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1485 lines
46 KiB
1485 lines
46 KiB
"""
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Computations with modules over polynomial rings.
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This module implements various classes that encapsulate groebner basis
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computations for modules. Most of them should not be instantiated by hand.
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Instead, use the constructing routines on objects you already have.
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For example, to construct a free module over ``QQ[x, y]``, call
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``QQ[x, y].free_module(rank)`` instead of the ``FreeModule`` constructor.
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In fact ``FreeModule`` is an abstract base class that should not be
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instantiated, the ``free_module`` method instead returns the implementing class
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``FreeModulePolyRing``.
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In general, the abstract base classes implement most functionality in terms of
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a few non-implemented methods. The concrete base classes supply only these
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non-implemented methods. They may also supply new implementations of the
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convenience methods, for example if there are faster algorithms available.
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"""
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from copy import copy
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from functools import reduce
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from sympy.polys.agca.ideals import Ideal
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from sympy.polys.domains.field import Field
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from sympy.polys.orderings import ProductOrder, monomial_key
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from sympy.polys.polyerrors import CoercionFailed
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from sympy.core.basic import _aresame
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from sympy.utilities.iterables import iterable
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# TODO
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# - module saturation
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# - module quotient/intersection for quotient rings
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# - free resoltutions / syzygies
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# - finding small/minimal generating sets
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# - ...
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##########################################################################
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## Abstract base classes #################################################
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##########################################################################
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class Module:
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"""
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Abstract base class for modules.
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Do not instantiate - use ring explicit constructors instead:
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>>> from sympy import QQ
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>>> from sympy.abc import x
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>>> QQ.old_poly_ring(x).free_module(2)
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QQ[x]**2
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Attributes:
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- dtype - type of elements
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- ring - containing ring
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Non-implemented methods:
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- submodule
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- quotient_module
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- is_zero
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- is_submodule
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- multiply_ideal
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The method convert likely needs to be changed in subclasses.
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"""
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def __init__(self, ring):
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self.ring = ring
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def convert(self, elem, M=None):
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"""
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Convert ``elem`` into internal representation of this module.
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If ``M`` is not None, it should be a module containing it.
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"""
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if not isinstance(elem, self.dtype):
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raise CoercionFailed
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return elem
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def submodule(self, *gens):
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"""Generate a submodule."""
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raise NotImplementedError
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def quotient_module(self, other):
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"""Generate a quotient module."""
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raise NotImplementedError
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def __truediv__(self, e):
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if not isinstance(e, Module):
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e = self.submodule(*e)
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return self.quotient_module(e)
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def contains(self, elem):
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"""Return True if ``elem`` is an element of this module."""
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try:
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self.convert(elem)
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return True
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except CoercionFailed:
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return False
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def __contains__(self, elem):
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return self.contains(elem)
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def subset(self, other):
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"""
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Returns True if ``other`` is is a subset of ``self``.
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Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> F = QQ.old_poly_ring(x).free_module(2)
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>>> F.subset([(1, x), (x, 2)])
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True
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>>> F.subset([(1/x, x), (x, 2)])
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False
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"""
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return all(self.contains(x) for x in other)
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def __eq__(self, other):
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return self.is_submodule(other) and other.is_submodule(self)
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def __ne__(self, other):
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return not (self == other)
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def is_zero(self):
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"""Returns True if ``self`` is a zero module."""
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raise NotImplementedError
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def is_submodule(self, other):
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"""Returns True if ``other`` is a submodule of ``self``."""
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raise NotImplementedError
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def multiply_ideal(self, other):
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"""
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Multiply ``self`` by the ideal ``other``.
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"""
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raise NotImplementedError
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def __mul__(self, e):
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if not isinstance(e, Ideal):
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try:
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e = self.ring.ideal(e)
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except (CoercionFailed, NotImplementedError):
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return NotImplemented
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return self.multiply_ideal(e)
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__rmul__ = __mul__
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def identity_hom(self):
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"""Return the identity homomorphism on ``self``."""
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raise NotImplementedError
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class ModuleElement:
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"""
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Base class for module element wrappers.
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Use this class to wrap primitive data types as module elements. It stores
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a reference to the containing module, and implements all the arithmetic
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operators.
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Attributes:
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- module - containing module
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- data - internal data
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Methods that likely need change in subclasses:
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- add
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- mul
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- div
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- eq
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"""
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def __init__(self, module, data):
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self.module = module
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self.data = data
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def add(self, d1, d2):
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"""Add data ``d1`` and ``d2``."""
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return d1 + d2
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def mul(self, m, d):
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"""Multiply module data ``m`` by coefficient d."""
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return m * d
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def div(self, m, d):
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"""Divide module data ``m`` by coefficient d."""
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return m / d
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def eq(self, d1, d2):
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"""Return true if d1 and d2 represent the same element."""
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return d1 == d2
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def __add__(self, om):
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if not isinstance(om, self.__class__) or om.module != self.module:
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try:
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om = self.module.convert(om)
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except CoercionFailed:
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return NotImplemented
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return self.__class__(self.module, self.add(self.data, om.data))
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__radd__ = __add__
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def __neg__(self):
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return self.__class__(self.module, self.mul(self.data,
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self.module.ring.convert(-1)))
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def __sub__(self, om):
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if not isinstance(om, self.__class__) or om.module != self.module:
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try:
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om = self.module.convert(om)
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except CoercionFailed:
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return NotImplemented
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return self.__add__(-om)
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def __rsub__(self, om):
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return (-self).__add__(om)
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def __mul__(self, o):
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if not isinstance(o, self.module.ring.dtype):
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try:
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o = self.module.ring.convert(o)
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except CoercionFailed:
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return NotImplemented
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return self.__class__(self.module, self.mul(self.data, o))
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__rmul__ = __mul__
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def __truediv__(self, o):
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if not isinstance(o, self.module.ring.dtype):
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try:
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o = self.module.ring.convert(o)
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except CoercionFailed:
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return NotImplemented
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return self.__class__(self.module, self.div(self.data, o))
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def __eq__(self, om):
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if not isinstance(om, self.__class__) or om.module != self.module:
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try:
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om = self.module.convert(om)
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except CoercionFailed:
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return False
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return self.eq(self.data, om.data)
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def __ne__(self, om):
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return not self == om
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##########################################################################
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## Free Modules ##########################################################
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##########################################################################
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class FreeModuleElement(ModuleElement):
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"""Element of a free module. Data stored as a tuple."""
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def add(self, d1, d2):
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return tuple(x + y for x, y in zip(d1, d2))
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def mul(self, d, p):
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return tuple(x * p for x in d)
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def div(self, d, p):
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return tuple(x / p for x in d)
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def __repr__(self):
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from sympy.printing.str import sstr
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return '[' + ', '.join(sstr(x) for x in self.data) + ']'
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def __iter__(self):
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return self.data.__iter__()
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def __getitem__(self, idx):
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return self.data[idx]
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class FreeModule(Module):
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"""
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Abstract base class for free modules.
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Additional attributes:
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- rank - rank of the free module
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Non-implemented methods:
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- submodule
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"""
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dtype = FreeModuleElement
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def __init__(self, ring, rank):
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Module.__init__(self, ring)
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self.rank = rank
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def __repr__(self):
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return repr(self.ring) + "**" + repr(self.rank)
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def is_submodule(self, other):
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"""
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Returns True if ``other`` is a submodule of ``self``.
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Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> F = QQ.old_poly_ring(x).free_module(2)
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>>> M = F.submodule([2, x])
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>>> F.is_submodule(F)
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True
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>>> F.is_submodule(M)
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True
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>>> M.is_submodule(F)
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False
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"""
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if isinstance(other, SubModule):
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return other.container == self
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if isinstance(other, FreeModule):
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return other.ring == self.ring and other.rank == self.rank
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return False
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def convert(self, elem, M=None):
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"""
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Convert ``elem`` into the internal representation.
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This method is called implicitly whenever computations involve elements
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not in the internal representation.
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Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> F = QQ.old_poly_ring(x).free_module(2)
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>>> F.convert([1, 0])
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[1, 0]
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"""
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if isinstance(elem, FreeModuleElement):
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if elem.module is self:
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return elem
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if elem.module.rank != self.rank:
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raise CoercionFailed
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return FreeModuleElement(self,
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tuple(self.ring.convert(x, elem.module.ring) for x in elem.data))
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elif iterable(elem):
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tpl = tuple(self.ring.convert(x) for x in elem)
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if len(tpl) != self.rank:
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raise CoercionFailed
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return FreeModuleElement(self, tpl)
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elif _aresame(elem, 0):
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return FreeModuleElement(self, (self.ring.convert(0),)*self.rank)
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else:
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raise CoercionFailed
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def is_zero(self):
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"""
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Returns True if ``self`` is a zero module.
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(If, as this implementation assumes, the coefficient ring is not the
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zero ring, then this is equivalent to the rank being zero.)
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Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> QQ.old_poly_ring(x).free_module(0).is_zero()
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True
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>>> QQ.old_poly_ring(x).free_module(1).is_zero()
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False
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"""
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return self.rank == 0
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def basis(self):
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"""
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|
Return a set of basis elements.
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|
|
Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> QQ.old_poly_ring(x).free_module(3).basis()
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([1, 0, 0], [0, 1, 0], [0, 0, 1])
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"""
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from sympy.matrices import eye
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M = eye(self.rank)
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return tuple(self.convert(M.row(i)) for i in range(self.rank))
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def quotient_module(self, submodule):
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"""
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Return a quotient module.
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|
|
Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> M = QQ.old_poly_ring(x).free_module(2)
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>>> M.quotient_module(M.submodule([1, x], [x, 2]))
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QQ[x]**2/<[1, x], [x, 2]>
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Or more conicisely, using the overloaded division operator:
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>>> QQ.old_poly_ring(x).free_module(2) / [[1, x], [x, 2]]
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QQ[x]**2/<[1, x], [x, 2]>
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"""
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return QuotientModule(self.ring, self, submodule)
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def multiply_ideal(self, other):
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"""
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Multiply ``self`` by the ideal ``other``.
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|
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Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> I = QQ.old_poly_ring(x).ideal(x)
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>>> F = QQ.old_poly_ring(x).free_module(2)
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>>> F.multiply_ideal(I)
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<[x, 0], [0, x]>
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"""
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return self.submodule(*self.basis()).multiply_ideal(other)
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|
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def identity_hom(self):
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"""
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Return the identity homomorphism on ``self``.
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|
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Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> QQ.old_poly_ring(x).free_module(2).identity_hom()
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Matrix([
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[1, 0], : QQ[x]**2 -> QQ[x]**2
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[0, 1]])
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"""
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from sympy.polys.agca.homomorphisms import homomorphism
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return homomorphism(self, self, self.basis())
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|
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class FreeModulePolyRing(FreeModule):
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"""
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Free module over a generalized polynomial ring.
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Do not instantiate this, use the constructor method of the ring instead:
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|
|
|
Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> F = QQ.old_poly_ring(x).free_module(3)
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>>> F
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QQ[x]**3
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>>> F.contains([x, 1, 0])
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True
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>>> F.contains([1/x, 0, 1])
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False
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"""
|
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def __init__(self, ring, rank):
|
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from sympy.polys.domains.old_polynomialring import PolynomialRingBase
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FreeModule.__init__(self, ring, rank)
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if not isinstance(ring, PolynomialRingBase):
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raise NotImplementedError('This implementation only works over '
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+ 'polynomial rings, got %s' % ring)
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if not isinstance(ring.dom, Field):
|
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raise NotImplementedError('Ground domain must be a field, '
|
|
+ 'got %s' % ring.dom)
|
|
|
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def submodule(self, *gens, **opts):
|
|
"""
|
|
Generate a submodule.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x, y
|
|
>>> from sympy import QQ
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>>> M = QQ.old_poly_ring(x, y).free_module(2).submodule([x, x + y])
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>>> M
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<[x, x + y]>
|
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>>> M.contains([2*x, 2*x + 2*y])
|
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True
|
|
>>> M.contains([x, y])
|
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False
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|
"""
|
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return SubModulePolyRing(gens, self, **opts)
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|
|
|
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class FreeModuleQuotientRing(FreeModule):
|
|
"""
|
|
Free module over a quotient ring.
|
|
|
|
Do not instantiate this, use the constructor method of the ring instead:
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = (QQ.old_poly_ring(x)/[x**2 + 1]).free_module(3)
|
|
>>> F
|
|
(QQ[x]/<x**2 + 1>)**3
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|
|
|
Attributes
|
|
|
|
- quot - the quotient module `R^n / IR^n`, where `R/I` is our ring
|
|
"""
|
|
|
|
def __init__(self, ring, rank):
|
|
from sympy.polys.domains.quotientring import QuotientRing
|
|
FreeModule.__init__(self, ring, rank)
|
|
if not isinstance(ring, QuotientRing):
|
|
raise NotImplementedError('This implementation only works over '
|
|
+ 'quotient rings, got %s' % ring)
|
|
F = self.ring.ring.free_module(self.rank)
|
|
self.quot = F / (self.ring.base_ideal*F)
|
|
|
|
def __repr__(self):
|
|
return "(" + repr(self.ring) + ")" + "**" + repr(self.rank)
|
|
|
|
def submodule(self, *gens, **opts):
|
|
"""
|
|
Generate a submodule.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x, y
|
|
>>> from sympy import QQ
|
|
>>> M = (QQ.old_poly_ring(x, y)/[x**2 - y**2]).free_module(2).submodule([x, x + y])
|
|
>>> M
|
|
<[x + <x**2 - y**2>, x + y + <x**2 - y**2>]>
|
|
>>> M.contains([y**2, x**2 + x*y])
|
|
True
|
|
>>> M.contains([x, y])
|
|
False
|
|
"""
|
|
return SubModuleQuotientRing(gens, self, **opts)
|
|
|
|
def lift(self, elem):
|
|
"""
|
|
Lift the element ``elem`` of self to the module self.quot.
|
|
|
|
Note that self.quot is the same set as self, just as an R-module
|
|
and not as an R/I-module, so this makes sense.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = (QQ.old_poly_ring(x)/[x**2 + 1]).free_module(2)
|
|
>>> e = F.convert([1, 0])
|
|
>>> e
|
|
[1 + <x**2 + 1>, 0 + <x**2 + 1>]
|
|
>>> L = F.quot
|
|
>>> l = F.lift(e)
|
|
>>> l
|
|
[1, 0] + <[x**2 + 1, 0], [0, x**2 + 1]>
|
|
>>> L.contains(l)
|
|
True
|
|
"""
|
|
return self.quot.convert([x.data for x in elem])
|
|
|
|
def unlift(self, elem):
|
|
"""
|
|
Push down an element of self.quot to self.
|
|
|
|
This undoes ``lift``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = (QQ.old_poly_ring(x)/[x**2 + 1]).free_module(2)
|
|
>>> e = F.convert([1, 0])
|
|
>>> l = F.lift(e)
|
|
>>> e == l
|
|
False
|
|
>>> e == F.unlift(l)
|
|
True
|
|
"""
|
|
return self.convert(elem.data)
|
|
|
|
##########################################################################
|
|
## Submodules and subquotients ###########################################
|
|
##########################################################################
|
|
|
|
|
|
class SubModule(Module):
|
|
"""
|
|
Base class for submodules.
|
|
|
|
Attributes:
|
|
|
|
- container - containing module
|
|
- gens - generators (subset of containing module)
|
|
- rank - rank of containing module
|
|
|
|
Non-implemented methods:
|
|
|
|
- _contains
|
|
- _syzygies
|
|
- _in_terms_of_generators
|
|
- _intersect
|
|
- _module_quotient
|
|
|
|
Methods that likely need change in subclasses:
|
|
|
|
- reduce_element
|
|
"""
|
|
|
|
def __init__(self, gens, container):
|
|
Module.__init__(self, container.ring)
|
|
self.gens = tuple(container.convert(x) for x in gens)
|
|
self.container = container
|
|
self.rank = container.rank
|
|
self.ring = container.ring
|
|
self.dtype = container.dtype
|
|
|
|
def __repr__(self):
|
|
return "<" + ", ".join(repr(x) for x in self.gens) + ">"
|
|
|
|
def _contains(self, other):
|
|
"""Implementation of containment.
|
|
Other is guaranteed to be FreeModuleElement."""
|
|
raise NotImplementedError
|
|
|
|
def _syzygies(self):
|
|
"""Implementation of syzygy computation wrt self generators."""
|
|
raise NotImplementedError
|
|
|
|
def _in_terms_of_generators(self, e):
|
|
"""Implementation of expression in terms of generators."""
|
|
raise NotImplementedError
|
|
|
|
def convert(self, elem, M=None):
|
|
"""
|
|
Convert ``elem`` into the internal represantition.
|
|
|
|
Mostly called implicitly.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> M = QQ.old_poly_ring(x).free_module(2).submodule([1, x])
|
|
>>> M.convert([2, 2*x])
|
|
[2, 2*x]
|
|
"""
|
|
if isinstance(elem, self.container.dtype) and elem.module is self:
|
|
return elem
|
|
r = copy(self.container.convert(elem, M))
|
|
r.module = self
|
|
if not self._contains(r):
|
|
raise CoercionFailed
|
|
return r
|
|
|
|
def _intersect(self, other):
|
|
"""Implementation of intersection.
|
|
Other is guaranteed to be a submodule of same free module."""
|
|
raise NotImplementedError
|
|
|
|
def _module_quotient(self, other):
|
|
"""Implementation of quotient.
|
|
Other is guaranteed to be a submodule of same free module."""
|
|
raise NotImplementedError
|
|
|
|
def intersect(self, other, **options):
|
|
"""
|
|
Returns the intersection of ``self`` with submodule ``other``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x, y
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x, y).free_module(2)
|
|
>>> F.submodule([x, x]).intersect(F.submodule([y, y]))
|
|
<[x*y, x*y]>
|
|
|
|
Some implementation allow further options to be passed. Currently, to
|
|
only one implemented is ``relations=True``, in which case the function
|
|
will return a triple ``(res, rela, relb)``, where ``res`` is the
|
|
intersection module, and ``rela`` and ``relb`` are lists of coefficient
|
|
vectors, expressing the generators of ``res`` in terms of the
|
|
generators of ``self`` (``rela``) and ``other`` (``relb``).
|
|
|
|
>>> F.submodule([x, x]).intersect(F.submodule([y, y]), relations=True)
|
|
(<[x*y, x*y]>, [(y,)], [(x,)])
|
|
|
|
The above result says: the intersection module is generated by the
|
|
single element `(-xy, -xy) = -y (x, x) = -x (y, y)`, where
|
|
`(x, x)` and `(y, y)` respectively are the unique generators of
|
|
the two modules being intersected.
|
|
"""
|
|
if not isinstance(other, SubModule):
|
|
raise TypeError('%s is not a SubModule' % other)
|
|
if other.container != self.container:
|
|
raise ValueError(
|
|
'%s is contained in a different free module' % other)
|
|
return self._intersect(other, **options)
|
|
|
|
def module_quotient(self, other, **options):
|
|
r"""
|
|
Returns the module quotient of ``self`` by submodule ``other``.
|
|
|
|
That is, if ``self`` is the module `M` and ``other`` is `N`, then
|
|
return the ideal `\{f \in R | fN \subset M\}`.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy import QQ
|
|
>>> from sympy.abc import x, y
|
|
>>> F = QQ.old_poly_ring(x, y).free_module(2)
|
|
>>> S = F.submodule([x*y, x*y])
|
|
>>> T = F.submodule([x, x])
|
|
>>> S.module_quotient(T)
|
|
<y>
|
|
|
|
Some implementations allow further options to be passed. Currently, the
|
|
only one implemented is ``relations=True``, which may only be passed
|
|
if ``other`` is principal. In this case the function
|
|
will return a pair ``(res, rel)`` where ``res`` is the ideal, and
|
|
``rel`` is a list of coefficient vectors, expressing the generators of
|
|
the ideal, multiplied by the generator of ``other`` in terms of
|
|
generators of ``self``.
|
|
|
|
>>> S.module_quotient(T, relations=True)
|
|
(<y>, [[1]])
|
|
|
|
This means that the quotient ideal is generated by the single element
|
|
`y`, and that `y (x, x) = 1 (xy, xy)`, `(x, x)` and `(xy, xy)` being
|
|
the generators of `T` and `S`, respectively.
|
|
"""
|
|
if not isinstance(other, SubModule):
|
|
raise TypeError('%s is not a SubModule' % other)
|
|
if other.container != self.container:
|
|
raise ValueError(
|
|
'%s is contained in a different free module' % other)
|
|
return self._module_quotient(other, **options)
|
|
|
|
def union(self, other):
|
|
"""
|
|
Returns the module generated by the union of ``self`` and ``other``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(1)
|
|
>>> M = F.submodule([x**2 + x]) # <x(x+1)>
|
|
>>> N = F.submodule([x**2 - 1]) # <(x-1)(x+1)>
|
|
>>> M.union(N) == F.submodule([x+1])
|
|
True
|
|
"""
|
|
if not isinstance(other, SubModule):
|
|
raise TypeError('%s is not a SubModule' % other)
|
|
if other.container != self.container:
|
|
raise ValueError(
|
|
'%s is contained in a different free module' % other)
|
|
return self.__class__(self.gens + other.gens, self.container)
|
|
|
|
def is_zero(self):
|
|
"""
|
|
Return True if ``self`` is a zero module.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> F.submodule([x, 1]).is_zero()
|
|
False
|
|
>>> F.submodule([0, 0]).is_zero()
|
|
True
|
|
"""
|
|
return all(x == 0 for x in self.gens)
|
|
|
|
def submodule(self, *gens):
|
|
"""
|
|
Generate a submodule.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> M = QQ.old_poly_ring(x).free_module(2).submodule([x, 1])
|
|
>>> M.submodule([x**2, x])
|
|
<[x**2, x]>
|
|
"""
|
|
if not self.subset(gens):
|
|
raise ValueError('%s not a subset of %s' % (gens, self))
|
|
return self.__class__(gens, self.container)
|
|
|
|
def is_full_module(self):
|
|
"""
|
|
Return True if ``self`` is the entire free module.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> F.submodule([x, 1]).is_full_module()
|
|
False
|
|
>>> F.submodule([1, 1], [1, 2]).is_full_module()
|
|
True
|
|
"""
|
|
return all(self.contains(x) for x in self.container.basis())
|
|
|
|
def is_submodule(self, other):
|
|
"""
|
|
Returns True if ``other`` is a submodule of ``self``.
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> M = F.submodule([2, x])
|
|
>>> N = M.submodule([2*x, x**2])
|
|
>>> M.is_submodule(M)
|
|
True
|
|
>>> M.is_submodule(N)
|
|
True
|
|
>>> N.is_submodule(M)
|
|
False
|
|
"""
|
|
if isinstance(other, SubModule):
|
|
return self.container == other.container and \
|
|
all(self.contains(x) for x in other.gens)
|
|
if isinstance(other, (FreeModule, QuotientModule)):
|
|
return self.container == other and self.is_full_module()
|
|
return False
|
|
|
|
def syzygy_module(self, **opts):
|
|
r"""
|
|
Compute the syzygy module of the generators of ``self``.
|
|
|
|
Suppose `M` is generated by `f_1, \ldots, f_n` over the ring
|
|
`R`. Consider the homomorphism `\phi: R^n \to M`, given by
|
|
sending `(r_1, \ldots, r_n) \to r_1 f_1 + \cdots + r_n f_n`.
|
|
The syzygy module is defined to be the kernel of `\phi`.
|
|
|
|
Examples
|
|
========
|
|
|
|
The syzygy module is zero iff the generators generate freely a free
|
|
submodule:
|
|
|
|
>>> from sympy.abc import x, y
|
|
>>> from sympy import QQ
|
|
>>> QQ.old_poly_ring(x).free_module(2).submodule([1, 0], [1, 1]).syzygy_module().is_zero()
|
|
True
|
|
|
|
A slightly more interesting example:
|
|
|
|
>>> M = QQ.old_poly_ring(x, y).free_module(2).submodule([x, 2*x], [y, 2*y])
|
|
>>> S = QQ.old_poly_ring(x, y).free_module(2).submodule([y, -x])
|
|
>>> M.syzygy_module() == S
|
|
True
|
|
"""
|
|
F = self.ring.free_module(len(self.gens))
|
|
# NOTE we filter out zero syzygies. This is for convenience of the
|
|
# _syzygies function and not meant to replace any real "generating set
|
|
# reduction" algorithm
|
|
return F.submodule(*[x for x in self._syzygies() if F.convert(x) != 0],
|
|
**opts)
|
|
|
|
def in_terms_of_generators(self, e):
|
|
"""
|
|
Express element ``e`` of ``self`` in terms of the generators.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> M = F.submodule([1, 0], [1, 1])
|
|
>>> M.in_terms_of_generators([x, x**2])
|
|
[-x**2 + x, x**2]
|
|
"""
|
|
try:
|
|
e = self.convert(e)
|
|
except CoercionFailed:
|
|
raise ValueError('%s is not an element of %s' % (e, self))
|
|
return self._in_terms_of_generators(e)
|
|
|
|
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 quotient_module(self, other, **opts):
|
|
"""
|
|
Return a quotient module.
|
|
|
|
This is the same as taking a submodule of a quotient of the containing
|
|
module.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> S1 = F.submodule([x, 1])
|
|
>>> S2 = F.submodule([x**2, x])
|
|
>>> S1.quotient_module(S2)
|
|
<[x, 1] + <[x**2, x]>>
|
|
|
|
Or more coincisely, using the overloaded division operator:
|
|
|
|
>>> F.submodule([x, 1]) / [(x**2, x)]
|
|
<[x, 1] + <[x**2, x]>>
|
|
"""
|
|
if not self.is_submodule(other):
|
|
raise ValueError('%s not a submodule of %s' % (other, self))
|
|
return SubQuotientModule(self.gens,
|
|
self.container.quotient_module(other), **opts)
|
|
|
|
def __add__(self, oth):
|
|
return self.container.quotient_module(self).convert(oth)
|
|
|
|
__radd__ = __add__
|
|
|
|
def multiply_ideal(self, I):
|
|
"""
|
|
Multiply ``self`` by the ideal ``I``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> I = QQ.old_poly_ring(x).ideal(x**2)
|
|
>>> M = QQ.old_poly_ring(x).free_module(2).submodule([1, 1])
|
|
>>> I*M
|
|
<[x**2, x**2]>
|
|
"""
|
|
return self.submodule(*[x*g for [x] in I._module.gens for g in self.gens])
|
|
|
|
def inclusion_hom(self):
|
|
"""
|
|
Return a homomorphism representing the inclusion map of ``self``.
|
|
|
|
That is, the natural map from ``self`` to ``self.container``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> QQ.old_poly_ring(x).free_module(2).submodule([x, x]).inclusion_hom()
|
|
Matrix([
|
|
[1, 0], : <[x, x]> -> QQ[x]**2
|
|
[0, 1]])
|
|
"""
|
|
return self.container.identity_hom().restrict_domain(self)
|
|
|
|
def identity_hom(self):
|
|
"""
|
|
Return the identity homomorphism on ``self``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> QQ.old_poly_ring(x).free_module(2).submodule([x, x]).identity_hom()
|
|
Matrix([
|
|
[1, 0], : <[x, x]> -> <[x, x]>
|
|
[0, 1]])
|
|
"""
|
|
return self.container.identity_hom().restrict_domain(
|
|
self).restrict_codomain(self)
|
|
|
|
|
|
class SubQuotientModule(SubModule):
|
|
"""
|
|
Submodule of a quotient module.
|
|
|
|
Equivalently, quotient module of a submodule.
|
|
|
|
Do not instantiate this, instead use the submodule or quotient_module
|
|
constructing methods:
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> S = F.submodule([1, 0], [1, x])
|
|
>>> Q = F/[(1, 0)]
|
|
>>> S/[(1, 0)] == Q.submodule([5, x])
|
|
True
|
|
|
|
Attributes:
|
|
|
|
- base - base module we are quotient of
|
|
- killed_module - submodule used to form the quotient
|
|
"""
|
|
def __init__(self, gens, container, **opts):
|
|
SubModule.__init__(self, gens, container)
|
|
self.killed_module = self.container.killed_module
|
|
# XXX it is important for some code below that the generators of base
|
|
# are in this particular order!
|
|
self.base = self.container.base.submodule(
|
|
*[x.data for x in self.gens], **opts).union(self.killed_module)
|
|
|
|
def _contains(self, elem):
|
|
return self.base.contains(elem.data)
|
|
|
|
def _syzygies(self):
|
|
# let N = self.killed_module be generated by e_1, ..., e_r
|
|
# let F = self.base be generated by f_1, ..., f_s and e_1, ..., e_r
|
|
# Then self = F/N.
|
|
# Let phi: R**s --> self be the evident surjection.
|
|
# Similarly psi: R**(s + r) --> F.
|
|
# We need to find generators for ker(phi). Let chi: R**s --> F be the
|
|
# evident lift of phi. For X in R**s, phi(X) = 0 iff chi(X) is
|
|
# contained in N, iff there exists Y in R**r such that
|
|
# psi(X, Y) = 0.
|
|
# Hence if alpha: R**(s + r) --> R**s is the projection map, then
|
|
# ker(phi) = alpha ker(psi).
|
|
return [X[:len(self.gens)] for X in self.base._syzygies()]
|
|
|
|
def _in_terms_of_generators(self, e):
|
|
return self.base._in_terms_of_generators(e.data)[:len(self.gens)]
|
|
|
|
def is_full_module(self):
|
|
"""
|
|
Return True if ``self`` is the entire free module.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> F.submodule([x, 1]).is_full_module()
|
|
False
|
|
>>> F.submodule([1, 1], [1, 2]).is_full_module()
|
|
True
|
|
"""
|
|
return self.base.is_full_module()
|
|
|
|
def quotient_hom(self):
|
|
"""
|
|
Return the quotient homomorphism to self.
|
|
|
|
That is, return the natural map from ``self.base`` to ``self``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> M = (QQ.old_poly_ring(x).free_module(2) / [(1, x)]).submodule([1, 0])
|
|
>>> M.quotient_hom()
|
|
Matrix([
|
|
[1, 0], : <[1, 0], [1, x]> -> <[1, 0] + <[1, x]>, [1, x] + <[1, x]>>
|
|
[0, 1]])
|
|
"""
|
|
return self.base.identity_hom().quotient_codomain(self.killed_module)
|
|
|
|
|
|
_subs0 = lambda x: x[0]
|
|
_subs1 = lambda x: x[1:]
|
|
|
|
|
|
class ModuleOrder(ProductOrder):
|
|
"""A product monomial order with a zeroth term as module index."""
|
|
|
|
def __init__(self, o1, o2, TOP):
|
|
if TOP:
|
|
ProductOrder.__init__(self, (o2, _subs1), (o1, _subs0))
|
|
else:
|
|
ProductOrder.__init__(self, (o1, _subs0), (o2, _subs1))
|
|
|
|
|
|
class SubModulePolyRing(SubModule):
|
|
"""
|
|
Submodule of a free module over a generalized polynomial ring.
|
|
|
|
Do not instantiate this, use the constructor method of FreeModule instead:
|
|
|
|
>>> from sympy.abc import x, y
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x, y).free_module(2)
|
|
>>> F.submodule([x, y], [1, 0])
|
|
<[x, y], [1, 0]>
|
|
|
|
Attributes:
|
|
|
|
- order - monomial order used
|
|
"""
|
|
|
|
#self._gb - cached groebner basis
|
|
#self._gbe - cached groebner basis relations
|
|
|
|
def __init__(self, gens, container, order="lex", TOP=True):
|
|
SubModule.__init__(self, gens, container)
|
|
if not isinstance(container, FreeModulePolyRing):
|
|
raise NotImplementedError('This implementation is for submodules of '
|
|
+ 'FreeModulePolyRing, got %s' % container)
|
|
self.order = ModuleOrder(monomial_key(order), self.ring.order, TOP)
|
|
self._gb = None
|
|
self._gbe = None
|
|
|
|
def __eq__(self, other):
|
|
if isinstance(other, SubModulePolyRing) and self.order != other.order:
|
|
return False
|
|
return SubModule.__eq__(self, other)
|
|
|
|
def _groebner(self, extended=False):
|
|
"""Returns a standard basis in sdm form."""
|
|
from sympy.polys.distributedmodules import sdm_groebner, sdm_nf_mora
|
|
if self._gbe is None and extended:
|
|
gb, gbe = sdm_groebner(
|
|
[self.ring._vector_to_sdm(x, self.order) for x in self.gens],
|
|
sdm_nf_mora, self.order, self.ring.dom, extended=True)
|
|
self._gb, self._gbe = tuple(gb), tuple(gbe)
|
|
if self._gb is None:
|
|
self._gb = tuple(sdm_groebner(
|
|
[self.ring._vector_to_sdm(x, self.order) for x in self.gens],
|
|
sdm_nf_mora, self.order, self.ring.dom))
|
|
if extended:
|
|
return self._gb, self._gbe
|
|
else:
|
|
return self._gb
|
|
|
|
def _groebner_vec(self, extended=False):
|
|
"""Returns a standard basis in element form."""
|
|
if not extended:
|
|
return [FreeModuleElement(self,
|
|
tuple(self.ring._sdm_to_vector(x, self.rank)))
|
|
for x in self._groebner()]
|
|
gb, gbe = self._groebner(extended=True)
|
|
return ([self.convert(self.ring._sdm_to_vector(x, self.rank))
|
|
for x in gb],
|
|
[self.ring._sdm_to_vector(x, len(self.gens)) for x in gbe])
|
|
|
|
def _contains(self, x):
|
|
from sympy.polys.distributedmodules import sdm_zero, sdm_nf_mora
|
|
return sdm_nf_mora(self.ring._vector_to_sdm(x, self.order),
|
|
self._groebner(), self.order, self.ring.dom) == \
|
|
sdm_zero()
|
|
|
|
def _syzygies(self):
|
|
"""Compute syzygies. See [SCA, algorithm 2.5.4]."""
|
|
# NOTE if self.gens is a standard basis, this can be done more
|
|
# efficiently using Schreyer's theorem
|
|
|
|
# First bullet point
|
|
k = len(self.gens)
|
|
r = self.rank
|
|
zero = self.ring.convert(0)
|
|
one = self.ring.convert(1)
|
|
Rkr = self.ring.free_module(r + k)
|
|
newgens = []
|
|
for j, f in enumerate(self.gens):
|
|
m = [0]*(r + k)
|
|
for i, v in enumerate(f):
|
|
m[i] = f[i]
|
|
for i in range(k):
|
|
m[r + i] = one if j == i else zero
|
|
m = FreeModuleElement(Rkr, tuple(m))
|
|
newgens.append(m)
|
|
# Note: we need *descending* order on module index, and TOP=False to
|
|
# get an elimination order
|
|
F = Rkr.submodule(*newgens, order='ilex', TOP=False)
|
|
|
|
# Second bullet point: standard basis of F
|
|
G = F._groebner_vec()
|
|
|
|
# Third bullet point: G0 = G intersect the new k components
|
|
G0 = [x[r:] for x in G if all(y == zero for y in x[:r])]
|
|
|
|
# Fourth and fifth bullet points: we are done
|
|
return G0
|
|
|
|
def _in_terms_of_generators(self, e):
|
|
"""Expression in terms of generators. See [SCA, 2.8.1]."""
|
|
# NOTE: if gens is a standard basis, this can be done more efficiently
|
|
M = self.ring.free_module(self.rank).submodule(*((e,) + self.gens))
|
|
S = M.syzygy_module(
|
|
order="ilex", TOP=False) # We want decreasing order!
|
|
G = S._groebner_vec()
|
|
# This list cannot not be empty since e is an element
|
|
e = [x for x in G if self.ring.is_unit(x[0])][0]
|
|
return [-x/e[0] for x in e[1:]]
|
|
|
|
def reduce_element(self, x, NF=None):
|
|
"""
|
|
Reduce the element ``x`` of our container modulo ``self``.
|
|
|
|
This applies the normal form ``NF`` to ``x``. If ``NF`` is passed
|
|
as none, the default Mora normal form is used (which is not unique!).
|
|
"""
|
|
from sympy.polys.distributedmodules import sdm_nf_mora
|
|
if NF is None:
|
|
NF = sdm_nf_mora
|
|
return self.container.convert(self.ring._sdm_to_vector(NF(
|
|
self.ring._vector_to_sdm(x, self.order), self._groebner(),
|
|
self.order, self.ring.dom),
|
|
self.rank))
|
|
|
|
def _intersect(self, other, relations=False):
|
|
# See: [SCA, section 2.8.2]
|
|
fi = self.gens
|
|
hi = other.gens
|
|
r = self.rank
|
|
ci = [[0]*(2*r) for _ in range(r)]
|
|
for k in range(r):
|
|
ci[k][k] = 1
|
|
ci[k][r + k] = 1
|
|
di = [list(f) + [0]*r for f in fi]
|
|
ei = [[0]*r + list(h) for h in hi]
|
|
syz = self.ring.free_module(2*r).submodule(*(ci + di + ei))._syzygies()
|
|
nonzero = [x for x in syz if any(y != self.ring.zero for y in x[:r])]
|
|
res = self.container.submodule(*([-y for y in x[:r]] for x in nonzero))
|
|
reln1 = [x[r:r + len(fi)] for x in nonzero]
|
|
reln2 = [x[r + len(fi):] for x in nonzero]
|
|
if relations:
|
|
return res, reln1, reln2
|
|
return res
|
|
|
|
def _module_quotient(self, other, relations=False):
|
|
# See: [SCA, section 2.8.4]
|
|
if relations and len(other.gens) != 1:
|
|
raise NotImplementedError
|
|
if len(other.gens) == 0:
|
|
return self.ring.ideal(1)
|
|
elif len(other.gens) == 1:
|
|
# We do some trickery. Let f be the (vector!) generating ``other``
|
|
# and f1, .., fn be the (vectors) generating self.
|
|
# Consider the submodule of R^{r+1} generated by (f, 1) and
|
|
# {(fi, 0) | i}. Then the intersection with the last module
|
|
# component yields the quotient.
|
|
g1 = list(other.gens[0]) + [1]
|
|
gi = [list(x) + [0] for x in self.gens]
|
|
# NOTE: We *need* to use an elimination order
|
|
M = self.ring.free_module(self.rank + 1).submodule(*([g1] + gi),
|
|
order='ilex', TOP=False)
|
|
if not relations:
|
|
return self.ring.ideal(*[x[-1] for x in M._groebner_vec() if
|
|
all(y == self.ring.zero for y in x[:-1])])
|
|
else:
|
|
G, R = M._groebner_vec(extended=True)
|
|
indices = [i for i, x in enumerate(G) if
|
|
all(y == self.ring.zero for y in x[:-1])]
|
|
return (self.ring.ideal(*[G[i][-1] for i in indices]),
|
|
[[-x for x in R[i][1:]] for i in indices])
|
|
# For more generators, we use I : <h1, .., hn> = intersection of
|
|
# {I : <hi> | i}
|
|
# TODO this can be done more efficiently
|
|
return reduce(lambda x, y: x.intersect(y),
|
|
(self._module_quotient(self.container.submodule(x)) for x in other.gens))
|
|
|
|
|
|
class SubModuleQuotientRing(SubModule):
|
|
"""
|
|
Class for submodules of free modules over quotient rings.
|
|
|
|
Do not instantiate this. Instead use the submodule methods.
|
|
|
|
>>> from sympy.abc import x, y
|
|
>>> from sympy import QQ
|
|
>>> M = (QQ.old_poly_ring(x, y)/[x**2 - y**2]).free_module(2).submodule([x, x + y])
|
|
>>> M
|
|
<[x + <x**2 - y**2>, x + y + <x**2 - y**2>]>
|
|
>>> M.contains([y**2, x**2 + x*y])
|
|
True
|
|
>>> M.contains([x, y])
|
|
False
|
|
|
|
Attributes:
|
|
|
|
- quot - the subquotient of `R^n/IR^n` generated by lifts of our generators
|
|
"""
|
|
|
|
def __init__(self, gens, container):
|
|
SubModule.__init__(self, gens, container)
|
|
self.quot = self.container.quot.submodule(
|
|
*[self.container.lift(x) for x in self.gens])
|
|
|
|
def _contains(self, elem):
|
|
return self.quot._contains(self.container.lift(elem))
|
|
|
|
def _syzygies(self):
|
|
return [tuple(self.ring.convert(y, self.quot.ring) for y in x)
|
|
for x in self.quot._syzygies()]
|
|
|
|
def _in_terms_of_generators(self, elem):
|
|
return [self.ring.convert(x, self.quot.ring) for x in
|
|
self.quot._in_terms_of_generators(self.container.lift(elem))]
|
|
|
|
##########################################################################
|
|
## Quotient Modules ######################################################
|
|
##########################################################################
|
|
|
|
|
|
class QuotientModuleElement(ModuleElement):
|
|
"""Element of a quotient module."""
|
|
|
|
def eq(self, d1, d2):
|
|
"""Equality comparison."""
|
|
return self.module.killed_module.contains(d1 - d2)
|
|
|
|
def __repr__(self):
|
|
return repr(self.data) + " + " + repr(self.module.killed_module)
|
|
|
|
|
|
class QuotientModule(Module):
|
|
"""
|
|
Class for quotient modules.
|
|
|
|
Do not instantiate this directly. For subquotients, see the
|
|
SubQuotientModule class.
|
|
|
|
Attributes:
|
|
|
|
- base - the base module we are a quotient of
|
|
- killed_module - the submodule used to form the quotient
|
|
- rank of the base
|
|
"""
|
|
|
|
dtype = QuotientModuleElement
|
|
|
|
def __init__(self, ring, base, submodule):
|
|
Module.__init__(self, ring)
|
|
if not base.is_submodule(submodule):
|
|
raise ValueError('%s is not a submodule of %s' % (submodule, base))
|
|
self.base = base
|
|
self.killed_module = submodule
|
|
self.rank = base.rank
|
|
|
|
def __repr__(self):
|
|
return repr(self.base) + "/" + repr(self.killed_module)
|
|
|
|
def is_zero(self):
|
|
"""
|
|
Return True if ``self`` is a zero module.
|
|
|
|
This happens if and only if the base module is the same as the
|
|
submodule being killed.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2)
|
|
>>> (F/[(1, 0)]).is_zero()
|
|
False
|
|
>>> (F/[(1, 0), (0, 1)]).is_zero()
|
|
True
|
|
"""
|
|
return self.base == self.killed_module
|
|
|
|
def is_submodule(self, other):
|
|
"""
|
|
Return True if ``other`` is a submodule of ``self``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> Q = QQ.old_poly_ring(x).free_module(2) / [(x, x)]
|
|
>>> S = Q.submodule([1, 0])
|
|
>>> Q.is_submodule(S)
|
|
True
|
|
>>> S.is_submodule(Q)
|
|
False
|
|
"""
|
|
if isinstance(other, QuotientModule):
|
|
return self.killed_module == other.killed_module and \
|
|
self.base.is_submodule(other.base)
|
|
if isinstance(other, SubQuotientModule):
|
|
return other.container == self
|
|
return False
|
|
|
|
def submodule(self, *gens, **opts):
|
|
"""
|
|
Generate a submodule.
|
|
|
|
This is the same as taking a quotient of a submodule of the base
|
|
module.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> Q = QQ.old_poly_ring(x).free_module(2) / [(x, x)]
|
|
>>> Q.submodule([x, 0])
|
|
<[x, 0] + <[x, x]>>
|
|
"""
|
|
return SubQuotientModule(gens, self, **opts)
|
|
|
|
def convert(self, elem, M=None):
|
|
"""
|
|
Convert ``elem`` into the internal representation.
|
|
|
|
This method is called implicitly whenever computations involve elements
|
|
not in the internal representation.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> F = QQ.old_poly_ring(x).free_module(2) / [(1, 2), (1, x)]
|
|
>>> F.convert([1, 0])
|
|
[1, 0] + <[1, 2], [1, x]>
|
|
"""
|
|
if isinstance(elem, QuotientModuleElement):
|
|
if elem.module is self:
|
|
return elem
|
|
if self.killed_module.is_submodule(elem.module.killed_module):
|
|
return QuotientModuleElement(self, self.base.convert(elem.data))
|
|
raise CoercionFailed
|
|
return QuotientModuleElement(self, self.base.convert(elem))
|
|
|
|
def identity_hom(self):
|
|
"""
|
|
Return the identity homomorphism on ``self``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> M = QQ.old_poly_ring(x).free_module(2) / [(1, 2), (1, x)]
|
|
>>> M.identity_hom()
|
|
Matrix([
|
|
[1, 0], : QQ[x]**2/<[1, 2], [1, x]> -> QQ[x]**2/<[1, 2], [1, x]>
|
|
[0, 1]])
|
|
"""
|
|
return self.base.identity_hom().quotient_codomain(
|
|
self.killed_module).quotient_domain(self.killed_module)
|
|
|
|
def quotient_hom(self):
|
|
"""
|
|
Return the quotient homomorphism to ``self``.
|
|
|
|
That is, return a homomorphism representing the natural map from
|
|
``self.base`` to ``self``.
|
|
|
|
Examples
|
|
========
|
|
|
|
>>> from sympy.abc import x
|
|
>>> from sympy import QQ
|
|
>>> M = QQ.old_poly_ring(x).free_module(2) / [(1, 2), (1, x)]
|
|
>>> M.quotient_hom()
|
|
Matrix([
|
|
[1, 0], : QQ[x]**2 -> QQ[x]**2/<[1, 2], [1, x]>
|
|
[0, 1]])
|
|
"""
|
|
return self.base.identity_hom().quotient_codomain(
|
|
self.killed_module)
|