463 lines
15 KiB
Python
463 lines
15 KiB
Python
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"""Implementation of :class:`PolynomialRing` class. """
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from sympy.polys.agca.modules import FreeModulePolyRing
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from sympy.polys.domains.characteristiczero import CharacteristicZero
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from sympy.polys.domains.compositedomain import CompositeDomain
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from sympy.polys.domains.old_fractionfield import FractionField
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from sympy.polys.domains.ring import Ring
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from sympy.polys.orderings import monomial_key, build_product_order
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from sympy.polys.polyclasses import DMP, DMF
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from sympy.polys.polyerrors import (GeneratorsNeeded, PolynomialError,
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CoercionFailed, ExactQuotientFailed, NotReversible)
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from sympy.polys.polyutils import dict_from_basic, basic_from_dict, _dict_reorder
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from sympy.utilities import public
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from sympy.utilities.iterables import iterable
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# XXX why does this derive from CharacteristicZero???
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@public
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class PolynomialRingBase(Ring, CharacteristicZero, CompositeDomain):
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"""
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Base class for generalized polynomial rings.
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This base class should be used for uniform access to generalized polynomial
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rings. Subclasses only supply information about the element storage etc.
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Do not instantiate.
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"""
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has_assoc_Ring = True
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has_assoc_Field = True
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default_order = "grevlex"
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def __init__(self, dom, *gens, **opts):
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if not gens:
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raise GeneratorsNeeded("generators not specified")
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lev = len(gens) - 1
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self.ngens = len(gens)
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self.zero = self.dtype.zero(lev, dom, ring=self)
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self.one = self.dtype.one(lev, dom, ring=self)
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self.domain = self.dom = dom
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self.symbols = self.gens = gens
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# NOTE 'order' may not be set if inject was called through CompositeDomain
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self.order = opts.get('order', monomial_key(self.default_order))
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def new(self, element):
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return self.dtype(element, self.dom, len(self.gens) - 1, ring=self)
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def __str__(self):
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s_order = str(self.order)
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orderstr = (
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" order=" + s_order) if s_order != self.default_order else ""
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return str(self.dom) + '[' + ','.join(map(str, self.gens)) + orderstr + ']'
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def __hash__(self):
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return hash((self.__class__.__name__, self.dtype, self.dom,
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self.gens, self.order))
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def __eq__(self, other):
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"""Returns ``True`` if two domains are equivalent. """
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return isinstance(other, PolynomialRingBase) and \
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self.dtype == other.dtype and self.dom == other.dom and \
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self.gens == other.gens and self.order == other.order
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def from_ZZ(K1, a, K0):
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"""Convert a Python ``int`` object to ``dtype``. """
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return K1(K1.dom.convert(a, K0))
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def from_ZZ_python(K1, a, K0):
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"""Convert a Python ``int`` object to ``dtype``. """
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return K1(K1.dom.convert(a, K0))
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def from_QQ(K1, a, K0):
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"""Convert a Python ``Fraction`` object to ``dtype``. """
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return K1(K1.dom.convert(a, K0))
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def from_QQ_python(K1, a, K0):
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"""Convert a Python ``Fraction`` object to ``dtype``. """
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return K1(K1.dom.convert(a, K0))
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def from_ZZ_gmpy(K1, a, K0):
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"""Convert a GMPY ``mpz`` object to ``dtype``. """
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return K1(K1.dom.convert(a, K0))
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def from_QQ_gmpy(K1, a, K0):
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"""Convert a GMPY ``mpq`` object to ``dtype``. """
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return K1(K1.dom.convert(a, K0))
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def from_RealField(K1, a, K0):
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"""Convert a mpmath ``mpf`` object to ``dtype``. """
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return K1(K1.dom.convert(a, K0))
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def from_AlgebraicField(K1, a, K0):
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"""Convert a ``ANP`` object to ``dtype``. """
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if K1.dom == K0:
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return K1(a)
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def from_PolynomialRing(K1, a, K0):
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"""Convert a ``PolyElement`` object to ``dtype``. """
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if K1.gens == K0.symbols:
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if K1.dom == K0.dom:
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return K1(dict(a)) # set the correct ring
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else:
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convert_dom = lambda c: K1.dom.convert_from(c, K0.dom)
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return K1({m: convert_dom(c) for m, c in a.items()})
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else:
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monoms, coeffs = _dict_reorder(a.to_dict(), K0.symbols, K1.gens)
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if K1.dom != K0.dom:
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coeffs = [ K1.dom.convert(c, K0.dom) for c in coeffs ]
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return K1(dict(zip(monoms, coeffs)))
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def from_GlobalPolynomialRing(K1, a, K0):
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"""Convert a ``DMP`` object to ``dtype``. """
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if K1.gens == K0.gens:
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if K1.dom == K0.dom:
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return K1(a.rep) # set the correct ring
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else:
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return K1(a.convert(K1.dom).rep)
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else:
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monoms, coeffs = _dict_reorder(a.to_dict(), K0.gens, K1.gens)
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if K1.dom != K0.dom:
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coeffs = [ K1.dom.convert(c, K0.dom) for c in coeffs ]
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return K1(dict(zip(monoms, coeffs)))
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def get_field(self):
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"""Returns a field associated with ``self``. """
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return FractionField(self.dom, *self.gens)
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def poly_ring(self, *gens):
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"""Returns a polynomial ring, i.e. ``K[X]``. """
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raise NotImplementedError('nested domains not allowed')
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def frac_field(self, *gens):
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"""Returns a fraction field, i.e. ``K(X)``. """
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raise NotImplementedError('nested domains not allowed')
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def revert(self, a):
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try:
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return 1/a
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except (ExactQuotientFailed, ZeroDivisionError):
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raise NotReversible('%s is not a unit' % a)
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def gcdex(self, a, b):
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"""Extended GCD of ``a`` and ``b``. """
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return a.gcdex(b)
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def gcd(self, a, b):
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"""Returns GCD of ``a`` and ``b``. """
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return a.gcd(b)
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def lcm(self, a, b):
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"""Returns LCM of ``a`` and ``b``. """
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return a.lcm(b)
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def factorial(self, a):
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"""Returns factorial of ``a``. """
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return self.dtype(self.dom.factorial(a))
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def _vector_to_sdm(self, v, order):
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"""
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For internal use by the modules class.
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Convert an iterable of elements of this ring into a sparse distributed
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module element.
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"""
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raise NotImplementedError
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def _sdm_to_dics(self, s, n):
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"""Helper for _sdm_to_vector."""
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from sympy.polys.distributedmodules import sdm_to_dict
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dic = sdm_to_dict(s)
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res = [{} for _ in range(n)]
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for k, v in dic.items():
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res[k[0]][k[1:]] = v
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return res
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def _sdm_to_vector(self, s, n):
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"""
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For internal use by the modules class.
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Convert a sparse distributed module into a list of length ``n``.
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Examples
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========
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>>> from sympy import QQ, ilex
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>>> from sympy.abc import x, y
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>>> R = QQ.old_poly_ring(x, y, order=ilex)
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>>> L = [((1, 1, 1), QQ(1)), ((0, 1, 0), QQ(1)), ((0, 0, 1), QQ(2))]
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>>> R._sdm_to_vector(L, 2)
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[x + 2*y, x*y]
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"""
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dics = self._sdm_to_dics(s, n)
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# NOTE this works for global and local rings!
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return [self(x) for x in dics]
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def free_module(self, rank):
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"""
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Generate a free module of rank ``rank`` over ``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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>>> QQ.old_poly_ring(x).free_module(2)
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QQ[x]**2
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"""
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return FreeModulePolyRing(self, rank)
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def _vector_to_sdm_helper(v, order):
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"""Helper method for common code in Global and Local poly rings."""
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from sympy.polys.distributedmodules import sdm_from_dict
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d = {}
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for i, e in enumerate(v):
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for key, value in e.to_dict().items():
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d[(i,) + key] = value
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return sdm_from_dict(d, order)
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@public
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class GlobalPolynomialRing(PolynomialRingBase):
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"""A true polynomial ring, with objects DMP. """
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is_PolynomialRing = is_Poly = True
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dtype = DMP
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def from_FractionField(K1, a, K0):
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"""
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Convert a ``DMF`` object to ``DMP``.
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Examples
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========
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>>> from sympy.polys.polyclasses import DMP, DMF
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>>> from sympy.polys.domains import ZZ
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>>> from sympy.abc import x
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>>> f = DMF(([ZZ(1), ZZ(1)], [ZZ(1)]), ZZ)
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>>> K = ZZ.old_frac_field(x)
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>>> F = ZZ.old_poly_ring(x).from_FractionField(f, K)
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>>> F == DMP([ZZ(1), ZZ(1)], ZZ)
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True
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>>> type(F)
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<class 'sympy.polys.polyclasses.DMP'>
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"""
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if a.denom().is_one:
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return K1.from_GlobalPolynomialRing(a.numer(), K0)
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def to_sympy(self, a):
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"""Convert ``a`` to a SymPy object. """
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return basic_from_dict(a.to_sympy_dict(), *self.gens)
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def from_sympy(self, a):
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"""Convert SymPy's expression to ``dtype``. """
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try:
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rep, _ = dict_from_basic(a, gens=self.gens)
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except PolynomialError:
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raise CoercionFailed("Cannot convert %s to type %s" % (a, self))
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for k, v in rep.items():
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rep[k] = self.dom.from_sympy(v)
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return self(rep)
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def is_positive(self, a):
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"""Returns True if ``LC(a)`` is positive. """
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return self.dom.is_positive(a.LC())
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def is_negative(self, a):
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"""Returns True if ``LC(a)`` is negative. """
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return self.dom.is_negative(a.LC())
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def is_nonpositive(self, a):
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"""Returns True if ``LC(a)`` is non-positive. """
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return self.dom.is_nonpositive(a.LC())
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def is_nonnegative(self, a):
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"""Returns True if ``LC(a)`` is non-negative. """
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return self.dom.is_nonnegative(a.LC())
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def _vector_to_sdm(self, v, order):
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"""
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Examples
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========
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>>> from sympy import lex, QQ
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>>> from sympy.abc import x, y
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>>> R = QQ.old_poly_ring(x, y)
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>>> f = R.convert(x + 2*y)
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>>> g = R.convert(x * y)
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>>> R._vector_to_sdm([f, g], lex)
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[((1, 1, 1), 1), ((0, 1, 0), 1), ((0, 0, 1), 2)]
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"""
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return _vector_to_sdm_helper(v, order)
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class GeneralizedPolynomialRing(PolynomialRingBase):
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"""A generalized polynomial ring, with objects DMF. """
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dtype = DMF
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def new(self, a):
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"""Construct an element of ``self`` domain from ``a``. """
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res = self.dtype(a, self.dom, len(self.gens) - 1, ring=self)
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# make sure res is actually in our ring
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if res.denom().terms(order=self.order)[0][0] != (0,)*len(self.gens):
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from sympy.printing.str import sstr
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raise CoercionFailed("denominator %s not allowed in %s"
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% (sstr(res), self))
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return res
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def __contains__(self, a):
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try:
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a = self.convert(a)
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except CoercionFailed:
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return False
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return a.denom().terms(order=self.order)[0][0] == (0,)*len(self.gens)
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def from_FractionField(K1, a, K0):
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dmf = K1.get_field().from_FractionField(a, K0)
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return K1((dmf.num, dmf.den))
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def to_sympy(self, a):
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"""Convert ``a`` to a SymPy object. """
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return (basic_from_dict(a.numer().to_sympy_dict(), *self.gens) /
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basic_from_dict(a.denom().to_sympy_dict(), *self.gens))
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def from_sympy(self, a):
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"""Convert SymPy's expression to ``dtype``. """
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p, q = a.as_numer_denom()
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num, _ = dict_from_basic(p, gens=self.gens)
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den, _ = dict_from_basic(q, gens=self.gens)
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for k, v in num.items():
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num[k] = self.dom.from_sympy(v)
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for k, v in den.items():
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den[k] = self.dom.from_sympy(v)
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return self((num, den)).cancel()
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def _vector_to_sdm(self, v, order):
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"""
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|
|
Turn an iterable into a sparse distributed module.
|
||
|
|
|
||
|
|
Note that the vector is multiplied by a unit first to make all entries
|
||
|
|
polynomials.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import ilex, QQ
|
||
|
|
>>> from sympy.abc import x, y
|
||
|
|
>>> R = QQ.old_poly_ring(x, y, order=ilex)
|
||
|
|
>>> f = R.convert((x + 2*y) / (1 + x))
|
||
|
|
>>> g = R.convert(x * y)
|
||
|
|
>>> R._vector_to_sdm([f, g], ilex)
|
||
|
|
[((0, 0, 1), 2), ((0, 1, 0), 1), ((1, 1, 1), 1), ((1,
|
||
|
|
2, 1), 1)]
|
||
|
|
"""
|
||
|
|
# NOTE this is quite inefficient...
|
||
|
|
u = self.one.numer()
|
||
|
|
for x in v:
|
||
|
|
u *= x.denom()
|
||
|
|
return _vector_to_sdm_helper([x.numer()*u/x.denom() for x in v], order)
|
||
|
|
|
||
|
|
|
||
|
|
@public
|
||
|
|
def PolynomialRing(dom, *gens, **opts):
|
||
|
|
r"""
|
||
|
|
Create a generalized multivariate polynomial ring.
|
||
|
|
|
||
|
|
A generalized polynomial ring is defined by a ground field `K`, a set
|
||
|
|
of generators (typically `x_1, \ldots, x_n`) and a monomial order `<`.
|
||
|
|
The monomial order can be global, local or mixed. In any case it induces
|
||
|
|
a total ordering on the monomials, and there exists for every (non-zero)
|
||
|
|
polynomial `f \in K[x_1, \ldots, x_n]` a well-defined "leading monomial"
|
||
|
|
`LM(f) = LM(f, >)`. One can then define a multiplicative subset
|
||
|
|
`S = S_> = \{f \in K[x_1, \ldots, x_n] | LM(f) = 1\}`. The generalized
|
||
|
|
polynomial ring corresponding to the monomial order is
|
||
|
|
`R = S^{-1}K[x_1, \ldots, x_n]`.
|
||
|
|
|
||
|
|
If `>` is a so-called global order, that is `1` is the smallest monomial,
|
||
|
|
then we just have `S = K` and `R = K[x_1, \ldots, x_n]`.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
A few examples may make this clearer.
|
||
|
|
|
||
|
|
>>> from sympy.abc import x, y
|
||
|
|
>>> from sympy import QQ
|
||
|
|
|
||
|
|
Our first ring uses global lexicographic order.
|
||
|
|
|
||
|
|
>>> R1 = QQ.old_poly_ring(x, y, order=(("lex", x, y),))
|
||
|
|
|
||
|
|
The second ring uses local lexicographic order. Note that when using a
|
||
|
|
single (non-product) order, you can just specify the name and omit the
|
||
|
|
variables:
|
||
|
|
|
||
|
|
>>> R2 = QQ.old_poly_ring(x, y, order="ilex")
|
||
|
|
|
||
|
|
The third and fourth rings use a mixed orders:
|
||
|
|
|
||
|
|
>>> o1 = (("ilex", x), ("lex", y))
|
||
|
|
>>> o2 = (("lex", x), ("ilex", y))
|
||
|
|
>>> R3 = QQ.old_poly_ring(x, y, order=o1)
|
||
|
|
>>> R4 = QQ.old_poly_ring(x, y, order=o2)
|
||
|
|
|
||
|
|
We will investigate what elements of `K(x, y)` are contained in the various
|
||
|
|
rings.
|
||
|
|
|
||
|
|
>>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
|
||
|
|
>>> test = lambda R: [f in R for f in L]
|
||
|
|
|
||
|
|
The first ring is just `K[x, y]`:
|
||
|
|
|
||
|
|
>>> test(R1)
|
||
|
|
[True, False, False, False, False]
|
||
|
|
|
||
|
|
The second ring is R1 localised at the maximal ideal (x, y):
|
||
|
|
|
||
|
|
>>> test(R2)
|
||
|
|
[True, False, True, True, True]
|
||
|
|
|
||
|
|
The third ring is R1 localised at the prime ideal (x):
|
||
|
|
|
||
|
|
>>> test(R3)
|
||
|
|
[True, False, True, False, True]
|
||
|
|
|
||
|
|
Finally the fourth ring is R1 localised at `S = K[x, y] \setminus yK[y]`:
|
||
|
|
|
||
|
|
>>> test(R4)
|
||
|
|
[True, False, False, True, False]
|
||
|
|
"""
|
||
|
|
|
||
|
|
order = opts.get("order", GeneralizedPolynomialRing.default_order)
|
||
|
|
if iterable(order):
|
||
|
|
order = build_product_order(order, gens)
|
||
|
|
order = monomial_key(order)
|
||
|
|
opts['order'] = order
|
||
|
|
|
||
|
|
if order.is_global:
|
||
|
|
return GlobalPolynomialRing(dom, *gens, **opts)
|
||
|
|
else:
|
||
|
|
return GeneralizedPolynomialRing(dom, *gens, **opts)
|