def __add__(self, other):
"""
Return the sum of two subgroups.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup()
sage: C.gens()
[[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
sage: A + B == C
True
An example where the parent abelian varieties are different::
A = J0(48); A[0].cuspidal_subgroup() + A[1].cuspidal_subgroup()
Finite subgroup with invariants [2, 4, 4] over QQ of Abelian subvariety of dimension 2 of J0(48)
"""
if not isinstance(other, FiniteSubgroup):
raise TypeError("only addition of two finite subgroups is defined")
A = self.abelian_variety()
B = other.abelian_variety()
if not A.in_same_ambient_variety(B):
raise ValueError("self and other must be in the same ambient Jacobian")
K = coercion_model.common_parent(self.field_of_definition(), other.field_of_definition())
lattice = self.lattice() + other.lattice()
if A != B:
C = A + B
lattice += C.lattice()
return FiniteSubgroup_lattice(C, lattice, field_of_definition=K)
else:
return FiniteSubgroup_lattice(self.abelian_variety(), lattice, field_of_definition=K)
def __add__(self, other):
"""
Return the sum of two subgroups.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup()
sage: C.gens()
[[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
sage: A + B == C
True
"""
if not isinstance(other, FiniteSubgroup):
raise TypeError("only addition of two finite subgroups is defined")
A = self.abelian_variety()
B = other.abelian_variety()
if not A.in_same_ambient_variety(B):
raise ValueError("self and other must be in the same ambient Jacobian")
K = coercion_model.common_parent(self.field_of_definition(), other.field_of_definition())
lattice = self.lattice() + other.lattice()
if A != B:
lattice += C.lattice()
return FiniteSubgroup_lattice(self.abelian_variety(), lattice, field_of_definition=K)
def __add__(self, other):
"""
Return the sum of two subgroups.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup()
sage: C.gens()
[[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
sage: A + B == C
True
"""
if not isinstance(other, FiniteSubgroup):
raise TypeError("only addition of two finite subgroups is defined")
A = self.abelian_variety()
B = other.abelian_variety()
if not A.in_same_ambient_variety(B):
raise ValueError(
"self and other must be in the same ambient Jacobian")
K = coercion_model.common_parent(self.field_of_definition(),
other.field_of_definition())
lattice = self.lattice() + other.lattice()
if A != B:
lattice += C.lattice()
return FiniteSubgroup_lattice(self.abelian_variety(),
lattice,
field_of_definition=K)
def cup_product(self, cochain):
"""
Return the cup product with another cochain
INPUT:
- ``cochain`` -- cochain over the same cell complex
EXAMPLES::
sage: T2 = simplicial_complexes.Torus()
sage: C1 = T2.n_chains(1, base_ring=ZZ, cochains=True)
sage: def l(i, j):
....: return C1(Simplex([i, j]))
sage: l1 = l(1, 3) + l(1, 4) + l(1, 6) + l(2, 4) - l(4, 5) + l(5, 6)
sage: l2 = l(1, 6) - l(2, 3) - l(2, 5) + l(3, 6) - l(4, 5) + l(5, 6)
The two one-cocycles are cohomology generators::
sage: l1.is_cocycle(), l1.is_coboundary()
(True, False)
sage: l2.is_cocycle(), l2.is_coboundary()
(True, False)
Their cup product is a two-cocycle that is again non-trivial in
cohomology::
sage: l12 = l1.cup_product(l2)
sage: l12
\chi_(1, 3, 6) - \chi_(2, 4, 5) - \chi_(4, 5, 6)
sage: l1.parent().degree(), l2.parent().degree(), l12.parent().degree()
(1, 1, 2)
sage: l12.is_cocycle(), l12.is_coboundary()
(True, False)
"""
if not isinstance(cochain.parent(), Cochains):
raise ValueError('argument must be a cochain')
if cochain.parent().cell_complex() != self.parent().cell_complex():
raise ValueError('cochain must be over the same cell complex')
left_deg = self.parent().degree()
right_deg = cochain.parent().degree()
left_chains = self.parent().dual()
right_chains = cochain.parent().dual()
base_ring = coercion_model.common_parent(left_chains.base_ring(),
right_chains.base_ring())
cx = self.parent().cell_complex()
codomain = cx.n_chains(left_deg + right_deg,
base_ring=base_ring,
cochains=True)
accumulator = codomain.zero()
for cell in codomain.indices():
for (coeff, left_cell,
right_cell) in cx.alexander_whitney(cell, left_deg):
if not coeff:
continue
left = left_chains(left_cell)
right = right_chains(right_cell)
accumulator += codomain(cell) * coeff * self.eval(
left) * cochain.eval(right)
return accumulator
def extend_scalars(self, *pts):
r"""
Extend the ground field so that the new field contains pts.
"""
Dops = self.parent()
Pols = Dops.base_ring()
Scalars = Pols.base_ring()
if all(Scalars.has_coerce_map_from(pt.parent()) for pt in pts):
return (self, ) + pts
gen = Scalars.gen()
try:
# Largely redundant with the other branch, but may do a better job
# in some cases, e.g. pushout(QQ, QQ(α)), where as_enf_elts() would
# invent new generator names.
NF0 = coercion_model.common_parent(Scalars, *pts)
if not is_NumberField(NF0):
raise CoercionException
NF, hom = utilities.good_number_field(NF0)
gen1 = hom(NF0.coerce(gen))
pts1 = tuple(hom(NF0.coerce(pt)) for pt in pts)
except (CoercionException, TypeError):
NF, val1 = as_embedded_number_field_elements((gen, ) + pts)
gen1, pts1 = val1[0], tuple(val1[1:])
hom = Scalars.hom([gen1], codomain=NF)
Dops1 = OreAlgebra(Pols.change_ring(NF), (Dops.variable_name(), {}, {
Pols.gen(): Pols.one()
}))
dop1 = Dops1([pol.map_coefficients(hom) for pol in self])
dop1 = PlainDifferentialOperator(dop1)
assert dop1.base_ring().base_ring() is NF
return (dop1, ) + pts1
def cup_product(self, cochain):
"""
Return the cup product with another cochain
INPUT:
- ``cochain`` -- cochain over the same cell complex
EXAMPLES::
sage: T2 = simplicial_complexes.Torus()
sage: C1 = T2.n_chains(1, base_ring=ZZ, cochains=True)
sage: def l(i, j):
....: return C1(Simplex([i, j]))
sage: l1 = l(1, 3) + l(1, 4) + l(1, 6) + l(2, 4) - l(4, 5) + l(5, 6)
sage: l2 = l(1, 6) - l(2, 3) - l(2, 5) + l(3, 6) - l(4, 5) + l(5, 6)
The two one-cocycles are cohomology generators::
sage: l1.is_cocycle(), l1.is_coboundary()
(True, False)
sage: l2.is_cocycle(), l2.is_coboundary()
(True, False)
Their cup product is a two-cocycle that is again non-trivial in
cohomology::
sage: l12 = l1.cup_product(l2)
sage: l12
\chi_(1, 3, 6) - \chi_(2, 4, 5) - \chi_(4, 5, 6)
sage: l1.parent().degree(), l2.parent().degree(), l12.parent().degree()
(1, 1, 2)
sage: l12.is_cocycle(), l12.is_coboundary()
(True, False)
"""
if not isinstance(cochain.parent(), Cochains):
raise ValueError('argument must be a cochain')
if cochain.parent().cell_complex() != self.parent().cell_complex():
raise ValueError('cochain must be over the same cell complex')
left_deg = self.parent().degree()
right_deg = cochain.parent().degree()
left_chains = self.parent().dual()
right_chains = cochain.parent().dual()
base_ring = coercion_model.common_parent(
left_chains.base_ring(), right_chains.base_ring())
cx = self.parent().cell_complex()
codomain = cx.n_chains(
left_deg + right_deg, base_ring=base_ring, cochains=True)
accumulator = codomain.zero()
for cell in codomain.indices():
for (coeff, left_cell, right_cell) in cx.alexander_whitney(cell, left_deg):
if not coeff:
continue
left = left_chains(left_cell)
right = right_chains(right_cell)
accumulator += codomain(cell) * coeff * self.eval(left) * cochain.eval(right)
return accumulator
def eval(self, other):
"""
Evaluate this cochain on the chain ``other``.
INPUT:
- ``other`` -- a chain for the same cell complex in the
same dimension with the same base ring
OUTPUT: scalar
EXAMPLES::
sage: S2 = simplicial_complexes.Sphere(2)
sage: C_2 = S2.n_chains(1)
sage: C_2_co = S2.n_chains(1, cochains=True)
sage: x = C_2.basis()[Simplex((0,2))]
sage: y = C_2.basis()[Simplex((1,3))]
sage: z = x+2*y
sage: a = C_2_co.basis()[Simplex((1,3))]
sage: b = C_2_co.basis()[Simplex((0,3))]
sage: c = 3*a-2*b
sage: z
(0, 2) + 2*(1, 3)
sage: c
-2*\chi_(0, 3) + 3*\chi_(1, 3)
sage: c.eval(z)
6
TESTS::
sage: z.eval(c) # z is not a cochain
Traceback (most recent call last):
...
AttributeError: 'Chains_with_category.element_class' object has no attribute 'eval'
sage: c.eval(c) # can't evaluate a cochain on a cochain
Traceback (most recent call last):
...
ValueError: argument is not a chain
"""
if not isinstance(other.parent(), Chains):
raise ValueError('argument is not a chain')
if other.parent().indices() != self.parent().indices():
raise ValueError('the cells are not compatible')
result = sum(coeff * other.coefficient(cell)
for cell, coeff in self)
R = self.base_ring()
if R != other.base_ring():
R = coercion_model.common_parent(R, other.base_ring())
return R(result)
def eval(self, other):
"""
Evaluate this cochain on the chain ``other``.
INPUT:
- ``other`` -- a chain for the same cell complex in the
same dimension with the same base ring
OUTPUT: scalar
EXAMPLES::
sage: S2 = simplicial_complexes.Sphere(2)
sage: C_2 = S2.n_chains(1)
sage: C_2_co = S2.n_chains(1, cochains=True)
sage: x = C_2.basis()[Simplex((0,2))]
sage: y = C_2.basis()[Simplex((1,3))]
sage: z = x+2*y
sage: a = C_2_co.basis()[Simplex((1,3))]
sage: b = C_2_co.basis()[Simplex((0,3))]
sage: c = 3*a-2*b
sage: z
(0, 2) + 2*(1, 3)
sage: c
-2*\chi_(0, 3) + 3*\chi_(1, 3)
sage: c.eval(z)
6
TESTS::
sage: z.eval(c) # z is not a cochain
Traceback (most recent call last):
...
AttributeError: 'Chains_with_category.element_class' object has no attribute 'eval'
sage: c.eval(c) # can't evaluate a cochain on a cochain
Traceback (most recent call last):
...
ValueError: argument is not a chain
"""
if not isinstance(other.parent(), Chains):
raise ValueError('argument is not a chain')
if other.parent().indices() != self.parent().indices():
raise ValueError('the cells are not compatible')
result = sum(coeff * other.coefficient(cell)
for cell, coeff in self)
R = self.base_ring()
if R != other.base_ring():
R = coercion_model.common_parent(R, other.base_ring())
return R(result)
def _from_polymake_polytope(cls, parent, polymake_polytope):
r"""
Initializes a polyhedron from a polymake Polytope object.
TESTS::
sage: from sage.geometry.polyhedron.backend_polymake import Polyhedron_polymake
sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: P=Polyhedron(ieqs=[[1, 0, 2], [3, 0, -2], [3, 2, -2]])
sage: PP = polymake(P) # optional - polymake
sage: parent = Polyhedra(QQ, 2, backend='polymake') # optional - polymake
sage: Q=Polyhedron_polymake._from_polymake_polytope(parent, PP) # optional - polymake
"""
if parent is None:
from .parent import Polyhedra
from sage.rings.rational_field import QQ
from sage.rings.qqbar import AA
if polymake_polytope.typeof(
)[0] == 'Polymake::polytope::Polytope__Rational':
base_ring = QQ
else:
from sage.structure.element import coercion_model
data = [
g.sage()
for g in itertools.chain(polymake_polytope.VERTICES,
polymake_polytope.LINEALITY_SPACE,
polymake_polytope.FACETS,
polymake_polytope.AFFINE_HULL)
]
if data:
base_ring = coercion_model.common_parent(*data).base_ring()
else:
base_ring = QQ
ambient_dim = polymake_polytope.AMBIENT_DIM().sage()
parent = Polyhedra(base_ring=base_ring,
ambient_dim=ambient_dim,
backend='polymake')
return cls(parent, None, None, polymake_polytope=polymake_polytope)
def __add__(self, other):
"""
Return the sum of two subgroups.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup()
sage: C.gens()
[[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
sage: A + B == C
True
An example where the parent abelian varieties are different::
A = J0(48); A[0].cuspidal_subgroup() + A[1].cuspidal_subgroup()
Finite subgroup with invariants [2, 4, 4] over QQ of Abelian subvariety of dimension 2 of J0(48)
"""
if not isinstance(other, FiniteSubgroup):
raise TypeError("only addition of two finite subgroups is defined")
A = self.abelian_variety()
B = other.abelian_variety()
if not A.in_same_ambient_variety(B):
raise ValueError(
"self and other must be in the same ambient Jacobian")
K = coercion_model.common_parent(self.field_of_definition(),
other.field_of_definition())
lattice = self.lattice() + other.lattice()
if A != B:
C = A + B
lattice += C.lattice()
return FiniteSubgroup_lattice(C, lattice, field_of_definition=K)
else:
return FiniteSubgroup_lattice(self.abelian_variety(),
lattice,
field_of_definition=K)
def intersection(self, other):
"""
Return the intersection of the finite subgroups self and other.
INPUT:
- ``other`` - a finite group
OUTPUT: a finite group
EXAMPLES::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
sage: G.intersection(H)
Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: W = E11a1.torsion_subgroup(15)
sage: G.intersection(W)
Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect subgroups of different abelian varieties.
::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
sage: G.intersection(H)
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect abelian varieties with subgroups::
sage: t = J0(33).hecke_operator(7)
sage: G = t.kernel()[0]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
sage: A = J0(33).old_subvariety()
sage: A.intersection(G)
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: A.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: B = J0(33).new_subvariety()
sage: B.intersection(G)
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: B.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: A.intersection(B)[0]
Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33)
"""
from .abvar import is_ModularAbelianVariety
A = self.abelian_variety()
if is_ModularAbelianVariety(other):
amb = other
B = other
M = B.lattice().scale(Integer(1)/self.exponent())
K = coercion_model.common_parent(self.field_of_definition(), other.base_field())
else:
amb = A
if not isinstance(other, FiniteSubgroup):
raise TypeError("only intersection with a finite subgroup or "
"modular abelian variety is defined")
B = other.abelian_variety()
if A.ambient_variety() != B.ambient_variety():
raise TypeError("finite subgroups must be in the same ambient product Jacobian")
M = other.lattice()
K = coercion_model.common_parent(self.field_of_definition(), other.field_of_definition())
L = self.lattice()
if A != B:
# TODO: This might be way slower than what we could do if
# we think more carefully.
C = A + B
L = L + C.lattice()
M = M + C.lattice()
W = L.intersection(M).intersection(amb.vector_space())
return FiniteSubgroup_lattice(amb, W, field_of_definition=K)
def intersection(self, other):
"""
Return the intersection of the finite subgroups self and other.
INPUT:
- ``other`` - a finite group
OUTPUT: a finite group
EXAMPLES::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
sage: G.intersection(H)
Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: W = E11a1.torsion_subgroup(15)
sage: G.intersection(W)
Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect subgroups of different abelian varieties.
::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
sage: G.intersection(H)
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect abelian varieties with subgroups::
sage: t = J0(33).hecke_operator(7)
sage: G = t.kernel()[0]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
sage: A = J0(33).old_subvariety()
sage: A.intersection(G)
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: A.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: B = J0(33).new_subvariety()
sage: B.intersection(G)
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: B.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: A.intersection(B)[0]
Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33)
"""
from .abvar import is_ModularAbelianVariety
A = self.abelian_variety()
if is_ModularAbelianVariety(other):
amb = other
B = other
M = B.lattice().scale(Integer(1) / self.exponent())
K = coercion_model.common_parent(self.field_of_definition(),
other.base_field())
else:
amb = A
if not isinstance(other, FiniteSubgroup):
raise TypeError("only intersection with a finite subgroup or "
"modular abelian variety is defined")
B = other.abelian_variety()
if A.ambient_variety() != B.ambient_variety():
raise TypeError(
"finite subgroups must be in the same ambient product Jacobian"
)
M = other.lattice()
K = coercion_model.common_parent(self.field_of_definition(),
other.field_of_definition())
L = self.lattice()
if A != B:
# TODO: This might be way slower than what we could do if
# we think more carefully.
C = A + B
L = L + C.lattice()
M = M + C.lattice()
W = L.intersection(M).intersection(amb.vector_space())
return FiniteSubgroup_lattice(amb, W, field_of_definition=K)
def __call__(self, *values):
"""
Replace the generators of the free group by corresponding
elements of the iterable ``values`` in the group element
``self``.
INPUT:
- ``*values`` -- a sequence of values, or a
list/tuple/iterable of the same length as the number of
generators of the free group.
OUTPUT:
The product of ``values`` in the order and with exponents
specified by ``self``.
EXAMPLES::
sage: G.<a,b> = FreeGroup()
sage: w = a^2 * b^-1 * a^3
sage: w(1, 2)
1/2
sage: w(2, 1)
32
sage: w.subs(b=1, a=2) # indirect doctest
32
TESTS::
sage: w([1, 2])
1/2
sage: w((1, 2))
1/2
sage: w(i+1 for i in range(2))
1/2
Check that :trac:`25017` is fixed::
sage: F = FreeGroup(2)
sage: x0, x1 = F.gens()
sage: u = F(1)
sage: parent(u.subs({x1:x0})) is F
True
sage: F = FreeGroup(2)
sage: x0, x1 = F.gens()
sage: u = x0*x1
sage: u.subs({x0:3, x1:2})
6
sage: u.subs({x0:1r, x1:2r})
2
sage: M0 = matrix(ZZ,2,[1,1,0,1])
sage: M1 = matrix(ZZ,2,[1,0,1,1])
sage: u.subs({x0: M0, x1: M1})
[2 1]
[1 1]
TESTS::
sage: F.<x,y> = FreeGroup()
sage: F.one().subs(x=x, y=1)
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents: 'Free Group on generators {x, y}' and 'Integer Ring'
"""
if len(values) == 1:
try:
values = list(values[0])
except TypeError:
pass
G = self.parent()
if len(values) != G.ngens():
raise ValueError('number of values has to match the number of generators')
replace = dict(zip(G.gens(), values))
new_parent = coercion_model.common_parent(*[parent(v) for v in values])
try:
return new_parent.prod(replace[gen] ** power
for gen, power in self.syllables())
except AttributeError:
return prod(new_parent(replace[gen]) ** power
for gen, power in self.syllables())
def __call__(self, *values):
"""
Replace the generators of the free group by corresponding
elements of the iterable ``values`` in the group element
``self``.
INPUT:
- ``*values`` -- a sequence of values, or a
list/tuple/iterable of the same length as the number of
generators of the free group.
OUTPUT:
The product of ``values`` in the order and with exponents
specified by ``self``.
EXAMPLES::
sage: G.<a,b> = FreeGroup()
sage: w = a^2 * b^-1 * a^3
sage: w(1, 2)
1/2
sage: w(2, 1)
32
sage: w.subs(b=1, a=2) # indirect doctest
32
TESTS::
sage: w([1, 2])
1/2
sage: w((1, 2))
1/2
sage: w(i+1 for i in range(2))
1/2
Check that :trac:`25017` is fixed::
sage: F = FreeGroup(2)
sage: x0, x1 = F.gens()
sage: u = F(1)
sage: parent(u.subs({x1:x0})) is F
True
sage: F = FreeGroup(2)
sage: x0, x1 = F.gens()
sage: u = x0*x1
sage: u.subs({x0:3, x1:2})
6
sage: u.subs({x0:1r, x1:2r})
2
sage: M0 = matrix(ZZ,2,[1,1,0,1])
sage: M1 = matrix(ZZ,2,[1,0,1,1])
sage: u.subs({x0: M0, x1: M1})
[2 1]
[1 1]
TESTS::
sage: F.<x,y> = FreeGroup()
sage: F.one().subs(x=x, y=1)
Traceback (most recent call last):
...
TypeError: no common canonical parent for objects with parents: 'Free Group on generators {x, y}' and 'Integer Ring'
"""
if len(values) == 1:
try:
values = list(values[0])
except TypeError:
pass
G = self.parent()
if len(values) != G.ngens():
raise ValueError(
'number of values has to match the number of generators')
replace = dict(zip(G.gens(), values))
new_parent = coercion_model.common_parent(*[parent(v) for v in values])
try:
return new_parent.prod(replace[gen]**power
for gen, power in self.syllables())
except AttributeError:
return prod(
new_parent(replace[gen])**power
for gen, power in self.syllables())
