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sagemathgh-38119: Fix hashing for Weyl algebra elements
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Weyl algebras elements cannot be hashed or compared, since the element
class is too barren.
This patch lets the element class inherit from IndexedFreeModuleElement,
which fixes this problem automatically.
    
URL: sagemath#38119
Reported by: Darij Grinberg
Reviewer(s): Travis Scrimshaw
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Release Manager committed Jun 7, 2024
2 parents e6c4f20 + 12162b2 commit b80817e
Showing 1 changed file with 66 additions and 124 deletions.
190 changes: 66 additions & 124 deletions src/sage/algebras/weyl_algebra.py
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Original file line number Diff line number Diff line change
Expand Up @@ -21,7 +21,6 @@
from sage.misc.latex import latex, LatexExpr
from sage.misc.lazy_attribute import lazy_attribute
from sage.misc.misc_c import prod
from sage.structure.richcmp import richcmp
from sage.structure.element import Element
from sage.structure.parent import Parent
from sage.structure.unique_representation import UniqueRepresentation
Expand All @@ -34,6 +33,7 @@
from sage.rings.polynomial.multi_polynomial_ring_base import MPolynomialRing_base
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.structure.global_options import GlobalOptions
from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement


def repr_from_monomials(monomials, term_repr, use_latex=False) -> str:
Expand Down Expand Up @@ -231,24 +231,68 @@ def repr_dx(k):
return ret


class DifferentialWeylAlgebraElement(Element):
class DifferentialWeylAlgebraElement(IndexedFreeModuleElement):
"""
An element in a differential Weyl algebra.
"""
def __init__(self, parent, monomials) -> None:
"""
Initialize ``self``.
TESTS::
TESTS:
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: elt = ((x^3-z)*dx + dy)^2
sage: TestSuite(elt).run()
"""
Element.__init__(self, parent)
self.__monomials = monomials
Some computations in the Weyl algebra, using the implicit
coercion from the polynomial ring into the Weyl algebra::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: elt = ((x^3-z)*dx + dy)^2
sage: TestSuite(elt).run()
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: dx,dy,dz = W.differentials()
sage: dy*(x^3-y*z)*dx == -z*dx + x^3*dx*dy - y*z*dx*dy
True
sage: W.zero() == 0
True
sage: W.one() == 1
True
sage: x == 1
False
sage: x + 1 == 1
False
sage: W(x^3 - y*z) == x^3 - y*z
True
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: dx != dy
True
sage: W.one() != 1
False
sage: dy - (3*x - z)*dx
dy + z*dx - 3*x*dx
sage: (dx*dy) + dz + x^3 - 2
dx*dy + dz + x^3 - 2
sage: elt = (dy - (3*x - z)*dx)
sage: sorted(elt.monomial_coefficients().items())
[(((0, 0, 0), (0, 1, 0)), 1),
(((0, 0, 1), (1, 0, 0)), 1),
(((1, 0, 0), (1, 0, 0)), -3)]
sage: elt = dy - (3*x - z)*dx + 1
sage: sorted(elt.support())
[((0, 0, 0), (0, 0, 0)),
((0, 0, 0), (0, 1, 0)),
((0, 0, 1), (1, 0, 0)),
((1, 0, 0), (1, 0, 0))]
Hashing works::
sage: hash(dx) == hash(dx) # hashing works
True
Comparison works, even if mathematically meaningless
(but useful e.g. for sorting)::
sage: dx < dy or dy < dx
True
"""
def _repr_(self) -> str:
r"""
Return a string representation of ``self``.
Expand Down Expand Up @@ -327,64 +371,6 @@ def half_term(mon, polynomial):
return p + ' ' + d
return repr_from_monomials(self.list(), term, True)

def _richcmp_(self, other, op) -> bool:
"""
Rich comparison for equal parents.
TESTS::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: dx,dy,dz = W.differentials()
sage: dy*(x^3-y*z)*dx == -z*dx + x^3*dx*dy - y*z*dx*dy
True
sage: W.zero() == 0
True
sage: W.one() == 1
True
sage: x == 1
False
sage: x + 1 == 1
False
sage: W(x^3 - y*z) == x^3 - y*z
True
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: dx != dy
True
sage: W.one() != 1
False
"""
return richcmp(self.__monomials, other.__monomials, op)

def __neg__(self):
"""
Return the negative of ``self``.
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: dy - (3*x - z)*dx
dy + z*dx - 3*x*dx
"""
return self.__class__(self.parent(),
{m: -c for m, c in self.__monomials.items()})

def _add_(self, other):
"""
Return ``self`` added to ``other``.
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: (dx*dy) + dz + x^3 - 2
dx*dy + dz + x^3 - 2
"""
F = self.parent()
return self.__class__(F, blas.add(self.__monomials, other.__monomials))

def _mul_(self, other):
"""
Return ``self`` multiplied by ``other``.
Expand All @@ -406,10 +392,10 @@ def add_tuples(x, y):
n = self.parent()._n
t = tuple([0] * n)
zero = self.parent().base_ring().zero()
for ml in self.__monomials:
cl = self.__monomials[ml]
for mr in other.__monomials:
cr = other.__monomials[mr]
for ml in self._monomial_coefficients:
cl = self._monomial_coefficients[ml]
for mr in other._monomial_coefficients:
cr = other._monomial_coefficients[mr]
cur = [((mr[0], t), cl * cr)]
for i, p in enumerate(ml[1]):
for _ in range(p):
Expand Down Expand Up @@ -447,7 +433,7 @@ def _rmul_(self, other):
"""
if other == 0:
return self.parent().zero()
M = self.__monomials
M = self._monomial_coefficients
return self.__class__(self.parent(), {t: other * M[t] for t in M})

def _lmul_(self, other):
Expand All @@ -464,36 +450,9 @@ def _lmul_(self, other):
"""
if other == 0:
return self.parent().zero()
M = self.__monomials
M = self._monomial_coefficients
return self.__class__(self.parent(), {t: M[t] * other for t in M})

def monomial_coefficients(self, copy=True) -> dict:
"""
Return a dictionary which has the basis keys in the support
of ``self`` as keys and their corresponding coefficients
as values.
INPUT:
- ``copy`` -- (default: ``True``) if ``self`` is internally
represented by a dictionary ``d``, then make a copy of ``d``;
if ``False``, then this can cause undesired behavior by
mutating ``d``
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: elt = (dy - (3*x - z)*dx)
sage: sorted(elt.monomial_coefficients().items())
[(((0, 0, 0), (0, 1, 0)), 1),
(((0, 0, 1), (1, 0, 0)), 1),
(((1, 0, 0), (1, 0, 0)), -3)]
"""
if copy:
return dict(self.__monomials)
return self.__monomials

def __iter__(self):
"""
Return an iterator of ``self``.
Expand Down Expand Up @@ -530,26 +489,9 @@ def list(self) -> list:
(((0, 0, 1), (1, 0, 0)), 1),
(((1, 0, 0), (1, 0, 0)), -3)]
"""
return sorted(self.__monomials.items(),
return sorted(self._monomial_coefficients.items(),
key=lambda x: (-sum(x[0][1]), x[0][1], -sum(x[0][0]), x[0][0]))

def support(self) -> list:
"""
Return the support of ``self``.
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: elt = dy - (3*x - z)*dx + 1
sage: sorted(elt.support())
[((0, 0, 0), (0, 0, 0)),
((0, 0, 0), (0, 1, 0)),
((0, 0, 1), (1, 0, 0)),
((1, 0, 0), (1, 0, 0))]
"""
return list(self.__monomials)

# This is essentially copied from
# sage.combinat.free_module.CombinatorialFreeModuleElement
def __truediv__(self, x):
Expand All @@ -567,7 +509,7 @@ def __truediv__(self, x):
2*y*z + x
"""
F = self.parent()
D = self.__monomials
D = self._monomial_coefficients
if F.base_ring().is_field():
x = F.base_ring()(x)
x_inv = x**-1
Expand Down

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