numbers else 53
+
+    if floats and complexes:
+        ground = ComplexField(prec=max_prec)
+    elif floats:
+        ground = RealField(prec=max_prec)
+    elif complexes:
+        if rationals:
+            ground = QQ_I
+        else:
+            ground = ZZ_I
+    elif rationals:
+        ground = QQ
+    else:
+        ground = ZZ
+
+    result = []
+
+    if not fractions:
+        domain = ground.poly_ring(*gens)
+
+        for numer in numers:
+            for monom, coeff in numer.items():
+                numer[monom] = ground.from_sympy(coeff)
+
+            result.append(domain(numer))
+    else:
+        domain = ground.frac_field(*gens)
+
+        for numer, denom in zip(numers, denoms):
+            for monom, coeff in numer.items():
+                numer[monom] = ground.from_sympy(coeff)
+
+            for monom, coeff in denom.items():
+                denom[monom] = ground.from_sympy(coeff)
+
+            result.append(domain((numer, denom)))
+
+    return domain, result
+
+
+def _construct_expression(coeffs, opt):
+    """The last resort case, i.e. use the expression domain. """
+    domain, result = EX, []
+
+    for coeff in coeffs:
+        result.append(domain.from_sympy(coeff))
+
+    return domain, result
+
+
+@public
+def construct_domain(obj, **args):
+    """Construct a minimal domain for a list of expressions.
+
+    Explanation
+    ===========
+
+    Given a list of normal SymPy expressions (of type :py:class:`~.Expr`)
+    ``construct_domain`` will find a minimal :py:class:`~.Domain` that can
+    represent those expressions. The expressions will be converted to elements
+    of the domain and both the domain and the domain elements are returned.
+
+    Parameters
+    ==========
+
+    obj: list or dict
+        The expressions to build a domain for.
+
+    **args: keyword arguments
+        Options that affect the choice of domain.
+
+    Returns
+    =======
+
+    (K, elements): Domain and list of domain elements
+        The domain K that can represent the expressions and the list or dict
+        of domain elements representing the same expressions as elements of K.
+
+    Examples
+    ========
+
+    Given a list of :py:class:`~.Integer` ``construct_domain`` will return the
+    domain :ref:`ZZ` and a list of integers as elements of :ref:`ZZ`.
+
+    >>> from sympy import construct_domain, S
+    >>> expressions = [S(2), S(3), S(4)]
+    >>> K, elements = construct_domain(expressions)
+    >>> K
+    ZZ
+    >>> elements
+    [2, 3, 4]
+    >>> type(elements[0])  # doctest: +SKIP
+    <class 'int'>
+    >>> type(expressions[0])
+    <class 'sympy.core.numbers.Integer'>
+
+    If there are any :py:class:`~.Rational` then :ref:`QQ` is returned
+    instead.
+
+    >>> construct_domain([S(1)/2, S(3)/4])
+    (QQ, [1/2, 3/4])
+
+    If there are symbols then a polynomial ring :ref:`K[x]` is returned.
+
+    >>> from sympy import symbols
+    >>> x, y = symbols('x, y')
+    >>> construct_domain([2*x + 1, S(3)/4])
+    (QQ[x], [2*x + 1, 3/4])
+    >>> construct_domain([2*x + 1, y])
+    (ZZ[x,y], [2*x + 1, y])
+
+    If any symbols appear with negative powers then a rational function field
+    :ref:`K(x)` will be returned.
+
+    >>> construct_domain([y/x, x/(1 - y)])
+    (ZZ(x,y), [y/x, -x/(y - 1)])
+
+    Irrational algebraic numbers will result in the :ref:`EX` domain by
+    default. The keyword argument ``extension=True`` leads to the construction
+    of an algebraic number field :ref:`QQ(a)`.
+
+    >>> from sympy import sqrt
+    >>> construct_domain([sqrt(2)])
+    (EX, [EX(sqrt(2))])
+    >>> construct_domain([sqrt(2)], extension=True)  # doctest: +SKIP
+    (QQ<sqrt(2)>, [ANP([1, 0], [1, 0, -2], QQ)])
+
+    See also
+    ========
+
+    Domain
+    Expr
+    """
+    opt = build_options(args)
+
+    if hasattr(obj, '__iter__'):
+        if isinstance(obj, dict):
+            if not obj:
+                monoms, coeffs = [], []
+            else:
+                monoms, coeffs = list(zip(*list(obj.items())))
+        else:
+            coeffs = obj
+    else:
+        coeffs = [obj]
+
+    coeffs = list(map(sympify, coeffs))
+    result = _construct_simple(coeffs, opt)
+
+    if result is not None:
+        if result is not False:
+            domain, coeffs = result
+        else:
+            domain, coeffs = _construct_expression(coeffs, opt)
+    else:
+        if opt.composite is False:
+            result = None
+        else:
+            result = _construct_composite(coeffs, opt)
+
+        if result is not None:
+            domain, coeffs = result
+        else:
+            domain, coeffs = _construct_expression(coeffs, opt)
+
+    if hasattr(obj, '__iter__'):
+        if isinstance(obj, dict):
+            return domain, dict(list(zip(monoms, coeffs)))
+        else:
+            return domain, coeffs
+    else:
+        return domain, coeffs[0]

rl/Lib/site-packages/sympy/polys/dispersion.py

@@ -0,0 +1,212 @@
+from sympy.core import S
+from sympy.polys import Poly
+
+
+def dispersionset(p, q=None, *gens, **args):
+    r"""Compute the *dispersion set* of two polynomials.
+
+    For two polynomials `f(x)` and `g(x)` with `\deg f > 0`
+    and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as:
+
+    .. math::
+        \operatorname{J}(f, g)
+        & := \{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\} \\
+        &  = \{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\}
+
+    For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`.
+
+    Examples
+    ========
+
+    >>> from sympy import poly
+    >>> from sympy.polys.dispersion import dispersion, dispersionset
+    >>> from sympy.abc import x
+
+    Dispersion set and dispersion of a simple polynomial:
+
+    >>> fp = poly((x - 3)*(x + 3), x)
+    >>> sorted(dispersionset(fp))
+    [0, 6]
+    >>> dispersion(fp)
+    6
+
+    Note that the definition of the dispersion is not symmetric:
+
+    >>> fp = poly(x**4 - 3*x**2 + 1, x)
+    >>> gp = fp.shift(-3)
+    >>> sorted(dispersionset(fp, gp))
+    [2, 3, 4]
+    >>> dispersion(fp, gp)
+    4
+    >>> sorted(dispersionset(gp, fp))
+    []
+    >>> dispersion(gp, fp)
+    -oo
+
+    Computing the dispersion also works over field extensions:
+
+    >>> from sympy import sqrt
+    >>> 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)>')
+    >>> sorted(dispersionset(fp, gp))
+    [2]
+    >>> sorted(dispersionset(gp, fp))
+    [1, 4]
+
+    We can even perform the computations for polynomials
+    having symbolic coefficients:
+
+    >>> from sympy.abc import a
+    >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
+    >>> sorted(dispersionset(fp))
+    [0, 1]
+
+    See Also
+    ========
+
+    dispersion
+
+    References
+    ==========
+
+    .. [1] [ManWright94]_
+    .. [2] [Koepf98]_
+    .. [3] [Abramov71]_
+    .. [4] [Man93]_
+    """
+    # Check for valid input
+    same = False if q is not None else True
+    if same:
+        q = p
+
+    p = Poly(p, *gens, **args)
+    q = Poly(q, *gens, **args)
+
+    if not p.is_univariate or not q.is_univariate:
+        raise ValueError("Polynomials need to be univariate")
+
+    # The generator
+    if not p.gen == q.gen:
+        raise ValueError("Polynomials must have the same generator")
+    gen = p.gen
+
+    # We define the dispersion of constant polynomials to be zero
+    if p.degree() < 1 or q.degree() < 1:
+        return {0}
+
+    # Factor p and q over the rationals
+    fp = p.factor_list()
+    fq = q.factor_list() if not same else fp
+
+    # Iterate over all pairs of factors
+    J = set()
+    for s, unused in fp[1]:
+        for t, unused in fq[1]:
+            m = s.degree()
+            n = t.degree()
+            if n != m:
+                continue
+            an = s.LC()
+            bn = t.LC()
+            if not (an - bn).is_zero:
+                continue
+            # Note that the roles of `s` and `t` below are switched
+            # w.r.t. the original paper. This is for consistency
+            # with the description in the book of W. Koepf.
+            anm1 = s.coeff_monomial(gen**(m-1))
+            bnm1 = t.coeff_monomial(gen**(n-1))
+            alpha = (anm1 - bnm1) / S(n*bn)
+            if not alpha.is_integer:
+                continue
+            if alpha < 0 or alpha in J:
+                continue
+            if n > 1 and not (s - t.shift(alpha)).is_zero:
+                continue
+            J.add(alpha)
+
+    return J
+
+
+def dispersion(p, q=None, *gens, **args):
+    r"""Compute the *dispersion* of polynomials.
+
+    For two polynomials `f(x)` and `g(x)` with `\deg f > 0`
+    and `\deg g > 0` the dispersion `\operatorname{dis}(f, g)` is defined as:
+
+    .. math::
+        \operatorname{dis}(f, g)
+        & := \max\{ J(f,g) \cup \{0\} \} \\
+        &  = \max\{ \{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\} \cup \{0\} \}
+
+    and for a single polynomial `\operatorname{dis}(f) := \operatorname{dis}(f, f)`.
+    Note that we make the definition `\max\{\} := -\infty`.
+
+    Examples
+    ========
+
+    >>> from sympy import poly
+    >>> from sympy.polys.dispersion import dispersion, dispersionset
+    >>> from sympy.abc import x
+
+    Dispersion set and dispersion of a simple polynomial:
+
+    >>> fp = poly((x - 3)*(x + 3), x)
+    >>> sorted(dispersionset(fp))
+    [0, 6]
+    >>> dispersion(fp)
+    6
+
+    Note that the definition of the dispersion is not symmetric:
+
+    >>> fp = poly(x**4 - 3*x**2 + 1, x)
+    >>> gp = fp.shift(-3)
+    >>> sorted(dispersionset(fp, gp))
+    [2, 3, 4]
+    >>> dispersion(fp, gp)
+    4
+    >>> sorted(dispersionset(gp, fp))
+    []
+    >>> dispersion(gp, fp)
+    -oo
+
+    The maximum of an empty set is defined to be `-\infty`
+    as seen in this example.
+
+    Computing the dispersion also works over field extensions:
+
+    >>> from sympy import sqrt
+    >>> 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)>')
+    >>> sorted(dispersionset(fp, gp))
+    [2]
+    >>> sorted(dispersionset(gp, fp))
+    [1, 4]
+
+    We can even perform the computations for polynomials
+    having symbolic coefficients:
+
+    >>> from sympy.abc import a
+    >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
+    >>> sorted(dispersionset(fp))
+    [0, 1]
+
+    See Also
+    ========
+
+    dispersionset
+
+    References
+    ==========
+
+    .. [1] [ManWright94]_
+    .. [2] [Koepf98]_
+    .. [3] [Abramov71]_
+    .. [4] [Man93]_
+    """
+    J = dispersionset(p, q, *gens, **args)
+    if not J:
+        # Definition for maximum of empty set
+        j = S.NegativeInfinity
+    else:
+        j = max(J)
+    return j

rl/Lib/site-packages/sympy/polys/distributedmodules.py

@@ -0,0 +1,739 @@
+r"""
+Sparse distributed elements of free modules over multivariate (generalized)
+polynomial rings.
+
+This code and its data structures are very much like the distributed
+polynomials, except that the first "exponent" of the monomial is
+a module generator index. That is, the multi-exponent ``(i, e_1, ..., e_n)``
+represents the "monomial" `x_1^{e_1} \cdots x_n^{e_n} f_i` of the free module
+`F` generated by `f_1, \ldots, f_r` over (a localization of) the ring
+`K[x_1, \ldots, x_n]`. A module element is simply stored as a list of terms
+ordered by the monomial order. Here a term is a pair of a multi-exponent and a
+coefficient. In general, this coefficient should never be zero (since it can
+then be omitted). The zero module element is stored as an empty list.
+
+The main routines are ``sdm_nf_mora`` and ``sdm_groebner`` which can be used
+to compute, respectively, weak normal forms and standard bases. They work with
+arbitrary (not necessarily global) monomial orders.
+
+In general, product orders have to be used to construct valid monomial orders
+for modules. However, ``lex`` can be used as-is.
+
+Note that the "level" (number of variables, i.e. parameter u+1 in
+distributedpolys.py) is never needed in this code.
+
+The main reference for this file is [SCA],
+"A Singular Introduction to Commutative Algebra".
+"""
+
+
+from itertools import permutations
+
+from sympy.polys.monomials import (
+    monomial_mul, monomial_lcm, monomial_div, monomial_deg
+)
+
+from sympy.polys.polytools import Poly
+from sympy.polys.polyutils import parallel_dict_from_expr
+from sympy.core.singleton import S
+from sympy.core.sympify import sympify
+
+# Additional monomial tools.
+
+
+def sdm_monomial_mul(M, X):
+    """
+    Multiply tuple ``X`` representing a monomial of `K[X]` into the tuple
+    ``M`` representing a monomial of `F`.
+
+    Examples
+    ========
+
+    Multiplying `xy^3` into `x f_1` yields `x^2 y^3 f_1`:
+
+    >>> from sympy.polys.distributedmodules import sdm_monomial_mul
+    >>> sdm_monomial_mul((1, 1, 0), (1, 3))
+    (1, 2, 3)
+    """
+    return (M[0],) + monomial_mul(X, M[1:])
+
+
+def sdm_monomial_deg(M):
+    """
+    Return the total degree of ``M``.
+
+    Examples
+    ========
+
+    For example, the total degree of `x^2 y f_5` is 3:
+
+    >>> from sympy.polys.distributedmodules import sdm_monomial_deg
+    >>> sdm_monomial_deg((5, 2, 1))
+    3
+    """
+    return monomial_deg(M[1:])
+
+
+def sdm_monomial_lcm(A, B):
+    r"""
+    Return the "least common multiple" of ``A`` and ``B``.
+
+    IF `A = M e_j` and `B = N e_j`, where `M` and `N` are polynomial monomials,
+    this returns `\lcm(M, N) e_j`. Note that ``A`` and ``B`` involve distinct
+    monomials.
+
+    Otherwise the result is undefined.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_monomial_lcm
+    >>> sdm_monomial_lcm((1, 2, 3), (1, 0, 5))
+    (1, 2, 5)
+    """
+    return (A[0],) + monomial_lcm(A[1:], B[1:])
+
+
+def sdm_monomial_divides(A, B):
+    """
+    Does there exist a (polynomial) monomial X such that XA = B?
+
+    Examples
+    ========
+
+    Positive examples:
+
+    In the following examples, the monomial is given in terms of x, y and the
+    generator(s), f_1, f_2 etc. The tuple form of that monomial is used in
+    the call to sdm_monomial_divides.
+    Note: the generator appears last in the expression but first in the tuple
+    and other factors appear in the same order that they appear in the monomial
+    expression.
+
+    `A = f_1` divides `B = f_1`
+
+    >>> from sympy.polys.distributedmodules import sdm_monomial_divides
+    >>> sdm_monomial_divides((1, 0, 0), (1, 0, 0))
+    True
+
+    `A = f_1` divides `B = x^2 y f_1`
+
+    >>> sdm_monomial_divides((1, 0, 0), (1, 2, 1))
+    True
+
+    `A = xy f_5` divides `B = x^2 y f_5`
+
+    >>> sdm_monomial_divides((5, 1, 1), (5, 2, 1))
+    True
+
+    Negative examples:
+
+    `A = f_1` does not divide `B = f_2`
+
+    >>> sdm_monomial_divides((1, 0, 0), (2, 0, 0))
+    False
+
+    `A = x f_1` does not divide `B = f_1`
+
+    >>> sdm_monomial_divides((1, 1, 0), (1, 0, 0))
+    False
+
+    `A = xy^2 f_5` does not divide `B = y f_5`
+
+    >>> sdm_monomial_divides((5, 1, 2), (5, 0, 1))
+    False
+    """
+    return A[0] == B[0] and all(a <= b for a, b in zip(A[1:], B[1:]))
+
+
+# The actual distributed modules code.
+
+def sdm_LC(f, K):
+    """Returns the leading coefficient of ``f``. """
+    if not f:
+        return K.zero
+    else:
+        return f[0][1]
+
+
+def sdm_to_dict(f):
+    """Make a dictionary from a distributed polynomial. """
+    return dict(f)
+
+
+def sdm_from_dict(d, O):
+    """
+    Create an sdm from a dictionary.
+
+    Here ``O`` is the monomial order to use.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_from_dict
+    >>> from sympy.polys import QQ, lex
+    >>> dic = {(1, 1, 0): QQ(1), (1, 0, 0): QQ(2), (0, 1, 0): QQ(0)}
+    >>> sdm_from_dict(dic, lex)
+    [((1, 1, 0), 1), ((1, 0, 0), 2)]
+    """
+    return sdm_strip(sdm_sort(list(d.items()), O))
+
+
+def sdm_sort(f, O):
+    """Sort terms in ``f`` using the given monomial order ``O``. """
+    return sorted(f, key=lambda term: O(term[0]), reverse=True)
+
+
+def sdm_strip(f):
+    """Remove terms with zero coefficients from ``f`` in ``K[X]``. """
+    return [ (monom, coeff) for monom, coeff in f if coeff ]
+
+
+def sdm_add(f, g, O, K):
+    """
+    Add two module elements ``f``, ``g``.
+
+    Addition is done over the ground field ``K``, monomials are ordered
+    according to ``O``.
+
+    Examples
+    ========
+
+    All examples use lexicographic order.
+
+    `(xy f_1) + (f_2) = f_2 + xy f_1`
+
+    >>> from sympy.polys.distributedmodules import sdm_add
+    >>> from sympy.polys import lex, QQ
+    >>> sdm_add([((1, 1, 1), QQ(1))], [((2, 0, 0), QQ(1))], lex, QQ)
+    [((2, 0, 0), 1), ((1, 1, 1), 1)]
+
+    `(xy f_1) + (-xy f_1)` = 0`
+
+    >>> sdm_add([((1, 1, 1), QQ(1))], [((1, 1, 1), QQ(-1))], lex, QQ)
+    []
+
+    `(f_1) + (2f_1) = 3f_1`
+
+    >>> sdm_add([((1, 0, 0), QQ(1))], [((1, 0, 0), QQ(2))], lex, QQ)
+    [((1, 0, 0), 3)]
+
+    `(yf_1) + (xf_1) = xf_1 + yf_1`
+
+    >>> sdm_add([((1, 0, 1), QQ(1))], [((1, 1, 0), QQ(1))], lex, QQ)
+    [((1, 1, 0), 1), ((1, 0, 1), 1)]
+    """
+    h = dict(f)
+
+    for monom, c in g:
+        if monom in h:
+            coeff = h[monom] + c
+
+            if not coeff:
+                del h[monom]
+            else:
+                h[monom] = coeff
+        else:
+            h[monom] = c
+
+    return sdm_from_dict(h, O)
+
+
+def sdm_LM(f):
+    r"""
+    Returns the leading monomial of ``f``.
+
+    Only valid if `f \ne 0`.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_LM, sdm_from_dict
+    >>> from sympy.polys import QQ, lex
+    >>> dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(1), (4, 0, 1): QQ(1)}
+    >>> sdm_LM(sdm_from_dict(dic, lex))
+    (4, 0, 1)
+    """
+    return f[0][0]
+
+
+def sdm_LT(f):
+    r"""
+    Returns the leading term of ``f``.
+
+    Only valid if `f \ne 0`.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_LT, sdm_from_dict
+    >>> from sympy.polys import QQ, lex
+    >>> dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(2), (4, 0, 1): QQ(3)}
+    >>> sdm_LT(sdm_from_dict(dic, lex))
+    ((4, 0, 1), 3)
+    """
+    return f[0]
+
+
+def sdm_mul_term(f, term, O, K):
+    """
+    Multiply a distributed module element ``f`` by a (polynomial) term ``term``.
+
+    Multiplication of coefficients is done over the ground field ``K``, and
+    monomials are ordered according to ``O``.
+
+    Examples
+    ========
+
+    `0 f_1 = 0`
+
+    >>> from sympy.polys.distributedmodules import sdm_mul_term
+    >>> from sympy.polys import lex, QQ
+    >>> sdm_mul_term([((1, 0, 0), QQ(1))], ((0, 0), QQ(0)), lex, QQ)
+    []
+
+    `x 0 = 0`
+
+    >>> sdm_mul_term([], ((1, 0), QQ(1)), lex, QQ)
+    []
+
+    `(x) (f_1) = xf_1`
+
+    >>> sdm_mul_term([((1, 0, 0), QQ(1))], ((1, 0), QQ(1)), lex, QQ)
+    [((1, 1, 0), 1)]
+
+    `(2xy) (3x f_1 + 4y f_2) = 8xy^2 f_2 + 6x^2y f_1`
+
+    >>> f = [((2, 0, 1), QQ(4)), ((1, 1, 0), QQ(3))]
+    >>> sdm_mul_term(f, ((1, 1), QQ(2)), lex, QQ)
+    [((2, 1, 2), 8), ((1, 2, 1), 6)]
+    """
+    X, c = term
+
+    if not f or not c:
+        return []
+    else:
+        if K.is_one(c):
+            return [ (sdm_monomial_mul(f_M, X), f_c) for f_M, f_c in f ]
+        else:
+            return [ (sdm_monomial_mul(f_M, X), f_c * c) for f_M, f_c in f ]
+
+
+def sdm_zero():
+    """Return the zero module element."""
+    return []
+
+
+def sdm_deg(f):
+    """
+    Degree of ``f``.
+
+    This is the maximum of the degrees of all its monomials.
+    Invalid if ``f`` is zero.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_deg
+    >>> sdm_deg([((1, 2, 3), 1), ((10, 0, 1), 1), ((2, 3, 4), 4)])
+    7
+    """
+    return max(sdm_monomial_deg(M[0]) for M in f)
+
+
+# Conversion
+
+def sdm_from_vector(vec, O, K, **opts):
+    """
+    Create an sdm from an iterable of expressions.
+
+    Coefficients are created in the ground field ``K``, and terms are ordered
+    according to monomial order ``O``. Named arguments are passed on to the
+    polys conversion code and can be used to specify for example generators.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_from_vector
+    >>> from sympy.abc import x, y, z
+    >>> from sympy.polys import QQ, lex
+    >>> sdm_from_vector([x**2+y**2, 2*z], lex, QQ)
+    [((1, 0, 0, 1), 2), ((0, 2, 0, 0), 1), ((0, 0, 2, 0), 1)]
+    """
+    dics, gens = parallel_dict_from_expr(sympify(vec), **opts)
+    dic = {}
+    for i, d in enumerate(dics):
+        for k, v in d.items():
+            dic[(i,) + k] = K.convert(v)
+    return sdm_from_dict(dic, O)
+
+
+def sdm_to_vector(f, gens, K, n=None):
+    """
+    Convert sdm ``f`` into a list of polynomial expressions.
+
+    The generators for the polynomial ring are specified via ``gens``. The rank
+    of the module is guessed, or passed via ``n``. The ground field is assumed
+    to be ``K``.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_to_vector
+    >>> from sympy.abc import x, y, z
+    >>> from sympy.polys import QQ
+    >>> f = [((1, 0, 0, 1), QQ(2)), ((0, 2, 0, 0), QQ(1)), ((0, 0, 2, 0), QQ(1))]
+    >>> sdm_to_vector(f, [x, y, z], QQ)
+    [x**2 + y**2, 2*z]
+    """
+    dic = sdm_to_dict(f)
+    dics = {}
+    for k, v in dic.items():
+        dics.setdefault(k[0], []).append((k[1:], v))
+    n = n or len(dics)
+    res = []
+    for k in range(n):
+        if k in dics:
+            res.append(Poly(dict(dics[k]), gens=gens, domain=K).as_expr())
+        else:
+            res.append(S.Zero)
+    return res
+
+# Algorithms.
+
+
+def sdm_spoly(f, g, O, K, phantom=None):
+    """
+    Compute the generalized s-polynomial of ``f`` and ``g``.
+
+    The ground field is assumed to be ``K``, and monomials ordered according to
+    ``O``.
+
+    This is invalid if either of ``f`` or ``g`` is zero.
+
+    If the leading terms of `f` and `g` involve different basis elements of
+    `F`, their s-poly is defined to be zero. Otherwise it is a certain linear
+    combination of `f` and `g` in which the leading terms cancel.
+    See [SCA, defn 2.3.6] for details.
+
+    If ``phantom`` is not ``None``, it should be a pair of module elements on
+    which to perform the same operation(s) as on ``f`` and ``g``. The in this
+    case both results are returned.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_spoly
+    >>> from sympy.polys import QQ, lex
+    >>> f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))]
+    >>> g = [((2, 3, 0), QQ(1))]
+    >>> h = [((1, 2, 3), QQ(1))]
+    >>> sdm_spoly(f, h, lex, QQ)
+    []
+    >>> sdm_spoly(f, g, lex, QQ)
+    [((1, 2, 1), 1)]
+    """
+    if not f or not g:
+        return sdm_zero()
+    LM1 = sdm_LM(f)
+    LM2 = sdm_LM(g)
+    if LM1[0] != LM2[0]:
+        return sdm_zero()
+    LM1 = LM1[1:]
+    LM2 = LM2[1:]
+    lcm = monomial_lcm(LM1, LM2)
+    m1 = monomial_div(lcm, LM1)
+    m2 = monomial_div(lcm, LM2)
+    c = K.quo(-sdm_LC(f, K), sdm_LC(g, K))
+    r1 = sdm_add(sdm_mul_term(f, (m1, K.one), O, K),
+                 sdm_mul_term(g, (m2, c), O, K), O, K)
+    if phantom is None:
+        return r1
+    r2 = sdm_add(sdm_mul_term(phantom[0], (m1, K.one), O, K),
+                 sdm_mul_term(phantom[1], (m2, c), O, K), O, K)
+    return r1, r2
+
+
+def sdm_ecart(f):
+    """
+    Compute the ecart of ``f``.
+
+    This is defined to be the difference of the total degree of `f` and the
+    total degree of the leading monomial of `f` [SCA, defn 2.3.7].
+
+    Invalid if f is zero.
+
+    Examples
+    ========
+
+    >>> from sympy.polys.distributedmodules import sdm_ecart
+    >>> sdm_ecart([((1, 2, 3), 1), ((1, 0, 1), 1)])
+    0
+    >>> sdm_ecart([((2, 2, 1), 1), ((1, 5, 1), 1)])
+    3
+    """
+    return sdm_deg(f) - sdm_monomial_deg(sdm_LM(f))
+
+
+def sdm_nf_mora(f, G, O, K, phantom=None):
+    r"""
+    Compute a weak normal form of ``f`` with respect to ``G`` and order ``O``.
+
+    The ground field is assumed to be ``K``, and monomials ordered according to
+    ``O``.
+
+    Weak normal forms are defined in [SCA, defn 2.3.3]. They are not unique.
+    This function deterministically computes a weak normal form, depending on
+    the order of `G`.
+
+    The most important property of a weak normal form is the following: if
+    `R` is the ring associated with the monomial ordering (if the ordering is
+    global, we just have `R = K[x_1, \ldots, x_n]`, otherwise it is a certain
+    localization thereof), `I` any ideal of `R` and `G` a standard basis for
+    `I`, then for any `f \in R`, we have `f \in I` if and only if
+    `NF(f | G) = 0`.
+
+    This is the generalized Mora algorithm for computing weak normal forms with
+    respect to arbitrary monomial orders [SCA, algorithm 2.3.9].
+
+    If ``phantom`` is not ``None``, it should be a pair of "phantom" arguments
+    on which to perform the same computations as on ``f``, ``G``, both results
+    are then returned.
+    """
+    from itertools import repeat
+    h = f
+    T = list(G)
+    if phantom is not None:
+        # "phantom" variables with suffix p
+        hp = phantom[0]
+        Tp = list(phantom[1])
+        phantom = True
+    else:
+        Tp = repeat([])
+        phantom = False
+    while h:
+        # TODO better data structure!!!
+        Th = [(g, sdm_ecart(g), gp) for g, gp in zip(T, Tp)
+              if sdm_monomial_divides(sdm_LM(g), sdm_LM(h))]
+        if not Th:
+            break
+        g, _, gp = min(Th, key=lambda x: x[1])
+        if sdm_ecart(g) > sdm_ecart(h):
+            T.append(h)
+            if phantom:
+                Tp.append(hp)
+        if phantom:
+            h, hp = sdm_spoly(h, g, O, K, phantom=(hp, gp))
+        else:
+            h = sdm_spoly(h, g, O, K)
+    if phantom:
+        return h, hp
+    return h
+
+
+def sdm_nf_buchberger(f, G, O, K, phantom=None):
+    r"""
+    Compute a weak normal form of ``f`` with respect to ``G`` and order ``O``.
+
+    The ground field is assumed to be ``K``, and monomials ordered according to
+    ``O``.
+
+    This is the standard Buchberger algorithm for computing weak normal forms with
+    respect to *global* monomial orders [SCA, algorithm 1.6.10].
+
+    If ``phantom`` is not ``None``, it should be a pair of "phantom" arguments
+    on which to perform the same computations as on ``f``, ``G``, both results
+    are then returned.
+    """
+    from itertools import repeat
+    h = f
+    T = list(G)
+    if phantom is not None:
+        # "phantom" variables with suffix p
+        hp = phantom[0]
+        Tp = list(phantom[1])
+        phantom = True
+    else:
+        Tp = repeat([])
+        phantom = False
+    while h:
+        try:
+            g, gp = next((g, gp) for g, gp in zip(T, Tp)
+                         if sdm_monomial_divides(sdm_LM(g), sdm_LM(h)))
+        except StopIteration:
+            break
+        if phantom:
+            h, hp = sdm_spoly(h, g, O, K, phantom=(hp, gp))
+        else:
+            h = sdm_spoly(h, g, O, K)
+    if phantom:
+        return h, hp
+    return h
+
+
+def sdm_nf_buchberger_reduced(f, G, O, K):
+    r"""
+    Compute a reduced normal form of ``f`` with respect to ``G`` and order ``O``.
+
+    The ground field is assumed to be ``K``, and monomials ordered according to
+    ``O``.
+
+    In contrast to weak normal forms, reduced normal forms *are* unique, but
+    their computation is more expensive.
+
+    This is the standard Buchberger algorithm for computing reduced normal forms
+    with respect to *global* monomial orders [SCA, algorithm 1.6.11].
+
+    The ``pantom`` option is not supported, so this normal form cannot be used
+    as a normal form for the "extended" groebner algorithm.
+    """
+    h = sdm_zero()
+    g = f
+    while g:
+        g = sdm_nf_buchberger(g, G, O, K)
+        if g:
+            h = sdm_add(h, [sdm_LT(g)], O, K)
+            g = g[1:]
+    return h
+
+
+def sdm_groebner(G, NF, O, K, extended=False):
+    """
+    Compute a minimal standard basis of ``G`` with respect to order ``O``.
+
+    The algorithm uses a normal form ``NF``, for example ``sdm_nf_mora``.
+    The ground field is assumed to be ``K``, and monomials ordered according
+    to ``O``.
+
+    Let `N` denote the submodule generated by elements of `G`. A standard
+    basis for `N` is a subset `S` of `N`, such that `in(S) = in(N)`, where for
+    any subset `X` of `F`, `in(X)` denotes the submodule generated by the
+    initial forms of elements of `X`. [SCA, defn 2.3.2]
+
+    A standard basis is called minimal if no subset of it is a standard basis.
+
+    One may show that standard bases are always generating sets.
+
+    Minimal standard bases are not unique. This algorithm computes a
+    deterministic result, depending on the particular order of `G`.
+
+    If ``extended=True``, also compute the transition matrix from the initial
+    generators to the groebner basis. That is, return a list of coefficient
+    vectors, expressing the elements of the groebner basis in terms of the
+    elements of ``G``.
+
+    This functions implements the "sugar" strategy, see
+
+    Giovini et al: "One sugar cube, please" OR Selection strategies in
+    Buchberger algorithm.
+    """
+
+    # The critical pair set.
+    # A critical pair is stored as (i, j, s, t) where (i, j) defines the pair
+    # (by indexing S), s is the sugar of the pair, and t is the lcm of their
+    # leading monomials.
+    P = []
+
+    # The eventual standard basis.
+    S = []
+    Sugars = []
+
+    def Ssugar(i, j):
+        """Compute the sugar of the S-poly corresponding to (i, j)."""
+        LMi = sdm_LM(S[i])
+        LMj = sdm_LM(S[j])
+        return max(Sugars[i] - sdm_monomial_deg(LMi),
+                   Sugars[j] - sdm_monomial_deg(LMj)) \
+            + sdm_monomial_deg(sdm_monomial_lcm(LMi, LMj))
+
+    ourkey = lambda p: (p[2], O(p[3]), p[1])
+
+    def update(f, sugar, P):
+        """Add f with sugar ``sugar`` to S, update P."""
+        if not f:
+            return P
+        k = len(S)
+        S.append(f)
+        Sugars.append(sugar)
+
+        LMf = sdm_LM(f)
+
+        def removethis(pair):
+            i, j, s, t = pair
+            if LMf[0] != t[0]:
+                return False
+            tik = sdm_monomial_lcm(LMf, sdm_LM(S[i]))
+            tjk = sdm_monomial_lcm(LMf, sdm_LM(S[j]))
+            return tik != t and tjk != t and sdm_monomial_divides(tik, t) and \
+                sdm_monomial_divides(tjk, t)
+        # apply the chain criterion
+        P = [p for p in P if not removethis(p)]
+
+        # new-pair set
+        N = [(i, k, Ssugar(i, k), sdm_monomial_lcm(LMf, sdm_LM(S[i])))
+             for i in range(k) if LMf[0] == sdm_LM(S[i])[0]]
+        # TODO apply the product criterion?
+        N.sort(key=ourkey)
+        remove = set()
+        for i, p in enumerate(N):
+            for j in range(i + 1, len(N)):
+                if sdm_monomial_divides(p[3], N[j][3]):
+                    remove.add(j)
+
+        # TODO mergesort?
+        P.extend(reversed([p for i, p in enumerate(N) if i not in remove]))
+        P.sort(key=ourkey, reverse=True)
+        # NOTE reverse-sort, because we want to pop from the end
+        return P
+
+    # Figure out the number of generators in the ground ring.
+    try:
+        # NOTE: we look for the first non-zero vector, take its first monomial
+        #       the number of generators in the ring is one less than the length
+        #       (since the zeroth entry is for the module generators)
+        numgens = len(next(x[0] for x in G if x)[0]) - 1
+    except StopIteration:
+        # No non-zero elements in G ...
+        if extended:
+            return [], []
+        return []
+
+    # This list will store expressions of the elements of S in terms of the
+    # initial generators
+    coefficients = []
+
+    # First add all the elements of G to S
+    for i, f in enumerate(G):
+        P = update(f, sdm_deg(f), P)
+        if extended and f:
+            coefficients.append(sdm_from_dict({(i,) + (0,)*numgens: K(1)}, O))
+
+    # Now carry out the buchberger algorithm.
+    while P:
+        i, j, s, t = P.pop()
+        f, g = S[i], S[j]
+        if extended:
+            sp, coeff = sdm_spoly(f, g, O, K,
+                                  phantom=(coefficients[i], coefficients[j]))
+            h, hcoeff = NF(sp, S, O, K, phantom=(coeff, coefficients))
+            if h:
+                coefficients.append(hcoeff)
+        else:
+            h = NF(sdm_spoly(f, g, O, K), S, O, K)
+        P = update(h, Ssugar(i, j), P)
+
+    # Finally interreduce the standard basis.
+    # (TODO again, better data structures)
+    S = {(tuple(f), i) for i, f in enumerate(S)}
+    for (a, ai), (b, bi) in permutations(S, 2):
+        A = sdm_LM(a)
+        B = sdm_LM(b)
+        if sdm_monomial_divides(A, B) and (b, bi) in S and (a, ai) in S:
+            S.remove((b, bi))
+
+    L = sorted(((list(f), i) for f, i in S), key=lambda p: O(sdm_LM(p[0])),
+               reverse=True)
+    res = [x[0] for x in L]
+    if extended:
+        return res, [coefficients[i] for _, i in L]
+    return res

rl/Lib/site-packages/sympy/polys/domainmatrix.py

@@ -0,0 +1,12 @@
+"""
+Stub module to expose DomainMatrix which has now moved to
+sympy.polys.matrices package. It should now be imported as:
+
+    >>> from sympy.polys.matrices import DomainMatrix
+
+This module might be removed in future.
+"""
+
+from sympy.polys.matrices.domainmatrix import DomainMatrix
+
+__all__ = ['DomainMatrix']

rl/Lib/site-packages/sympy/polys/domains/__init__.py

@@ -0,0 +1,57 @@
+"""Implementation of mathematical domains. """
+
+__all__ = [
+    'Domain', 'FiniteField', 'IntegerRing', 'RationalField', 'RealField',
+    'ComplexField', 'AlgebraicField', 'PolynomialRing', 'FractionField',
+    'ExpressionDomain', 'PythonRational',
+
+    'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR', 'CC', 'EX', 'EXRAW',
+]
+
+from .domain import Domain
+from .finitefield import FiniteField, FF, GF
+from .integerring import IntegerRing, ZZ
+from .rationalfield import RationalField, QQ
+from .algebraicfield import AlgebraicField
+from .gaussiandomains import ZZ_I, QQ_I
+from .realfield import RealField, RR
+from .complexfield import ComplexField, CC
+from .polynomialring import PolynomialRing
+from .fractionfield import FractionField
+from .expressiondomain import ExpressionDomain, EX
+from .expressionrawdomain import EXRAW
+from .pythonrational import PythonRational
+
+
+# This is imported purely for backwards compatibility because some parts of
+# the codebase used to import this from here and it's possible that downstream
+# does as well:
+from sympy.external.gmpy import GROUND_TYPES  # noqa: F401
+
+#
+# The rest of these are obsolete and provided only for backwards
+# compatibility:
+#
+
+from .pythonfinitefield import PythonFiniteField
+from .gmpyfinitefield import GMPYFiniteField
+from .pythonintegerring import PythonIntegerRing
+from .gmpyintegerring import GMPYIntegerRing
+from .pythonrationalfield import PythonRationalField
+from .gmpyrationalfield import GMPYRationalField
+
+FF_python = PythonFiniteField
+FF_gmpy = GMPYFiniteField
+
+ZZ_python = PythonIntegerRing
+ZZ_gmpy = GMPYIntegerRing
+
+QQ_python = PythonRationalField
+QQ_gmpy = GMPYRationalField
+
+__all__.extend((
+    'PythonFiniteField', 'GMPYFiniteField', 'PythonIntegerRing',
+    'GMPYIntegerRing', 'PythonRational', 'GMPYRationalField',
+
+    'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy', 'QQ_python', 'QQ_gmpy',
+))