def direct_sum(self, M):
r"""
Return the direct sum of this lattice with ``M``.
INPUT:
- ``M`` -- a module over `\ZZ`
EXAMPLES::
sage: A = IntegralLattice(1)
sage: A.direct_sum(A)
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[1 0]
[0 1]
"""
IM = matrix.block_diagonal([self.inner_product_matrix(),
M.inner_product_matrix()])
ambient = FreeQuadraticModule(ZZ,
self.degree() + M.degree(), IM)
smzero = matrix.zero(self.rank(), M.degree())
mszero = matrix.zero(M.rank(), self.degree())
basis = self.basis_matrix().augment(smzero).stack(
mszero.augment(M.basis_matrix()))
ipm = ambient.inner_product_matrix()
return FreeQuadraticModule_integer_symmetric(ambient=ambient,
basis=basis,
inner_product_matrix=ipm,
already_echelonized=False)
def direct_sum(self, M):
r"""
Return the direct sum of this lattice with ``M``.
INPUT:
- ``M`` -- a module over `\ZZ`
EXAMPLES::
sage: A = IntegralLattice(1)
sage: A.direct_sum(A)
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[1 0]
[0 1]
"""
IM = matrix.block_diagonal(
[self.inner_product_matrix(),
M.inner_product_matrix()])
ambient = FreeQuadraticModule(ZZ, self.degree() + M.degree(), IM)
smzero = matrix.zero(self.rank(), M.degree())
mszero = matrix.zero(M.rank(), self.degree())
basis = self.basis_matrix().augment(smzero).stack(
mszero.augment(M.basis_matrix()))
ipm = ambient.inner_product_matrix()
return FreeQuadraticModule_integer_symmetric(ambient=ambient,
basis=basis,
inner_product_matrix=ipm,
already_echelonized=False)
def gram_matrix_quadratic(self):
r"""
The Gram matrix of the quadratic form with respect to the generators.
OUTPUT:
- a rational matrix ``Gq`` with ``Gq[i,j] = gens[i]*gens[j]``
and ``G[i,i] = gens[i].q()``
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: D4_gram = Matrix(ZZ, [[2,0,0,-1],[0,2,0,-1],[0,0,2,-1],[-1,-1,-1,2]])
sage: D4 = FreeQuadraticModule(ZZ, 4, D4_gram)
sage: D4dual = D4.span(D4_gram.inverse())
sage: discrForm = TorsionQuadraticModule(D4dual, D4)
sage: discrForm.gram_matrix_quadratic()
[ 1 1/2]
[1/2 1]
sage: discrForm.gram_matrix_bilinear()
[ 0 1/2]
[1/2 0]
"""
gens = self.gens()
n = len(gens)
Q = self.base_ring().fraction_field()
G = matrix.zero(Q, n)
for i in range(n):
for j in range(i):
G[i, j] = G[j, i] = (gens[i] * gens[j]).lift()
G[i, i] = gens[i].q().lift()
return G
def gram_matrix_bilinear(self):
r"""
Return the Gram matrix with respect to the generators.
OUTPUT:
A rational matrix ``G`` with ``G[i,j]`` given by the inner product
of the `i`-th and `j`-th generator. Its entries are only well
defined `\mod (V, W)`.
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ, 3, matrix.identity(3)*5)
sage: T = TorsionQuadraticModule((1/5)*V, V)
sage: T.gram_matrix_bilinear()
[1/5 0 0]
[ 0 1/5 0]
[ 0 0 1/5]
"""
gens = self.gens()
n = len(gens)
Q = self.base_ring().fraction_field()
G = matrix.zero(Q, n)
for i in range(n):
for j in range(i + 1):
G[i, j] = G[j, i] = (gens[i] * gens[j]).lift()
return G
def gram_matrix_quadratic(self):
r"""
The gram matrix of the quadratic form with respect to the generators.
OUTPUT:
- a rational matrix ``Gq`` with ``Gq[i,j] = gens[i]*gens[j]``
and ``G[i,i] = gens[i].q()``
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: D4_gram = Matrix(ZZ, [[2,0,0,-1],[0,2,0,-1],[0,0,2,-1],[-1,-1,-1,2]])
sage: D4 = FreeQuadraticModule(ZZ, 4, D4_gram)
sage: D4dual = D4.span(D4_gram.inverse())
sage: discrForm = TorsionQuadraticModule(D4dual, D4)
sage: discrForm.gram_matrix_quadratic()
[ 1 1/2]
[1/2 1]
sage: discrForm.gram_matrix_bilinear()
[ 0 1/2]
[1/2 0]
"""
gens = self._gens
n = len(gens)
Q = self.base_ring().fraction_field()
G = matrix.zero(Q, n)
for i in range(n):
for j in range(i):
G[i,j] = G[j,i] = (gens[i] * gens[j]).lift()
G[i,i] = gens[i].q().lift()
return G
def gram_matrix_bilinear(self):
r"""
Return the gram matrix with respect to the generators.
OUTPUT:
A rational matrix ``G`` with ``G[i,j]`` given by the inner product
of the `i`-th and `j`-th generator. Its entries are only well
defined `\mod (V, W)`.
EXAMPLES::
sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ, 3, matrix.identity(3)*5)
sage: T = TorsionQuadraticModule((1/5)*V, V)
sage: T.gram_matrix_bilinear()
[1/5 0 0]
[ 0 1/5 0]
[ 0 0 1/5]
"""
gens = self._gens
n = len(gens)
Q = self.base_ring().fraction_field()
G = matrix.zero(Q, n)
for i in range(n):
for j in range(i+1):
G[i,j] = G[j,i] = (gens[i] * gens[j]).lift()
return G
def compacte_liste(listP, FractionField):
"""
Take a list of rational fractions
and return the an equivalent list with minimal degrees.
"""
check = matrix.zero(ZZ, len(listP))
Res = [P for P in listP]
for i, P in enumerate(listP):
P = FractionField(P)
Q, [case, ind] = replace_p(P, listP, FractionField)
check[i, i] = 1
if case != 0:
check2 = check
check2[i, ind] = case
if check2.rank() < check.rank(
): #On vérifie qu'on n'oublie pas une équation!
Q = P
else:
check = check2
if not (Q in Res):
Res.remove(P)
if Q.numerator().degree() + Q.denominator().degree() == 0:
#(in case we had to proportional equations
pass
else:
Res += [Q]
return Res
def zero(self):
"""
Construct the zero morphism of ``self``.
EXAMPLES::
sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: H = Hom(A, B)
sage: H.zero()
Morphism from Finite-dimensional algebra of degree 1 over Rational Field to
Finite-dimensional algebra of degree 2 over Rational Field given by matrix
[0 0]
"""
from sage.matrix.constructor import matrix
return FiniteDimensionalAlgebraMorphism(
self, matrix.zero(self.domain().ngens(),
self.codomain().ngens()), False, False)
def zero(self):
"""
Construct the zero morphism of ``self``.
EXAMPLES::
sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([1])])
sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
sage: H = Hom(A, B)
sage: H.zero()
Morphism from Finite-dimensional algebra of degree 1 over Rational Field to
Finite-dimensional algebra of degree 2 over Rational Field given by matrix
[0 0]
"""
from sage.matrix.constructor import matrix
return FiniteDimensionalAlgebraMorphism(self, matrix.zero(self.domain().ngens(),
self.codomain().ngens()),
False, False)
def IntegralLatticeDirectSum(Lattices, return_embeddings=False):
r"""
Return the direct sum of the lattices contained in the list ``Lattices``.
INPUT:
- ``Lattices`` -- a list of lattices ``[L_1,...,L_n]``
- ``return_embeddings`` -- (default: ``False``) a boolean
OUTPUT:
The direct sum of the `L_i` if ``return_embeddings`` is ``False`` or
the tuple ``[L, phi]`` where `L` is the direct sum of `L_i` and ``phi``
is the list of embeddings from `L_i` to `L`.
EXAMPLES::
sage: from sage.modules.free_quadratic_module_integer_symmetric import IntegralLatticeDirectSum
sage: L1 = IntegralLattice("D4")
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: L3 = IntegralLattice("A4", [[0, 1, 1, 2], [1, 2, 3, 1]])
sage: Lattices = [L1, L2, L3]
sage: IntegralLatticeDirectSum([L1, L2, L3])
Lattice of degree 11 and rank 7 over Integer Ring
Basis matrix:
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 1 2 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 2]
[0 0 0 0 0 0 0 1 2 3 1]
Inner product matrix:
[ 2 -1 0 0 0 0 0 0 0 0 0]
[-1 2 -1 -1 0 0 0 0 0 0 0]
[ 0 -1 2 0 0 0 0 0 0 0 0]
[ 0 -1 0 2 0 0 0 0 0 0 0]
[ 0 0 0 0 2 -1 0 0 0 0 0]
[ 0 0 0 0 -1 2 -1 0 0 0 0]
[ 0 0 0 0 0 -1 2 0 0 0 0]
[ 0 0 0 0 0 0 0 2 -1 0 0]
[ 0 0 0 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 0 0 0 0 -1 2]
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2, L3], True)
sage: LL3 = L.sublattice(phi[2].image().basis_matrix())
sage: L3.discriminant() == LL3.discriminant()
True
sage: x = L3([1, 2, 3, 1])
sage: phi[2](x).inner_product(phi[2](x)) == x.inner_product(x)
True
TESTS::
sage: IntegralLatticeDirectSum([IntegralLattice("D4")])
Lattice of degree 4 and rank 4 over Integer Ring
Basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
Inner product matrix:
[ 2 -1 0 0]
[-1 2 -1 -1]
[ 0 -1 2 0]
[ 0 -1 0 2]
sage: L1 = IntegralLattice(2 * matrix.identity(2), [[1/2, 1/2]])
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2], True)
sage: L
Lattice of degree 5 and rank 2 over Integer Ring
Basis matrix:
[1/2 1/2 0 0 0]
[ 0 0 1 1 2]
Inner product matrix:
[ 2 0 0 0 0]
[ 0 2 0 0 0]
[ 0 0 2 -1 0]
[ 0 0 -1 2 -1]
[ 0 0 0 -1 2]
"""
for L in Lattices:
if not isinstance(L, FreeQuadraticModule_integer_symmetric):
raise ValueError("Lattices must be a list of lattices")
N = len(Lattices)
dims = [L_i.dimension() for L_i in Lattices]
degrees = [L_i.degree() for L_i in Lattices]
degree_tot = sum(degrees)
sum_degree = [sum(degrees[:i]) for i in range(N + 1)]
inner_product_list = [copy(L_i.inner_product_matrix()) for L_i in Lattices]
IM = matrix.block_diagonal(inner_product_list)
ambient = FreeQuadraticModule(ZZ, degree_tot, inner_product_matrix=IM)
basis = [
matrix.block(1, 3, [
matrix.zero(dims[i], sum_degree[i]), Lattices[i].basis_matrix(),
matrix.zero(dims[i], sum_degree[-1] - sum_degree[i + 1])
]) for i in range(N)
]
basis_matrix = matrix.block(N, 1, basis)
ipm = ambient.inner_product_matrix()
direct_sum = FreeQuadraticModule_integer_symmetric(
ambient=ambient,
basis=basis_matrix,
inner_product_matrix=ipm,
already_echelonized=False)
if not return_embeddings:
return direct_sum
sum_dims = [sum(dims[:i]) for i in range(N + 1)]
phi = [
Lattices[i].hom(direct_sum.basis()[sum_dims[i]:sum_dims[i + 1]])
for i in range(N)
]
return [direct_sum, phi]
def IntegralLatticeDirectSum(Lattices, return_embeddings=False):
r"""
Return the direct sum of the lattices contained in the list ``Lattices``.
INPUT:
- ``Lattices`` -- a list of lattices ``[L_1,...,L_n]``
- ``return_embeddings`` -- (default: ``False``) a boolean
OUTPUT:
The direct sum of the `L_i` if ``return_embeddings`` is ``False`` or
the tuple ``[L, phi]`` where `L` is the direct sum of `L_i` and ``phi``
is the list of embeddings from `L_i` to `L`.
EXAMPLES::
sage: from sage.modules.free_quadratic_module_integer_symmetric import IntegralLatticeDirectSum
sage: L1 = IntegralLattice("D4")
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: L3 = IntegralLattice("A4", [[0, 1, 1, 2], [1, 2, 3, 1]])
sage: Lattices = [L1, L2, L3]
sage: IntegralLatticeDirectSum([L1, L2, L3])
Lattice of degree 11 and rank 7 over Integer Ring
Basis matrix:
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 1 2 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 2]
[0 0 0 0 0 0 0 1 2 3 1]
Inner product matrix:
[ 2 -1 0 0 0 0 0 0 0 0 0]
[-1 2 -1 -1 0 0 0 0 0 0 0]
[ 0 -1 2 0 0 0 0 0 0 0 0]
[ 0 -1 0 2 0 0 0 0 0 0 0]
[ 0 0 0 0 2 -1 0 0 0 0 0]
[ 0 0 0 0 -1 2 -1 0 0 0 0]
[ 0 0 0 0 0 -1 2 0 0 0 0]
[ 0 0 0 0 0 0 0 2 -1 0 0]
[ 0 0 0 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 0 0 0 0 -1 2]
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2, L3], True)
sage: LL3 = L.sublattice(phi[2].image().basis_matrix())
sage: L3.discriminant() == LL3.discriminant()
True
sage: x = L3([1, 2, 3, 1])
sage: phi[2](x).inner_product(phi[2](x)) == x.inner_product(x)
True
TESTS::
sage: IntegralLatticeDirectSum([IntegralLattice("D4")])
Lattice of degree 4 and rank 4 over Integer Ring
Basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
Inner product matrix:
[ 2 -1 0 0]
[-1 2 -1 -1]
[ 0 -1 2 0]
[ 0 -1 0 2]
sage: L1 = IntegralLattice(2 * matrix.identity(2), [[1/2, 1/2]])
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2], True)
sage: L
Lattice of degree 5 and rank 2 over Integer Ring
Basis matrix:
[1/2 1/2 0 0 0]
[ 0 0 1 1 2]
Inner product matrix:
[ 2 0 0 0 0]
[ 0 2 0 0 0]
[ 0 0 2 -1 0]
[ 0 0 -1 2 -1]
[ 0 0 0 -1 2]
"""
for L in Lattices:
if not isinstance(L, FreeQuadraticModule_integer_symmetric):
raise ValueError("Lattices must be a list of lattices")
N = len(Lattices)
dims = [L_i.dimension() for L_i in Lattices]
degrees = [L_i.degree() for L_i in Lattices]
dim_tot = sum(dims)
degree_tot = sum(degrees)
sum_degree = [sum(degrees[:i]) for i in range(N+1)]
inner_product_list = [copy(L_i.inner_product_matrix()) for L_i in Lattices]
IM = matrix.block_diagonal(inner_product_list)
ambient = FreeQuadraticModule(ZZ,
degree_tot,
inner_product_matrix=IM)
basis = [matrix.block(1, 3, [matrix.zero(dims[i], sum_degree[i]),
Lattices[i].basis_matrix(),
matrix.zero(dims[i], sum_degree[-1] - sum_degree[i+1])
]) for i in range(N)]
basis_matrix = matrix.block(N, 1, basis)
ipm = ambient.inner_product_matrix()
direct_sum = FreeQuadraticModule_integer_symmetric(ambient=ambient,
basis=basis_matrix,
inner_product_matrix=ipm,
already_echelonized=False)
if not return_embeddings:
return direct_sum
sum_dims = [sum(dims[:i]) for i in range(N+1)]
phi = [Lattices[i].hom(direct_sum.basis()[sum_dims[i]:sum_dims[i+1]])
for i in range(N)]
return [direct_sum, phi]
