def space(self):
r'''
Calculates the space of cocyles modulo coboundaries, as a Z-module.
TESTS:
sage: from darmonpoints.sarithgroup import *
sage: from darmonpoints.cohomology_abstract import *
sage: from darmonpoints.ocmodule import *
sage: GS = BigArithGroup(5, 6,1,use_shapiro=False,outfile='/tmp/darmonpoints.tmp') # optional - magma
sage: G = GS.large_group() # optional - magma
sage: V = OCVn(5,1) # optional - magma
sage: Coh = CohomologyGroup(G,V,trivial_action = False) # optional - magma
'''
verb = get_verbose()
set_verbose(0)
V = self.coefficient_module()
R = V.base_ring()
Vdim = V.dimension()
G = self.group()
gens = G.gens()
ambient = R**(Vdim * len(gens))
# Now find the subspace of cocycles
A = Matrix(R, Vdim * len(gens), 0)
for r in G.get_relation_words():
Alist = self.fox_gradient(r)
newA = block_matrix(Alist, nrows = 1)
A = A.augment(newA.transpose())
A = A.transpose()
cocycles = ambient.submodule([ambient(o) for o in A.right_kernel_matrix().rows()])
gmat = block_matrix([self._gen_pows[i][1] - 1 for i in range(len(G.gens()))], nrows = len(G.gens()))
coboundaries = cocycles.submodule([ambient(o) for o in gmat.columns()])
ans = cocycles.quotient(coboundaries)
set_verbose(verb)
return ans
def diagonal_matrix(self):
r"""
Returns a diagonal matrix `D` and a matrix `T` such that `T^t A T = D`
holds, where `(x, y, z) A (x, y, z)^t` is the defining polynomial
of the conic ``self``.
EXAMPLES:
::
sage: c = Conic(QQ, [1,2,3,4,5,6])
sage: d, t = c.diagonal_matrix(); d, t
(
[ 1 0 0] [ 1 -1 -7/6]
[ 0 3 0] [ 0 1 -1/3]
[ 0 0 41/12], [ 0 0 1]
)
sage: t.transpose()*c.symmetric_matrix()*t
[ 1 0 0]
[ 0 3 0]
[ 0 0 41/12]
Diagonal matrices are only defined in characteristic different
from `2`:
::
sage: c = Conic(GF(4, 'a'), [0, 1, 1, 1, 1, 1])
sage: c.is_smooth()
True
sage: c.diagonal_matrix()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field in a of size 2^2 defined by x*y + y^2 + x*z + y*z + z^2) has no symmetric matrix because the base field has characteristic 2
"""
A = self.symmetric_matrix()
B = self.base_ring()
basis = [vector(B,{2:0,i:1}) for i in range(3)]
for i in range(3):
zerovalue = (basis[i]*A*basis[i].column()== 0)
if zerovalue:
for j in range(i+1,3):
if basis[j]*A*basis[j].column() != 0:
b = basis[i]
basis[i] = basis[j]
basis[j] = b
zerovalue = False
if zerovalue:
for j in range(i+1,3):
if basis[i]*A*basis[j].column() != 0:
basis[i] = basis[i]+basis[j]
zerovalue = False
if not zerovalue:
l = (basis[i]*A*basis[i].column())
for j in range(i+1,3):
basis[j] = basis[j] - \
(basis[i]*A*basis[j].column())/l * basis[i]
T = Matrix(basis).transpose()
return T.transpose()*A*T, T
def diagonal_matrix(self):
r"""
Returns a diagonal matrix `D` and a matrix `T` such that `T^t A T = D`
holds, where `(x, y, z) A (x, y, z)^t` is the defining polynomial
of the conic ``self``.
EXAMPLES:
::
sage: c = Conic(QQ, [1,2,3,4,5,6])
sage: d, t = c.diagonal_matrix(); d, t
(
[ 1 0 0] [ 1 -1 -7/6]
[ 0 3 0] [ 0 1 -1/3]
[ 0 0 41/12], [ 0 0 1]
)
sage: t.transpose()*c.symmetric_matrix()*t
[ 1 0 0]
[ 0 3 0]
[ 0 0 41/12]
Diagonal matrices are only defined in characteristic different
from `2`:
::
sage: c = Conic(GF(4, 'a'), [0, 1, 1, 1, 1, 1])
sage: c.is_smooth()
True
sage: c.diagonal_matrix()
Traceback (most recent call last):
...
ValueError: The conic self (= Projective Conic Curve over Finite Field in a of size 2^2 defined by x*y + y^2 + x*z + y*z + z^2) has no symmetric matrix because the base field has characteristic 2
"""
A = self.symmetric_matrix()
B = self.base_ring()
basis = [vector(B,{2:0,i:1}) for i in range(3)]
for i in range(3):
zerovalue = (basis[i]*A*basis[i].column()== 0)
if zerovalue:
for j in range(i+1,3):
if basis[j]*A*basis[j].column() != 0:
b = basis[i]
basis[i] = basis[j]
basis[j] = b
zerovalue = False
if zerovalue:
for j in range(i+1,3):
if basis[i]*A*basis[j].column() != 0:
basis[i] = basis[i]+basis[j]
zerovalue = False
if not zerovalue:
l = (basis[i]*A*basis[i].column())
for j in range(i+1,3):
basis[j] = basis[j] - \
(basis[i]*A*basis[j].column())/l * basis[i]
T = Matrix(basis).transpose()
return T.transpose()*A*T, T
def LocalizedMomentMatrix(self, idx):
"""
Returns localized moment matrix corresponding to $g_{idx}$
"""
tmp_vec = self.MonomialsVec(self.Relaxation - self.HalfDegs[idx])
m = Matrix(1, len(tmp_vec), tmp_vec)
return self.Constraints[idx] * (m.transpose() * m)
def LocalizedMomentMatrix(self, idx):
"""
Returns localized moment matrix corresponding to $g_{idx}$
"""
tmp_vec = self.MonomialsVec(self.Relaxation-self.HalfDegs[idx])
m = Matrix(1, len(tmp_vec), tmp_vec)
return self.Constraints[idx]* (m.transpose() * m)
def decompose(self):
"""
Gives an SOS decomposition of f if exists as a list of polynomials
of at most half degree of f.
This method also fills the 'Info' as 'minimize' does. In addition,
returns Info['is sos'] which is Boolean depending on the status of
sdp solver.
"""
n = self.NumVars
N0 = self.NumMonomials(n, self.MainPolyHlfDeg)
N1 = self.NumMonomials(n, self.MainPolyTotDeg)
self.MatSize = [N0, N1]
vec = self.MonomialsVec(self.MainPolyHlfDeg)
m = Matrix(1, N0, vec)
Mmnt = m.transpose() * m
Blck = [[] for i in range(N1)]
C = []
h = Matrix(self.Field, N0, N0, 0)
C.append(h)
decomp = []
for i in range(N1):
p = self.Monomials[i]
A = self.Calpha(p, Mmnt)
Blck[i].append(A)
from SDP import SDP
sos_sdp = SDP.sdp(solver=self.solver, Settings={'detail': self.detail})
sos_sdp.solve(C, self.PolyCoefFullVec(), Blck)
if sos_sdp.Info['Status'] == 'Optimal':
self.Info['status'] = 'Feasible'
GramMtx = Matrix(sos_sdp.Info['X'][0])
try:
self.Info[
'Message'] = "A SOS decomposition of the polynomial were found."
self.Info['is sos'] = True
H1 = GramMtx.cholesky()
tmpM = Matrix(1, N0, vec)
decomp = list(tmpM * H1)[0]
self.Info['Wall'] = sos_sdp.Info['Wall']
self.Info['CPU'] = sos_sdp.Info['CPU']
except:
self.Info[
'Message'] = "The given polynomial seems to be a sum of squares, but no SOS decomposition were extracted."
self.Info['is sos'] = False
self.Info['Wall'] = sos_sdp.Info['Wall']
self.Info['CPU'] = sos_sdp.Info['CPU']
else:
self.Info[
'Message'] = "The given polynomial is not a sum of squares."
self.Info['status'] = 'Infeasible'
self.Info['is sos'] = False
self.Info["Size"] = self.MatSize
return decomp
def linear_approximation_matrix(self):
"""
Return linear approximation matrix ``A`` for this S-box.
Let ``i_b`` be the ``b``-th bit of ``i`` and ``o_b`` the
``b``-th bit of ``o``. Then ``v = A[i,o]`` encodes the bias of
the equation ``sum( i_b * x_i ) = sum( o_b * y_i )`` if
``x_i`` and ``y_i`` represent the input and output variables
of the S-box.
See [He2002]_ for an introduction to linear cryptanalysis.
EXAMPLES::
sage: from sage.crypto.sbox import SBox
sage: S = SBox(7,6,0,4,2,5,1,3)
sage: S.linear_approximation_matrix()
[ 4 0 0 0 0 0 0 0]
[ 0 0 0 0 2 2 2 -2]
[ 0 0 -2 -2 -2 2 0 0]
[ 0 0 -2 2 0 0 -2 -2]
[ 0 2 0 2 -2 0 2 0]
[ 0 -2 0 2 0 2 0 2]
[ 0 -2 -2 0 0 -2 2 0]
[ 0 -2 2 0 -2 0 0 -2]
According to this matrix the first bit of the input is equal
to the third bit of the output 6 out of 8 times::
sage: for i in srange(8): print(S.to_bits(i)[0] == S.to_bits(S(i))[2])
False
True
True
True
False
True
True
True
"""
m = self.m
n = self.n
nrows = 1<<m
ncols = 1<<n
B = BooleanFunction(self.m)
L = []
for j in range(ncols):
for i in range(nrows):
B[i] = ZZ(self(i)&j).popcount()
L.append(B.walsh_hadamard_transform())
A = Matrix(ZZ, ncols, nrows, L)
A = -A.transpose()/2
A.set_immutable()
return A
def angle_equations(M):
"""
Given a snappy manifold M, returns the matrix of left-hand-sides
of angle equations (that is, the tet, edge, and cusp equations).
"""
num_tet = M.num_tetrahedra()
G = M.gluing_equations()
T = Matrix(ZZ, [tet_vector(i, num_tet) for i in range(num_tet)])
return T.transpose().augment(G.transpose()).transpose() # sigh
def linear_approximation_matrix(self):
"""
Return linear approximation matrix ``A`` for this S-box.
Let ``i_b`` be the ``b``-th bit of ``i`` and ``o_b`` the
``b``-th bit of ``o``. Then ``v = A[i,o]`` encodes the bias of
the equation ``sum( i_b * x_i ) = sum( o_b * y_i )`` if
``x_i`` and ``y_i`` represent the input and output variables
of the S-box.
See [He2002]_ for an introduction to linear cryptanalysis.
EXAMPLES::
sage: from sage.crypto.sbox import SBox
sage: S = SBox(7,6,0,4,2,5,1,3)
sage: S.linear_approximation_matrix()
[ 4 0 0 0 0 0 0 0]
[ 0 0 0 0 2 2 2 -2]
[ 0 0 -2 -2 -2 2 0 0]
[ 0 0 -2 2 0 0 -2 -2]
[ 0 2 0 2 -2 0 2 0]
[ 0 -2 0 2 0 2 0 2]
[ 0 -2 -2 0 0 -2 2 0]
[ 0 -2 2 0 -2 0 0 -2]
According to this matrix the first bit of the input is equal
to the third bit of the output 6 out of 8 times::
sage: for i in srange(8): print(S.to_bits(i)[0] == S.to_bits(S(i))[2])
False
True
True
True
False
True
True
True
"""
m = self.m
n = self.n
nrows = 1<<m
ncols = 1<<n
B = BooleanFunction(self.m)
L = []
for j in range(ncols):
for i in range(nrows):
B[i] = ZZ(self(i)&j).popcount()
L.append(B.walsh_hadamard_transform())
A = Matrix(ZZ, ncols, nrows, L)
A = -A.transpose()/2
A.set_immutable()
return A
def linear_relation(self, List, Psi, verbose=True, prec=None):
for Phi in List:
assert Phi.valuation() >= 0, "Symbols must be integral"
assert Psi.valuation() >= 0
R = self.base()
Rbase = R.base()
w = R.gen()
d = len(List)
if d == 0:
if Psi.is_zero():
return [None, R(1)]
else:
return [None, 0]
if prec is None:
M, var_prec = Psi.precision_absolute()
else:
M, var_prec = prec
p = self.prime()
V = R**d
RSR = LaurentSeriesRing(Rbase, R.variable_name())
VSR = RSR**self.source().ngens()
List_TMs = [VSR(Phi.list_of_total_measures()) for Phi in List]
Psi_TMs = VSR(Psi.list_of_total_measures())
A = Matrix(RSR, List_TMs).transpose()
try:
sol = V([vv.power_series() for vv in A.solve_right(Psi_TMs)])
except ValueError:
#try "least squares"
if verbose:
print "Trying least squares."
sol = (A.transpose() * A).solve_right(A.transpose() * Psi_TMs)
#check precision (could make this better, checking each power of w)
p_prec = M
diff = Psi_TMs - sum([sol[i] * List_TMs[i] for i in range(len(List_TMs))])
for i in diff:
for j in i.list():
if p_prec > j.valuation():
p_prec = j.valuation()
if verbose:
print "p-adic precision is now", p_prec
#Is this right?
sol = V([R([j.add_bigoh(p_prec) for j in i.list()]) for i in sol])
return [sol, R(-1)]
def projection_to_homology(tri,angle):
non_tree_as_cycles = non_tree_edge_cycles(tri, angle)
Q = Matrix(non_tree_as_cycles)
S, U, V = faces_in_smith(tri,angle,[])
rank, dimU, dimV = rank_of_quotient(S)
U = Matrix(U)
P = U.transpose().inverse()
P = P.delete_rows(range(0, dimU-rank))
A = P*Q
return A
def decompose(self):
"""
Gives an SOS decomposition of f if exists as a list of polynomials
of at most half degree of f.
This method also fills the 'Info' as 'minimize' does. In addition,
returns Info['is sos'] which is Boolean depending on the status of
sdp solver.
"""
n = self.NumVars
N0 = self.NumMonomials(n, self.MainPolyHlfDeg)
N1 = self.NumMonomials(n, self.MainPolyTotDeg)
self.MatSize = [N0, N1]
vec = self.MonomialsVec(self.MainPolyHlfDeg)
m = Matrix(1, N0, vec)
Mmnt = m.transpose() * m
Blck = [[] for i in range(N1)]
C = []
h = Matrix(self.Field, N0, N0, 0)
C.append(h)
decomp = []
for i in range(N1):
p = self.Monomials[i]
A = self.Calpha(p, Mmnt)
Blck[i].append(A)
from SDP import SDP
sos_sdp = SDP.sdp(solver = self.solver, Settings = {'detail':self.detail})
sos_sdp.solve(C, self.PolyCoefFullVec(), Blck)
if sos_sdp.Info['Status'] == 'Optimal':
self.Info['status'] = 'Feasible'
GramMtx = Matrix(sos_sdp.Info['X'][0])
try:
self.Info['Message'] = "A SOS decomposition of the polynomial were found."
self.Info['is sos'] = True
H1 = GramMtx.cholesky();
tmpM = Matrix(1, N0, vec)
decomp = list(tmpM*H1)[0]
self.Info['Wall'] = sos_sdp.Info['Wall']
self.Info['CPU'] = sos_sdp.Info['CPU']
except:
self.Info['Message'] = "The given polynomial seems to be a sum of squares, but no SOS decomposition were extracted."
self.Info['is sos'] = False
self.Info['Wall'] = sos_sdp.Info['Wall']
self.Info['CPU'] = sos_sdp.Info['CPU']
else:
self.Info['Message'] = "The given polynomial is not a sum of squares."
self.Info['status'] = 'Infeasible'
self.Info['is sos']= False
self.Info["Size"] = self.MatSize
return decomp
def monomial_multiplier(elts, ZH):
if all(elt == 0 for elt in elts):
# Zero (unlike other constants) has valuation -\infty. This
# can show up as an issue when computing the big polynomial.
# For an example, compute ET for
# "kLLLMPPkcdgfehijjijhshassqhdqr_1222011022"
return ZH(1)
elts = [ZH(elt) for elt in elts]
A = Matrix(ZZ, join_lists([uniform_exponents(p) for p in elts]))
min_exp = tuple([min(row) for row in A.transpose()])
return ZH({min_exp: 1})
def poly_dual_basis(P, poly_basis):
r"""
Return a collection of polynomials which are dual under the differential
bilinear form to a given homogeneous collection
INPUT:
- ``P`` -- a polynomial ring
- ``poly_basis`` -- a collection of polynomials in ``P`` which are
homogeneous and linearly independent
OUTPUT:
- the dual basis of the polynomials in ``poly_basis`` in their span
EXAMPLES:
sage: P.<x, y> = PolynomialRing(QQ)
sage: poly_basis = (1, x, x+y)
sage: poly_dual_basis(P, poly_basis)
[1, x - y, y]
sage: poly_basis = (1, 2*x - y, x^2, x^2 + x*y)
sage: poly_dual_basis(P, poly_basis)
[1, 2/5*x - 1/5*y, 1/2*x^2 - x*y, x*y]
"""
# recast poly_basis to ensure elements are all from P
poly_basis = [P(p) for p in poly_basis]
# compute max degree of basis polynomials for linear algebra computations
deg = max([p.degree() for p in poly_basis])
# construct polynomial free module for linear algebra computations
monoms = Monomials(P, (0, deg))
poly_module = PolynomialFreeModule(P, basis=monoms)
# compute the values of the bilinear form <m|m> for basis monomials m
bilinear_form_coeffs = []
for b in poly_module.basis().keys():
# each b is a monomial in P of degree at most deg
b = P(b)
bilinear_form_coeffs.append(prod(map(factorial, b.degrees())))
# compute dual basis
A = Matrix([poly_module(p).to_vector() for p in poly_basis])
D = Matrix.diagonal(bilinear_form_coeffs, sparse=False)
B = (A * D * A.transpose()).inverse()
# reconstruct dual basis polynomials from corresponding vectors
dual_basis = []
for col in B.columns():
q = sum([coeff * p for coeff, p in zip(col, poly_basis)])
dual_basis.append(q)
return dual_basis
def space(self):
r'''
Calculates the space of cocyles modulo coboundaries, as a Z-module.
TESTS:
sage: from darmonpoints.sarithgroup import *
sage: from darmonpoints.cohomology_abstract import *
sage: from darmonpoints.ocmodule import *
sage: GS = BigArithGroup(5, 6,1,use_shapiro=False,outfile='/tmp/darmonpoints.tmp') # optional - magma
sage: G = GS.large_group() # optional - magma
sage: V = OCVn(5,1) # optional - magma
sage: Coh = CohomologyGroup(G,V,trivial_action = False) # optional - magma
'''
verb = get_verbose()
set_verbose(0)
V = self.coefficient_module()
R = V.base_ring()
Vdim = V.dimension()
G = self.group()
gens = G.gens()
ambient = R**(Vdim * len(gens))
# Now find the subspace of cocycles
A = Matrix(R, Vdim * len(gens), 0)
for nr, r in enumerate(G.get_relation_words()):
set_verbose(verb)
verbose('Processing relation word %s' % nr)
set_verbose(0)
Alist = [
MatrixSpace(R, Vdim, Vdim)(self.GA_to_local(o))
for o in self.fox_gradient(tuple(r))
]
newA = block_matrix(Alist, nrows=1)
A = A.augment(newA.transpose())
A = A.transpose()
cocycles = ambient.submodule(
[ambient(o) for o in A.right_kernel_matrix().rows()])
gmat = block_matrix(
[self._acting_matrix(g, Vdim) - 1 for g in G.gens()],
nrows=len(G.gens()))
coboundaries = cocycles.submodule([ambient(o) for o in gmat.columns()])
ans = cocycles.quotient(coboundaries)
set_verbose(verb)
return ans
def taut_cone_homological_dim(tri, angle):
# find the dimension of the projection of the taut cone into
# homology
# boundaries of tets
bdys = zeroth_coboundary(tri)
bdys = matrix_transpose(bdys)
rays = taut_rays(tri, angle)
# but these are all 'upwards', so we need to fix the
# co-orientations
coorient = is_transverse_taut(tri, angle, return_type = "face_coorientations")
rays = [[int(a * b) for a, b in zip(coorient, ray)] for ray in rays]
# now work in the space of two-chains
Rays = IntegerLattice(rays + bdys)
Bdys = Matrix(bdys)
Cobs = Bdys.transpose()
Anns = Cobs.kernel()
return Rays.intersection(Anns).dimension()
def to_elt(alp):
ref = self.domain().reflection(alp)
m = Matrix([ref(x).to_vector() for x in self.domain().basis()])
return self(m.transpose())
def make_regular_matroid_from_matroid(matroid):
r"""
Attempt to construct a regular representation of a matroid.
INPUT:
- ``matroid`` -- a matroid.
OUTPUT:
Return a `(0, 1, -1)`-matrix over the integers such that, if the input is
a regular matroid, then the output is a totally unimodular matrix
representing that matroid.
EXAMPLES::
sage: from sage.matroids.utilities import make_regular_matroid_from_matroid
sage: make_regular_matroid_from_matroid(
....: matroids.CompleteGraphic(6)).is_isomorphic(
....: matroids.CompleteGraphic(6))
True
"""
import sage.matroids.linear_matroid
M = matroid
if isinstance(M, sage.matroids.linear_matroid.RegularMatroid):
return M
rk = M.full_rank()
# First create a reduced 0-1 matrix
B = list(M.basis())
NB = list(M.groundset().difference(B))
dB = {}
i = 0
for e in B:
dB[e] = i
i += 1
dNB = {}
i = 0
for e in NB:
dNB[e] = i
i += 1
A = Matrix(ZZ, len(B), len(NB), 0)
G = BipartiteGraph(
A.transpose()
) # Sage's BipartiteGraph uses the column set as first color class. This is an edgeless graph.
for e in NB:
C = M.circuit(B + [e])
for f in C.difference([e]):
A[dB[f], dNB[e]] = 1
# Change some entries from -1 to 1
entries = BipartiteGraph(A.transpose()).edges(labels=False)
while len(entries) > 0:
L = [G.shortest_path(u, v) for u, v in entries]
mindex, minval = min(enumerate(L), key=lambda x: len(x[1]))
# if minval = 0, there is an edge not spanned by the current subgraph. Its entry is free to be scaled any way.
if len(minval) > 0:
# Check the subdeterminant
S = frozenset(L[mindex])
rows = []
cols = []
for i in S:
if i < rk:
rows.append(i)
else:
cols.append(i - rk)
if A[rows, cols].det() != 0:
A[entries[mindex][0], entries[mindex][1] - rk] = -1
G.add_edge(entries[mindex][0], entries[mindex][1])
entries.pop(mindex)
return sage.matroids.linear_matroid.RegularMatroid(groundset=B + NB,
reduced_matrix=A)
def automorphisms(self):
"""
Return a list of the automorphisms of the quadratic form.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.number_of_automorphisms() # optional -- souvigner
48
sage: 2^3 * factorial(3)
48
sage: len(Q.automorphisms())
48
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.number_of_automorphisms() # optional -- souvigner
16
sage: aut = Q.automorphisms()
sage: len(aut)
16
sage: print([Q(M) == Q for M in aut])
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
sage: Q = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q.automorphisms()
[
[1 0 0] [-1 0 0]
[0 1 0] [ 0 -1 0]
[0 0 1], [ 0 0 -1]
]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.automorphisms()
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.automorphisms()
"""
## only for definite forms
if not self.is_definite():
raise ValueError("not a definite form in QuadraticForm.automorphisms()")
## Check for a cached value
try:
return self.__automorphisms
except AttributeError:
pass
## Find a basis of short vectors, and their lengths
basis, pivot_lengths = self.basis_of_short_vectors(show_lengths=True)
## List the relevant vectors by length
max_len = max(pivot_lengths)
vector_list_by_length = self.short_primitive_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis).transpose()
Ainv = A.inverse()
# A1 = A.inverse() * A.det()
# Q1 = A1.transpose() * self.matrix() * A1 ## This is the matrix of Q
# Q = self.matrix() * A.det()**2
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of Q in the new basis
Q3 = self.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
# ct = 0
## DIAGNOSTIC
# print "n = " + str(n)
# print "pivot_lengths = " + str(pivot_lengths)
# print "vector_list_by_length = " + str(vector_list_by_length)
# print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(vector_list_by_length[pivot_lengths[i]]) for i in range(n)]):
M = Matrix([vector_list_by_length[pivot_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
# Q1 = self.matrix()
# if self(M) == self:
# ct += 1
# print "ct = ", ct, " M = "
# print M
# print
if M.transpose() * Q3 * M == Q2: ## THIS DOES THE SAME THING! =(
Auto_list.append(M * Ainv)
## Cache the answer and return the list
self.__automorphisms = Auto_list
self.__number_of_automorphisms = len(Auto_list)
return Auto_list
def make_regular_matroid_from_matroid(matroid):
r"""
Attempt to construct a regular representation of a matroid.
INPUT:
- ``matroid`` -- a matroid.
OUTPUT:
Return a `(0, 1, -1)`-matrix over the integers such that, if the input is
a regular matroid, then the output is a totally unimodular matrix
representing that matroid.
EXAMPLES::
sage: from sage.matroids.utilities import make_regular_matroid_from_matroid
sage: make_regular_matroid_from_matroid(
....: matroids.CompleteGraphic(6)).is_isomorphic(
....: matroids.CompleteGraphic(6))
True
"""
import sage.matroids.linear_matroid
M = matroid
if isinstance(M, sage.matroids.linear_matroid.RegularMatroid):
return M
rk = M.full_rank()
# First create a reduced 0-1 matrix
B = list(M.basis())
NB = list(M.groundset().difference(B))
dB = {}
i = 0
for e in B:
dB[e] = i
i += 1
dNB = {}
i = 0
for e in NB:
dNB[e] = i
i += 1
A = Matrix(ZZ, len(B), len(NB), 0)
G = BipartiteGraph(A.transpose()) # Sage's BipartiteGraph uses the column set as first color class. This is an edgeless graph.
for e in NB:
C = M.circuit(B + [e])
for f in C.difference([e]):
A[dB[f], dNB[e]] = 1
# Change some entries from -1 to 1
entries = BipartiteGraph(A.transpose()).edges(labels=False)
while len(entries) > 0:
L = [G.shortest_path(u, v) for u, v in entries]
mindex, minval = min(enumerate(L), key=lambda x: len(x[1]))
# if minval = 0, there is an edge not spanned by the current subgraph. Its entry is free to be scaled any way.
if len(minval) > 0:
# Check the subdeterminant
S = frozenset(L[mindex])
rows = []
cols = []
for i in S:
if i < rk:
rows.append(i)
else:
cols.append(i - rk)
if A[rows, cols].det() != 0:
A[entries[mindex][0], entries[mindex][1] - rk] = -1
G.add_edge(entries[mindex][0], entries[mindex][1])
entries.pop(mindex)
return sage.matroids.linear_matroid.RegularMatroid(groundset=B + NB, reduced_matrix=A)
def is_globally_equivalent_to(self, other, return_matrix=False, check_theta_to_precision='sturm', check_local_equivalence=True):
"""
Determines if the current quadratic form is equivalent to the
given form over ZZ. If return_matrix is True, then we also return
the transformation matrix M so that self(M) == other.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: M = Matrix(ZZ, 4, 4, [1,2,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1])
sage: Q1 = Q(M)
sage: Q.(Q1) # optional -- souvigner
True
sage: MM = Q.is_globally_equivalent_to(Q1, return_matrix=True) # optional -- souvigner
sage: Q(MM) == Q1 # optional -- souvigner
True
::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q3 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: Q1.is_globally_equivalent_to(Q2) # optional -- souvigner
False
sage: Q1.is_globally_equivalent_to(Q3) # optional -- souvigner
True
sage: M = Q1.is_globally_equivalent_to(Q3, True) ; M # optional -- souvigner
[-1 -1 0]
[ 1 1 1]
[-1 0 0]
sage: Q1(M) == Q3 # optional -- souvigner
True
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.is_globally_equivalent_to(Q)
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.is_globally_equivalent_to()
"""
## only for definite forms
if not self.is_definite():
raise ValueError, "not a definite form in QuadraticForm.is_globally_equivalent_to()"
## Check that other is a QuadraticForm
#if not isinstance(other, QuadraticForm):
if not is_QuadraticForm(other):
raise TypeError, "Oops! You must compare two quadratic forms, but the argument is not a quadratic form. =("
## Now use the Souvigner code by default! =)
return other.is_globally_equivalent__souvigner(self, return_matrix) ## Note: We switch this because the Souvigner code has the opposite mapping convention to us. (It takes the second argument to the first!)
## ---------------------------------- Unused Code below ---------------------------------------------------------
## Check if the forms are locally equivalent
if (check_local_equivalence == True):
if not self.is_locally_equivalent_to(other):
return False
## Check that the forms have the same theta function up to the desired precision (this can be set so that it determines the cusp form)
if check_theta_to_precision != None:
if self.theta_series(check_theta_to_precision, var_str='', safe_flag=False) != other.theta_series(check_theta_to_precision, var_str='', safe_flag=False):
return False
## Make all possible matrices which give an isomorphism -- can we do this more intelligently?
## ------------------------------------------------------------------------------------------
## Find a basis of short vectors for one form, and try to match them with vectors of that length in the other one.
basis_for_self, self_lengths = self.basis_of_short_vectors(show_lengths=True)
max_len = max(self_lengths)
short_vectors_of_other = other.short_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis_for_self).transpose()
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of 'self' in the new basis
Q3 = other.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(short_vectors_of_other[self_lengths[i]]) for i in range(n)]):
M = Matrix([short_vectors_of_other[self_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
if M.transpose() * Q3 * M == Q2:
if return_matrix:
return A * M.inverse()
else:
return True
## If we got here, then there is no isomorphism
return False
def is_globally_equivalent__souvigner(self, other, return_transformation=False):
"""
Uses the Souvigner code to compute the number of automorphisms.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q1 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: M = Q.is_globally_equivalent__souvigner(Q1, True) ; M # optional -- souvigner
[ 0 0 -1]
[ 1 0 0]
[-1 1 1]
sage: Q1(M) == Q # optional -- souvigner
True
"""
## Write an input text file
F_filename = '/tmp/tmp_isom_input' + str(random()) + ".txt"
F = open(F_filename, 'w')
#F = tempfile.NamedTemporaryFile(prefix='tmp_isom_input', suffix=".txt") ## This failed because it may have hyphens, which are interpreted badly by the Souvigner code.
F.write("\n #1 \n")
## Write the first form
n = self.dim()
F.write(str(n) + "x0 \n") ## Use the lower-triangular form
for i in range(n):
for j in range(i+1):
if i == j:
F.write(str(2 * self[i,j]) + " ")
else:
F.write(str(self[i,j]) + " ")
F.write("\n")
## Write the second form
F.write("\n")
n = self.dim()
F.write(str(n) + "x0 \n") ## Use the lower-triangular form
for i in range(n):
for j in range(i+1):
if i == j:
F.write(str(2 * other[i,j]) + " ")
else:
F.write(str(other[i,j]) + " ")
F.write("\n")
F.flush()
#print "Input filename = ", F.name
#os.system("less " + F.name)
## Call the Souvigner automorphism code
souvigner_isom_path = os.path.join(SAGE_LOCAL,'bin','Souvigner_ISOM')
G1 = tempfile.NamedTemporaryFile(prefix='tmp_isom_ouput', suffix=".txt")
#print "Output filename = ", G1.name
#print "Executing the shell command: " + souvigner_isom_path + " '" + F.name + "' > '" + G1.name + "'"
os.system(souvigner_isom_path + " '" + F.name + "' > '" + G1.name +"'")
## Read the output
G2 = open(G1.name, 'r')
line = G2.readline()
if line.startswith("Error:"):
raise RuntimeError, "There is a problem using the souvigner code... " + line
elif line.find("not isomorphic") != -1: ## Checking if this text appears, if so then they're not isomorphic!
return False
else:
## Decide whether to read the transformation matrix, and return true
if not return_transformation:
F.close()
G1.close()
G2.close()
os.system("rm -f " + F_filename)
return True
else:
## Try to read the isomorphism matrix
M = Matrix(ZZ, n, n)
for i in range(n):
new_row_text = G2.readline().split()
#print new_row_text
for j in range(n):
M[i,j] = new_row_text[j]
## Remove temporary files and return the value
F.close()
G1.close()
G2.close()
os.system("rm -f " + F_filename)
if return_transformation:
return M.transpose()
else:
return True
#return True, M
## Raise and error if we're here:
raise RuntimeError, "Oops! There is a problem..."
def to_elt(alp):
ref = self.domain().reflection(alp)
m = Matrix([ref(x).to_vector() for x in self.domain().basis()])
return self(m.transpose())
def automorphisms(self):
"""
Return a list of the automorphisms of the quadratic form.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.number_of_automorphisms() # optional -- souvigner
48
sage: 2^3 * factorial(3)
48
sage: len(Q.automorphisms())
48
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.number_of_automorphisms() # optional -- souvigner
16
sage: aut = Q.automorphisms()
sage: len(aut)
16
sage: print([Q(M) == Q for M in aut])
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
sage: Q = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q.automorphisms()
[
[1 0 0] [-1 0 0]
[0 1 0] [ 0 -1 0]
[0 0 1], [ 0 0 -1]
]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.automorphisms()
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.automorphisms()
"""
## only for definite forms
if not self.is_definite():
raise ValueError("not a definite form in QuadraticForm.automorphisms()")
## Check for a cached value
try:
return self.__automorphisms
except AttributeError:
pass
## Find a basis of short vectors, and their lengths
basis, pivot_lengths = self.basis_of_short_vectors(show_lengths=True)
## List the relevant vectors by length
max_len = max(pivot_lengths)
vector_list_by_length = self.short_primitive_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis).transpose()
Ainv = A.inverse()
#A1 = A.inverse() * A.det()
#Q1 = A1.transpose() * self.matrix() * A1 ## This is the matrix of Q
#Q = self.matrix() * A.det()**2
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of Q in the new basis
Q3 = self.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
#ct = 0
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(vector_list_by_length[pivot_lengths[i]]) for i in range(n)]):
M = Matrix([vector_list_by_length[pivot_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
#Q1 = self.matrix()
#if self(M) == self:
#ct += 1
#print "ct = ", ct, " M = "
#print M
#print
if M.transpose() * Q3 * M == Q2: ## THIS DOES THE SAME THING! =(
Auto_list.append(M * Ainv)
## Cache the answer and return the list
self.__automorphisms = Auto_list
self.__number_of_automorphisms = len(Auto_list)
return Auto_list
def faces_in_smith(triangulation, angle_structure, cycles):
N = edge_equation_matrix_taut_reduced(triangulation, angle_structure,
cycles)
N = Matrix(N)
N = N.transpose()
return N.smith_form()
def is_globally_equivalent__souvigner(self,
other,
return_transformation=False):
"""
Uses the Souvigner code to compute the number of automorphisms.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q1 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: M = Q.is_globally_equivalent__souvigner(Q1, True) ; M # optional -- souvigner
[ 0 0 -1]
[ 1 0 0]
[-1 1 1]
sage: Q1(M) == Q # optional -- souvigner
True
"""
## Write an input text file
F_filename = '/tmp/tmp_isom_input' + str(random()) + ".txt"
F = open(F_filename, 'w')
#F = tempfile.NamedTemporaryFile(prefix='tmp_isom_input', suffix=".txt") ## This failed because it may have hyphens, which are interpreted badly by the Souvigner code.
F.write("\n #1 \n")
## Write the first form
n = self.dim()
F.write(str(n) + "x0 \n") ## Use the lower-triangular form
for i in range(n):
for j in range(i + 1):
if i == j:
F.write(str(2 * self[i, j]) + " ")
else:
F.write(str(self[i, j]) + " ")
F.write("\n")
## Write the second form
F.write("\n")
n = self.dim()
F.write(str(n) + "x0 \n") ## Use the lower-triangular form
for i in range(n):
for j in range(i + 1):
if i == j:
F.write(str(2 * other[i, j]) + " ")
else:
F.write(str(other[i, j]) + " ")
F.write("\n")
F.flush()
#print "Input filename = ", F.name
#os.system("less " + F.name)
## Call the Souvigner automorphism code
souvigner_isom_path = os.path.join(SAGE_LOCAL, 'bin', 'Souvigner_ISOM')
G1 = tempfile.NamedTemporaryFile(prefix='tmp_isom_ouput', suffix=".txt")
#print "Output filename = ", G1.name
#print "Executing the shell command: " + souvigner_isom_path + " '" + F.name + "' > '" + G1.name + "'"
os.system(souvigner_isom_path + " '" + F.name + "' > '" + G1.name + "'")
## Read the output
G2 = open(G1.name, 'r')
line = G2.readline()
if line.startswith("Error:"):
raise RuntimeError("There is a problem using the souvigner code... " +
line)
elif line.find(
"not isomorphic"
) != -1: ## Checking if this text appears, if so then they're not isomorphic!
return False
else:
## Decide whether to read the transformation matrix, and return true
if not return_transformation:
F.close()
G1.close()
G2.close()
os.system("rm -f " + F_filename)
return True
else:
## Try to read the isomorphism matrix
M = Matrix(ZZ, n, n)
for i in range(n):
new_row_text = G2.readline().split()
#print new_row_text
for j in range(n):
M[i, j] = new_row_text[j]
## Remove temporary files and return the value
F.close()
G1.close()
G2.close()
os.system("rm -f " + F_filename)
if return_transformation:
return M.transpose()
else:
return True
#return True, M
## Raise and error if we're here:
raise RuntimeError("Oops! There is a problem...")
def is_globally_equivalent_to(self,
other,
return_matrix=False,
check_theta_to_precision='sturm',
check_local_equivalence=True):
"""
Determines if the current quadratic form is equivalent to the
given form over ZZ. If return_matrix is True, then we also return
the transformation matrix M so that self(M) == other.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: M = Matrix(ZZ, 4, 4, [1,2,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1])
sage: Q1 = Q(M)
sage: Q.(Q1) # optional -- souvigner
True
sage: MM = Q.is_globally_equivalent_to(Q1, return_matrix=True) # optional -- souvigner
sage: Q(MM) == Q1 # optional -- souvigner
True
::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q3 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: Q1.is_globally_equivalent_to(Q2) # optional -- souvigner
False
sage: Q1.is_globally_equivalent_to(Q3) # optional -- souvigner
True
sage: M = Q1.is_globally_equivalent_to(Q3, True) ; M # optional -- souvigner
[-1 -1 0]
[ 1 1 1]
[-1 0 0]
sage: Q1(M) == Q3 # optional -- souvigner
True
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.is_globally_equivalent_to(Q)
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.is_globally_equivalent_to()
"""
## only for definite forms
if not self.is_definite():
raise ValueError(
"not a definite form in QuadraticForm.is_globally_equivalent_to()")
## Check that other is a QuadraticForm
#if not isinstance(other, QuadraticForm):
if not is_QuadraticForm(other):
raise TypeError(
"Oops! You must compare two quadratic forms, but the argument is not a quadratic form. =("
)
## Now use the Souvigner code by default! =)
return other.is_globally_equivalent__souvigner(
self, return_matrix
) ## Note: We switch this because the Souvigner code has the opposite mapping convention to us. (It takes the second argument to the first!)
## ---------------------------------- Unused Code below ---------------------------------------------------------
## Check if the forms are locally equivalent
if (check_local_equivalence == True):
if not self.is_locally_equivalent_to(other):
return False
## Check that the forms have the same theta function up to the desired precision (this can be set so that it determines the cusp form)
if check_theta_to_precision is not None:
if self.theta_series(
check_theta_to_precision, var_str='',
safe_flag=False) != other.theta_series(
check_theta_to_precision, var_str='', safe_flag=False):
return False
## Make all possible matrices which give an isomorphism -- can we do this more intelligently?
## ------------------------------------------------------------------------------------------
## Find a basis of short vectors for one form, and try to match them with vectors of that length in the other one.
basis_for_self, self_lengths = self.basis_of_short_vectors(
show_lengths=True)
max_len = max(self_lengths)
short_vectors_of_other = other.short_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis_for_self).transpose()
Q2 = A.transpose() * self.matrix(
) * A ## This is the matrix of 'self' in the new basis
Q3 = other.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange(
[len(short_vectors_of_other[self_lengths[i]]) for i in range(n)]):
M = Matrix([
short_vectors_of_other[self_lengths[i]][index_vec[i]]
for i in range(n)
]).transpose()
if M.transpose() * Q3 * M == Q2:
if return_matrix:
return A * M.inverse()
else:
return True
## If we got here, then there is no isomorphism
return False
def _borcherds_product_polyhedron(self, pole_order, prec, verbose=False):
r"""
Construct a polyhedron representing a cone of Heegner divisors. For internal use in the methods borcherds_input_basis() and borcherds_input_Qbasis().
INPUT:
- ``pole_order`` -- pole order
- ``prec`` -- precision
OUTPUT: a tuple consisting of an integral matrix M, a Polyhedron p, and a WeilRepModularFormsBasis X
"""
K = self.weilrep().base_field()
O_K = K.maximal_order()
S = self.gram_matrix()
wt = self.input_wt()
w = self.weilrep()
rds = w.rds()
norm_dict = w.norm_dict()
X = w.nearly_holomorphic_modular_forms_basis(wt,
pole_order,
prec,
verbose=verbose)
N = len([g for g in rds if not norm_dict[tuple(g)]])
v_list = w.coefficient_vector_exponents(0,
1,
starting_from=-pole_order,
include_vectors=True)
exp_list = [v[1] for v in v_list]
d = w._ds_to_hds()
v_list = [vector(d[tuple(v[0])]) for v in v_list]
d = K.discriminant()
positive = []
zero = vector([0] * (len(exp_list) + 1))
M = Matrix([
x.coefficient_vector(starting_from=-pole_order, ending_with=0)[:-N]
for x in X
])
vs = M.transpose().kernel().basis()
prec = floor(min(exp_list) / max(filter(bool, exp_list)))
norm_list = w._norm_form().short_vector_list_up_to_length(prec + 1)
units = w._units()
_w = w._w()
norm_list = [[a + b * _w for a, b in x] for x in norm_list]
excluded_list = set([])
if d >= -4:
if d == -4:
f = w.multiplication_by_i()
f2 = None
else:
f = w.multiplication_by_zeta()
f2 = f * f
ys = []
mult = len(units) // 2
for i, n in enumerate(exp_list):
if i not in excluded_list:
ieq = copy(zero)
ieq[i + 1] = 1
v1 = v_list[i]
if d >= -4:
v2 = f(v1)
j = next(j for j, x in enumerate(v_list)
if exp_list[i] == exp_list[j] and (all(
t in O_K
for t in x - v2) or all(t in O_K
for t in x + v2)))
ieq[j + 1] += 1
if i != j:
excluded_list.add(j)
y = copy(zero)
y[i + 1] = 1
y[j + 1] = -1
ys.append(y)
if f2 is not None:
v2 = f2(v1)
j = next(j for j, x in enumerate(v_list)
if exp_list[i] == exp_list[j] and (all(
t in O_K
for t in x - v2) or all(t in O_K
for t in x + v2)))
ieq[j + 1] += 1
if i != j:
excluded_list.add(j)
y = copy(zero)
y[i + 1] = 1
y[j + 1] = -1
ys.append(y)
for j, m in enumerate(exp_list[:i]):
if j not in excluded_list:
N = m / n
if N in ZZ and N > 1:
v2 = v_list[j]
ieq[j + 1] = mult * any(
all(t in O_K for t in x * v1 + u * v2)
for x in norm_list[N] for u in units)
positive.append(ieq) # * denominator(ieq)
p = Polyhedron(ieqs=positive,
eqns=[vector([0] + list(v)) for v in vs] + ys)
return M, p, X
