def signature_function_of_integral_matrix(V, prec=53):
"""
Computes the signature function sigma of V via numerical methods.
Returns two lists, the first representing a partition of [0, 1]:
x_0 = 0 < x_1 < x_2 < ... < x_n = 1
and the second list consisting of the values [v_0, ... , v_(n-1)]
of sigma on the interval (x_i, x_(i+1)). Currently, the value of
sigma *at* x_i is not computed.
"""
poly = alexander_poly_from_seifert(V)
RR = RealField(prec)
CC = ComplexField(prec)
pi = RR.pi()
I = CC.gen()
partition = [RR(0)] + [a for a, e in roots_on_unit_circle(poly, prec)
] + [RR(1)]
n = len(partition) - 1
values = []
for i in range(n):
omega = exp((2 * pi * I) * (partition[i] + partition[i + 1]) / 2)
A = (1 - omega) * V + (1 - omega.conjugate()) * V.transpose()
values.append(signature_via_numpy(A))
assert list(reversed(values)) == values
return partition, values
def from_conjugacy_class_index_to_polynomial(self, index):
""" A function converting from a conjugacy class index (starting at 1) to the local Euler polynomial.
Saves a sequence of processed polynomials, obtained from the local factors table, so it can reuse computations from prime to prime
This sequence is indexed by conjugacy class indices (starting at 1, filled with dummy first) and gives the corresponding polynomials in the form
[coeff_deg_0, coeff_deg_1, ...], where coeff_deg_i is in ComplexField(). This could be changed later, or made parametrizable
"""
try:
return self._from_conjugacy_class_index_to_polynomial_fn(index)
except AttributeError:
local_factors = self.local_factors_table()
field = ComplexField()
root_of_unity = exp(
(field.gen()) * 2 * field.pi() / int(self.character_field()))
local_factor_processed_pols = [
0
] # dummy to account for the shift in indices
for pol in local_factors:
local_factor_processed_pols.append(
process_polynomial_over_algebraic_integer(
pol, field, root_of_unity))
def tmp(conjugacy_class_index_start_1):
return local_factor_processed_pols[
conjugacy_class_index_start_1]
self._from_conjugacy_class_index_to_polynomial_fn = tmp
return self._from_conjugacy_class_index_to_polynomial_fn(index)
def from_conjugacy_class_index_to_polynomial(self, index):
""" A function converting from a conjugacy class index (starting at 1) to the local Euler polynomial.
Saves a sequence of processed polynomials, obtained from the local factors table, so it can reuse computations from prime to prime
This sequence is indexed by conjugacy class indices (starting at 1, filled with dummy first) and gives the corresponding polynomials in the form
[coeff_deg_0, coeff_deg_1, ...], where coeff_deg_i is in ComplexField(). This could be changed later, or made parametrizable
"""
try:
return self._from_conjugacy_class_index_to_polynomial_fn(index)
except AttributeError:
local_factors = self.local_factors_table()
field = ComplexField()
root_of_unity = exp((field.gen()) * 2 * field.pi() / int(self.character_field()))
local_factor_processed_pols = [0] # dummy to account for the shift in indices
for pol in local_factors:
local_factor_processed_pols.append(
process_polynomial_over_algebraic_integer(pol, field, root_of_unity))
def tmp(conjugacy_class_index_start_1):
return local_factor_processed_pols[conjugacy_class_index_start_1]
self._from_conjugacy_class_index_to_polynomial_fn = tmp
return self._from_conjugacy_class_index_to_polynomial_fn(index)
