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rec_coeff.py
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# -*- coding: utf-8; mode: sage -*-
from .local_invariants import (xi_p, eta_p,
delta_p, small_d, xi_to_xi_dash)
from sage.all import PolynomialRing, QQ, cached_function, ZZ, hilbert_symbol
from .jordan_block import JordanBlocks
@cached_function
def _pol_ring():
R = PolynomialRing(QQ, names="x")
return R
def cbb2_1(q, q2):
'''
A polynomial defined before Theorem 4.1 in the Katsurada's paper.
q, q2: instances of jordan_block.JordanBlocks.
q: degree n
q2: degree n - 1
'''
p = q.p
X = _pol_ring().gens()[0]
n = q.dim()
if n % 2 == 0:
xi = xi_p(q)
return (1 - p ** (n // 2) * xi * X) / (1 - p ** (n + 1) * X ** 2)
else:
if n > 1:
xi_tilde = xi_p(q2)
else:
xi_tilde = 1
return ZZ(1) / (1 - p ** ((n + 1) // 2) * xi_tilde * X)
def _invariants_1_even(q, q2):
xi = xi_p(q)
xi_dash = xi_to_xi_dash(xi)
eta_tilde = eta_p(q2)
delta = delta_p(q)
delta_tilde = delta_p(q2)
return {"xi_dash": xi_dash,
"eta_tilde": eta_tilde,
"xi": xi,
"delta": delta,
"delta_tilde": delta_tilde}
def _invariants_1_odd(q, q2):
n = q.dim()
eta = eta_p(q)
delta = delta_p(q)
if n > 1:
xi_tilde = xi_p(q2)
xi_tilde_dash = xi_to_xi_dash(xi_tilde)
delta_tilde = delta_p(q2)
else:
xi_tilde = xi_tilde_dash = 1
delta_tilde = 0
return {"eta": eta,
"delta": delta,
"xi_tilde": xi_tilde,
"xi_tilde_dash": xi_tilde_dash,
"delta_tilde": delta_tilde}
def _rat_func_1_even(p, n,
xi_dash=None, eta_tilde=None, xi=None,
delta=None, delta_tilde=None):
X = _pol_ring().gens()[0]
num = ((-1) ** (xi + 1) * xi_dash * eta_tilde *
(1 - p ** (n // 2 + 1) * xi * X) *
(p ** (n // 2) * X) ** (delta - delta_tilde + xi ** 2) *
p ** (delta / 2))
denom = 1 - p ** (n + 1) * X ** 2
return num / denom
def _rat_func_1_odd(p, n,
delta_tilde=None,
eta=None,
xi_tilde_dash=None,
xi_tilde=None,
delta=None):
X = _pol_ring().gens()[0]
num = ((-1) ** xi_tilde * xi_tilde_dash * eta *
(p ** ((n - 1) // 2) * X) ** (delta - delta_tilde + 2 - xi_tilde ** 2) *
p ** ((2 * delta - delta_tilde + 2) / 2))
denom = 1 - p ** ((n + 1) / 2) * X * xi_tilde
return num / denom
def cbb2_0(q, q2):
'''
A polynomial defined before Theorem 4.1 in the Katsurada's paper.
q: degree n
q2: degree n - 1
'''
p = q.p
n = q.dim()
if n % 2 == 0:
d = _invariants_1_even(q, q2)
return _rat_func_1_even(p, n, **d)
else:
d = _invariants_1_odd(q, q2)
return _rat_func_1_odd(p, n, **d)
two = ZZ(2)
def _invariants_2_common(b1, q2):
'''
b1 is an instance of JordanBlock2.
q2: an instance of jordan_block.JordanBlocks of dim n - 2
'''
n = q2.dim() + 2
m = b1.m
q = b1 + q2
q3 = JordanBlocks([(m, 1)], two) + q2
delta = delta_p(q)
delta_tilde = delta_p(q3)
delta_hat = delta_p(q2)
# Definition of sigma
if ((n % 2 == 0 and b1.type == 'u' and small_d(q) % 2 == 1)
or
(n % 2 == 0 and b1.type != 'u' and xi_p(q2) == 0)):
sigma = (2 * delta_tilde - delta - delta_hat + 2) / two
elif n % 2 == 1 and b1.type != 'u' and small_d(q3) % 2 == 0:
sigma = ZZ(2)
else:
sigma = ZZ(0)
return {'sigma': sigma,
'delta': delta,
'delta_tilde': delta_tilde,
'delta_hat': delta_hat}
def _invariants_2_even(b1, q2):
n = q2.dim() + 2
m = b1.m
q = b1 + q2
xi = xi_p(q)
xi_dash = xi_to_xi_dash(xi)
xi_hat = xi_p(q2)
xi_hat_dash = xi_to_xi_dash(xi_hat)
# Definition of eta_tilde
if b1.type == 'u' and small_d(q2) % 2 == 0:
_q = JordanBlocks([(m, b1._mat_prim[(1, 1)])], two)
eta_tilde = eta_p(_q + q2)
elif b1.type != 'u' and xi_hat != 0:
eta_tilde = ((-1) ** (((n - 1) ** 2 - 1) // 8) * q2.hasse_invariant__OMeara()
* hilbert_symbol(two ** m,
(-1) ** (n // 2 - 1) * q2.Gram_det(),
2))
else:
eta_tilde = ZZ(1)
res = {"xi": xi,
"xi_dash": xi_dash,
"xi_hat": xi_hat,
"xi_hat_dash": xi_hat_dash,
"eta_tilde": eta_tilde}
res.update(_invariants_2_common(b1, q2))
return res
def _invariants_2_odd(b1, q2):
m = b1.m
q = b1 + q2
eta = eta_p(q)
eta_hat = eta_p(q2)
q3 = JordanBlocks([(m, 1)], two) + q2
if b1.type != 'u' and small_d(q3) % 2 == 0:
xi_tilde = 1
else:
xi_tilde = 0
res = {'eta': eta,
'eta_hat': eta_hat,
'xi_tilde': xi_tilde}
res.update(_invariants_2_common(b1, q2))
return res
def _rat_funcs_even(n, xi=None, xi_dash=None, xi_hat=None,
xi_hat_dash=None, eta_tilde=None, sigma=None,
delta=None, delta_tilde=None, delta_hat=None):
res = {}
X = _pol_ring().gens()[0]
# 11
num = 1 - two ** (n // 2) * xi * X
denom = 1 - two ** (n + 1) * X ** 2
res['11'] = num / denom
# 10
expt = delta - delta_tilde + xi ** 2 + sigma
num = ((-1) ** (xi + 1) * xi_dash * eta_tilde *
(1 - two ** (n // 2 + 1) * X * xi) *
X ** expt * 2 ** (delta / two + (n // 2) * expt))
denom = 1 - two ** (n + 1) * X ** 2
res['10'] = num / denom
# 21
num = ZZ(1)
denom = 1 - 2 ** (n // 2) * xi_hat * X
res['21'] = num / denom
# 20
expt = delta_tilde - delta_hat + 2 - xi_hat ** 2 - sigma
num = ((-1) ** xi_hat * xi_hat_dash * eta_tilde *
X ** expt *
2 ** (((n - 2) // 2) * expt +
(2 * delta_tilde - delta_hat + 2 - 2 * sigma) / two))
denom = 1 - 2 ** (n // 2) * X * xi_hat
res['20'] = num / denom
return res
def _rat_funcs_odd(n, sigma=None, delta=None, delta_tilde=None,
delta_hat=None, eta=None, eta_hat=None, xi_tilde=None):
res = {}
X = _pol_ring().gens()[0]
# 11
num = ZZ(1)
denom = 1 - 2 ** ((n + 1) // 2) * xi_tilde * X
res['11'] = num / denom
# 10
expt = delta - delta_tilde + 2 - xi_tilde ** 2 + sigma
num = ((-1) ** xi_tilde * eta *
X ** expt *
2 ** ((n - 1) // 2 * expt + (2 * delta - delta_tilde + 2 + sigma) / 2))
denom = 1 - 2 ** ((n + 1) // 2) * X * xi_tilde
res['10'] = num / denom
# 21
num = 1 - 2 ** ((n - 1) // 2) * xi_tilde * X
denom = 1 - 2 ** n * X ** 2
res['21'] = num / denom
# 20
expt = delta_tilde - delta_hat + xi_tilde ** 2 - sigma
num = ((-1) ** (xi_tilde + 1) * eta_hat *
(1 - 2 ** ((n + 1) // 2) * X * xi_tilde) *
X ** expt * 2 ** ((n - 1) // 2 * expt + (delta_tilde - sigma) / 2))
denom = 1 - 2 ** n * X ** 2
res['20'] = num / denom
return res
def cbb2_dict(b1, q2):
n = q2.dim() + 2
if n % 2 == 0:
d = _invariants_2_even(b1, q2)
return _rat_funcs_even(n, **d)
if n % 2 == 1:
d = _invariants_2_odd(b1, q2)
return _rat_funcs_odd(n, **d)