def __init__(self, N, P, p, e):
self._N = N
self._F = P.number_field()
self._p = p
from sage.rings.all import Integers
f = e // 2 + 1
assert f * 2 - 1 == e
R0 = Integers(p**f)
R1 = Integers(p**(f - 1))
self._ring = RamifiedProductRing(R0, R1)
self._im_gen = self._ring.gen()
self._sqrt5 = (self._F.gen() * 2 - 1)**2
def _delta_poly_modulo(N, prec=10):
"""
Return the q-expansion of `\Delta` modulo `N`. Used internally by
the :func:`~delta_qexp` function. See the docstring of :func:`~delta_qexp`
for more information.
INPUT:
- `N` -- positive integer modulo which we want to compute `\Delta`
- ``prec`` -- integer; the absolute precision of the output
OUTPUT:
the polynomial of degree ``prec``-1 which is the truncation
of `\Delta` modulo `N`, as an element of the polynomial
ring in `q` over the integers modulo `N`.
EXAMPLES::
sage: from sage.modular.modform.vm_basis import _delta_poly_modulo
sage: _delta_poly_modulo(5, 7)
2*q^6 + 3*q^4 + 2*q^3 + q^2 + q
sage: _delta_poly_modulo(10, 12)
2*q^11 + 7*q^9 + 6*q^7 + 2*q^6 + 8*q^4 + 2*q^3 + 6*q^2 + q
"""
if prec <= 0:
raise ValueError("prec must be positive")
v = [0] * prec
# Let F = \sum_{n >= 0} (-1)^n (2n+1) q^(floor(n(n+1)/2)).
# Then delta is F^8.
stop = int((-1 + math.sqrt(8 * prec)) / 2.0)
for n in xrange(stop + 1):
v[n * (n + 1) // 2] = ((N - 1) * (2 * n + 1) if (n & 1) else
(2 * n + 1))
from sage.rings.all import Integers
P = PolynomialRing(Integers(N), 'q')
f = P(v)
t = verbose('made series')
# fast way of computing f*f truncated at prec
f = f._mul_trunc_(f, prec)
t = verbose('squared (1 of 3)', t)
f = f._mul_trunc_(f, prec)
t = verbose('squared (2 of 3)', t)
f = f._mul_trunc_(f, prec - 1)
t = verbose('squared (3 of 3)', t)
f = f.shift(1)
t = verbose('shifted', t)
return f
def __init__(self, N, P, p, e):
self._N = N
self._F = P.number_field()
self._p = p
from sage.rings.all import Integers
assert e % 2 == 0
R = Integers(p**(e // 2))
modulus = self._F.defining_polynomial().change_ring(R)
S = R['x']
self._ring = S.quotient_by_principal_ideal(modulus)
self._im_gen = self._ring.gen()
def solve_mod(eqns, modulus, solution_dict=False):
r"""
Return all solutions to an equation or list of equations modulo the
given integer modulus. Each equation must involve only polynomials
in 1 or many variables.
By default the solutions are returned as `n`-tuples, where `n`
is the number of variables appearing anywhere in the given
equations. The variables are in alphabetical order.
INPUT:
- ``eqns`` - equation or list of equations
- ``modulus`` - an integer
- ``solution_dict`` - bool (default: False); if True or non-zero,
return a list of dictionaries containing the solutions. If there
are no solutions, return an empty list (rather than a list containing
an empty dictionary). Likewise, if there's only a single solution,
return a list containing one dictionary with that solution.
EXAMPLES::
sage: var('x,y')
(x, y)
sage: solve_mod([x^2 + 2 == x, x^2 + y == y^2], 14)
[(4, 2), (4, 6), (4, 9), (4, 13)]
sage: solve_mod([x^2 == 1, 4*x == 11], 15)
[(14,)]
Fermat's equation modulo 3 with exponent 5::
sage: var('x,y,z')
(x, y, z)
sage: solve_mod([x^5 + y^5 == z^5], 3)
[(0, 0, 0), (0, 1, 1), (0, 2, 2), (1, 0, 1), (1, 1, 2), (1, 2, 0), (2, 0, 2), (2, 1, 0), (2, 2, 1)]
We can solve with respect to a bigger modulus if it consists only of small prime factors::
sage: [d] = solve_mod([5*x + y == 3, 2*x - 3*y == 9], 3*5*7*11*19*23*29, solution_dict = True)
sage: d[x]
12915279
sage: d[y]
8610183
For cases where there are relatively few solutions and the prime
factors are small, this can be efficient even if the modulus itself
is large::
sage: sorted(solve_mod([x^2 == 41], 10^20))
[(4538602480526452429,), (11445932736758703821,), (38554067263241296179,),
(45461397519473547571,), (54538602480526452429,), (61445932736758703821,),
(88554067263241296179,), (95461397519473547571,)]
We solve a simple equation modulo 2::
sage: x,y = var('x,y')
sage: solve_mod([x == y], 2)
[(0, 0), (1, 1)]
.. warning::
The current implementation splits the modulus into prime
powers, then naively enumerates all possible solutions
(starting modulo primes and then working up through prime
powers), and finally combines the solution using the Chinese
Remainder Theorem. The interface is good, but the algorithm is
very inefficient if the modulus has some larger prime factors! Sage
*does* have the ability to do something much faster in certain
cases at least by using Groebner basis, linear algebra
techniques, etc. But for a lot of toy problems this function as
is might be useful. At least it establishes an interface.
TESTS:
Make sure that we short-circuit in at least some cases::
sage: solve_mod([2*x==1], 2*next_prime(10^50))
[]
Try multi-equation cases::
sage: x, y, z = var("x y z")
sage: solve_mod([2*x^2 + x*y, -x*y+2*y^2+x-2*y, -2*x^2+2*x*y-y^2-x-y], 12)
[(0, 0), (4, 4), (0, 3), (4, 7)]
sage: eqs = [-y^2+z^2, -x^2+y^2-3*z^2-z-1, -y*z-z^2-x-y+2, -x^2-12*z^2-y+z]
sage: solve_mod(eqs, 11)
[(8, 5, 6)]
Confirm that modulus 1 now behaves as it should::
sage: x, y = var("x y")
sage: solve_mod([x==1], 1)
[(0,)]
sage: solve_mod([2*x^2+x*y, -x*y+2*y^2+x-2*y, -2*x^2+2*x*y-y^2-x-y], 1)
[(0, 0)]
"""
from sage.rings.all import Integer, Integers, crt_basis
from sage.symbolic.expression import is_Expression
from sage.misc.all import cartesian_product_iterator
from sage.modules.all import vector
from sage.matrix.all import matrix
if not isinstance(eqns, (list, tuple)):
eqns = [eqns]
eqns = [eq if is_Expression(eq) else (eq.lhs() - eq.rhs()) for eq in eqns]
modulus = Integer(modulus)
if modulus < 1:
raise ValueError("the modulus must be a positive integer")
vars = list(set(sum([list(e.variables()) for e in eqns], [])))
vars.sort(key=repr)
if modulus == 1: # degenerate case
ans = [tuple(Integers(1)(0) for v in vars)]
return ans
factors = modulus.factor()
crt_basis = vector(Integers(modulus),
crt_basis([p**i for p, i in factors]))
solutions = []
has_solution = True
for p, i in factors:
solution = _solve_mod_prime_power(eqns, p, i, vars)
if len(solution) > 0:
solutions.append(solution)
else:
has_solution = False
break
ans = []
if has_solution:
for solution in cartesian_product_iterator(solutions):
solution_mat = matrix(Integers(modulus), solution)
ans.append(
tuple(
c.dot_product(crt_basis) for c in solution_mat.columns()))
# if solution_dict == True:
# Relaxed form suggested by Mike Hansen (#8553):
if solution_dict:
sol_dict = [dict(zip(vars, solution)) for solution in ans]
return sol_dict
else:
return ans
def _solve_mod_prime_power(eqns, p, m, vars):
r"""
Internal help function for solve_mod, does little checking since it expects
solve_mod to do that
Return all solutions to an equation or list of equations modulo p^m.
Each equation must involve only polynomials
in 1 or many variables.
The solutions are returned as `n`-tuples, where `n`
is the number of variables in vars.
INPUT:
- ``eqns`` - equation or list of equations
- ``p`` - a prime
- ``i`` - an integer > 0
- ``vars`` - a list of variables to solve for
EXAMPLES::
sage: var('x,y')
(x, y)
sage: solve_mod([x^2 + 2 == x, x^2 + y == y^2], 14)
[(4, 2), (4, 6), (4, 9), (4, 13)]
sage: solve_mod([x^2 == 1, 4*x == 11], 15)
[(14,)]
Fermat's equation modulo 3 with exponent 5::
sage: var('x,y,z')
(x, y, z)
sage: solve_mod([x^5 + y^5 == z^5], 3)
[(0, 0, 0), (0, 1, 1), (0, 2, 2), (1, 0, 1), (1, 1, 2), (1, 2, 0), (2, 0, 2), (2, 1, 0), (2, 2, 1)]
We solve a simple equation modulo 2::
sage: x,y = var('x,y')
sage: solve_mod([x == y], 2)
[(0, 0), (1, 1)]
.. warning::
Currently this constructs possible solutions by building up
from the smallest prime factor of the modulus. The interface
is good, but the algorithm is horrible if the modulus isn't the
product of many small primes! Sage *does* have the ability to
do something much faster in certain cases at least by using the
Chinese Remainder Theorem, Groebner basis, linear algebra
techniques, etc. But for a lot of toy problems this function as
is might be useful. At the very least, it establishes an
interface.
TESTS:
Confirm we can reproduce the first few terms of :oeis:`A187719`::
sage: from sage.symbolic.relation import _solve_mod_prime_power
sage: [sorted(_solve_mod_prime_power([x^2==41], 10, i, [x]))[0][0] for i in [1..13]]
[1, 21, 71, 1179, 2429, 47571, 1296179, 8703821, 26452429, 526452429,
13241296179, 19473547571, 2263241296179]
"""
from sage.rings.all import Integers, PolynomialRing
from sage.modules.all import vector
from sage.misc.all import cartesian_product_iterator
mrunning = 1
ans = []
for mi in range(m):
mrunning *= p
R = Integers(mrunning)
S = PolynomialRing(R, len(vars), vars)
eqns_mod = [S(eq) for eq in eqns]
if mi == 0:
possibles = cartesian_product_iterator(
[range(len(R)) for _ in range(len(vars))])
else:
shifts = cartesian_product_iterator(
[range(p) for _ in range(len(vars))])
pairs = cartesian_product_iterator([shifts, ans])
possibles = (tuple(vector(t) + vector(shift) * (mrunning // p))
for shift, t in pairs)
ans = list(t for t in possibles if all(e(*t) == 0 for e in eqns_mod))
if not ans: return ans
return ans
def __startup__():
import sage.algebras.steenrod.steenrod_algebra
import types
def resolution(self, memory=None, filename=None):
"""
The minimal resolution of the ground field as a module over 'self'.
TESTS::
sage: from yacop.resolutions.minres import MinimalResolution
sage: MinimalResolution(SteenrodAlgebra(7)) is SteenrodAlgebra(7).resolution()
True
"""
from yacop.resolutions.minres import MinimalResolution
return MinimalResolution(self, memory=memory, filename=filename)
setattr(
sage.algebras.steenrod.steenrod_algebra.SteenrodAlgebra_generic,
"resolution",
resolution,
)
def Ext(algebra, M, N=None, filename=None):
"""
``Ext(M,N)`` over ``algebra``.
"""
from yacop.resolutions.smashres import SmashResolution
assert N is None
return SmashResolution(M, algebra.resolution(),
filename=filename).Homology()
setattr(sage.algebras.steenrod.steenrod_algebra.SteenrodAlgebra_generic,
"Ext", Ext)
# there are known issues in Sage with pickling of morphisms
# and we don't want to document the same failures in our test
# suite.
from sage.combinat.free_module import CombinatorialFreeModule
def p_test_pickling(self, tester=None, **options):
if not hasattr(self,
"_can_test_pickling") or self._can_test_pickling():
tester = self._tester(**options)
from sage.misc.all import loads, dumps
tester.assertEqual(loads(dumps(self)), self)
else:
tester.info(" (skipped, not picklable) ", newline=False)
def e_test_pickling(self, tester=None, **options):
if not hasattr(self,
"_can_test_pickling") or self._can_test_pickling():
tester = self._tester(**options)
from sage.misc.all import loads, dumps
tester.assertEqual(loads(dumps(self)), self)
else:
tester.info(" (skipped, not picklable) ", newline=False)
CombinatorialFreeModule._test_pickling = p_test_pickling
# CombinatorialFreeModule.Element._test_pickling = e_test_pickling (no longer possible, by ticket/22632)
# workaround for Sage Ticket #13814
from sage.sets.family import LazyFamily
from sage.rings.all import Integers
if LazyFamily(Integers(),
lambda i: 2 * i) == LazyFamily(Integers(), lambda i: 2 * i):
def noteq(self, other):
if self is other:
return True
return False
LazyFamily.__eq__ = noteq
try:
16 in LazyFamily(Integers(), lambda i: 2 * i)
except:
# we need to overwrite _contains_ in our module base to fix
# the test suite of SteenrodModuleBase.basis()
def __contains__(self, x):
try:
return self._contains_(x)
except AttributeError:
return super(LazyFamily, self).__contains__(x)
LazyFamily.__contains__ = __contains__
# workaround for #13811
from sage.sets.family import AbstractFamily
def __copy__(self):
return self
AbstractFamily.__copy__ = __copy__
# LazyFamilies cannot be pickled... turn off the resulting noise
def _test_pickling(self, tester=None, **options):
pass
LazyFamily._test_pickling = _test_pickling
# workaround for Sage ticket #13833
from sage.algebras.steenrod.steenrod_algebra import SteenrodAlgebra
B = SteenrodAlgebra(2, profile=(3, 2, 1))
A = SteenrodAlgebra(2)
id = A.module_morphism(codomain=A, on_basis=lambda x: A.monomial(x))
try:
x = id(B.an_element())
except AssertionError:
from sage.categories.modules_with_basis import ModuleMorphismByLinearity
origcall = ModuleMorphismByLinearity.__call__
def call(self, *args):
before = args[0:self._position]
after = args[self._position + 1:len(args)]
x = args[self._position]
nargs = before + (self.domain()(x), ) + after
return origcall(self, *nargs)
ModuleMorphismByLinearity.__call__ = call
# workaround for Sage ticket #18449
from sage.sets.set_from_iterator import EnumeratedSetFromIterator
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
from sage.categories.enumerated_sets import EnumeratedSets
from sage.rings.integer_ring import ZZ
C = CombinatorialFreeModule(ZZ, EnumeratedSetFromIterator(Integers))
if False and sage.categories.tensor.tensor(
(C, )).basis() in FiniteEnumeratedSets():
def __init_18449__(self, set, function, name=None):
"""
patched __init__ function from ticket #18449
"""
from sage.combinat.combinat import CombinatorialClass # workaround #12482
from sage.structure.parent import Parent
category = EnumeratedSets()
if set in FiniteEnumeratedSets():
category = FiniteEnumeratedSets()
elif set in InfiniteEnumeratedSets():
category = InfiniteEnumeratedSets()
elif isinstance(set, (list, tuple)):
category = FiniteEnumeratedSets()
elif isinstance(set, CombinatorialClass):
try:
if set.is_finite():
category = FiniteEnumeratedSets()
except NotImplementedError:
pass
Parent.__init__(self, category=category)
from copy import copy
self.set = copy(set)
self.function = function
self.function_name = name
LazyFamily.__init__ = __init_18449__
# use a lower max_runs value for the TestSuite
from sage.misc.sage_unittest import InstanceTester
InstanceTester.__init_original__ = InstanceTester.__init__
def newinit(self, *args, **kwds):
self.__init_original__(*args, **kwds)
self._max_runs = 40
InstanceTester.__init__ = newinit
# fix a problem for __contain__ in Categories_over_base_ring
# when more than one base is involved:
from sage.categories.category import Category
from sage.categories.category_types import Category_over_base_ring
from sage.categories.modules import Modules
from sage.categories.algebras import Algebras
from sage.categories.modules_with_basis import ModulesWithBasis
from sage.rings.finite_rings.finite_field_constructor import GF
C = CombinatorialFreeModule(GF(5),
ZZ,
category=(ModulesWithBasis(GF(5)),
Algebras(ZZ)))
if C not in Modules(ZZ):
_sagecode = Category_over_base_ring.__contains__
Category_over_base_ring.__contains_sage__ = _sagecode
def __contains_yacop__(self, Z):
ans = self.__contains_sage__(Z)
if not ans:
try:
ans = self in Z.categories()
except:
pass
return ans
Category_over_base_ring.__contains__ = __contains_yacop__
# Sage insists that Subquotients of CartesianProducts are again CartesianProducts
# (and similarly for TensorProducts). We forcefully disagree:
from sage.categories.covariant_functorial_construction import RegressiveCovariantConstructionCategory
@classmethod
def default_super_categories_yacop(cls, category, *args):
"""
TESTS::
sage: import yacop
sage: # an earlier version of this hack broke the MRO for quasi symmetric functions:
sage: QuasiSymmetricFunctions(GF(3))
Quasisymmetric functions over the Finite Field of size 3
"""
sageresult = Category.join([
category,
super(RegressiveCovariantConstructionCategory,
cls).default_super_categories(category, *args)
])
ans = sageresult
if isinstance(ans, sage.categories.category.JoinCategory):
j = [
cat for cat in sageresult.super_categories()
if not hasattr(cat, "yacop_no_default_inheritance")
]
ans = Category.join(j)
return ans
RegressiveCovariantConstructionCategory.default_super_categories = default_super_categories_yacop