def get_bound_dynamical(F, f, m=1, dynatomic=True, prec=53, emb=None):
"""
The hyperbolic distance from `j` which must contain the smallest map.
This defines the maximum possible distance from `j` to the `z_0` covariant
of the associated binary form `F` in the hyperbolic 3-space
for which the map `f`` could have smaller coefficients.
INPUT:
- ``F`` -- binary form of degree at least 3 with no multiple roots associated
to ``f``
- ``f`` -- a dynamical system on `P^1`
- ``m`` - positive integer. the period used to create ``F``
- ``dynatomic`` -- boolean. whether ``F`` is the periodic points or the
formal periodic points of period ``m`` for ``f``
- ``prec``-- positive integer. precision to use in CC
- ``emb`` -- embedding into CC
OUTPUT: a positive real number
EXAMPLES::
sage: from sage.dynamics.arithmetic_dynamics.endPN_minimal_model import get_bound_dynamical
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: f = DynamicalSystem([50*x^2 + 795*x*y + 2120*y^2, 265*x^2 + 106*y^2])
sage: get_bound_dynamical(f.dynatomic_polynomial(1), f)
35.5546923182219
"""
def coshdelta(z):
#The cosh of the hyperbolic distance from z = t+uj to j
return (z.norm() + 1) / (2 * z.imag())
if F.base_ring() != ComplexField(prec=prec):
if emb is None:
compF = F.change_ring(ComplexField(prec=prec))
else:
compF = F.change_ring(emb)
else:
compF = F
n = F.degree()
z0F, thetaF = covariant_z0(compF, prec=prec, emb=emb)
cosh_delta = coshdelta(z0F)
d = f.degree()
hF = e**f.global_height(prec=prec)
#get precomputed constants C,k
if m == 1:
C = 4 * d + 2
k = 2
else:
Ck_values = {(False, 2, 2): (322, 6), (False, 2, 3): (385034, 14),\
(False, 2, 4): (4088003923454, 30), (False, 3, 2): (18044, 8),\
(False, 4, 2): (1761410, 10), (False, 5, 2): (269283820, 12),\
(True, 2, 2): (43, 4), (True, 2, 3): (106459, 12),\
(True, 2, 4): (39216735905, 24), (True, 3, 2): (1604, 6),\
(True, 4, 2): (114675, 8), (True, 5, 2): (14158456, 10)}
try:
C, k = Ck_values[(dynatomic, d, m)]
except KeyError:
raise ValueError("constants not computed for this (m,d) pair")
if n == 2 and d == 2:
#bound with epsilonF = 1
bound = 2 * ((2 * C * (hF**k)) / (thetaF))
else:
bound = cosh(epsinv(F, (2**(n - 1)) * C * (hF**k) / thetaF, prec=prec))
return bound
def get_bound_dynamical(F, f, m=1, dynatomic=True, prec=53, emb=None):
"""
The hyperbolic distance from `j` which must contain the smallest map.
This defines the maximum possible distance from `j` to the `z_0` covariant
of the assocaited binary form `F` in the hyperbolic 3-space
for which the map `f`` could have smaller coefficients.
INPUT:
- ``F`` -- binary form of degree at least 3 with no multiple roots associated
to ``f``
- ``f`` -- a dynamical system on `P^1`
- ``m`` - positive integer. the period used to create ``F``
- ``dyantomic`` -- boolean. whether ``F`` is the periodic points or the
formal periodic points of period ``m`` for ``f``
- ``prec``-- positive integer. precision to use in CC
- ``emb`` -- embedding into CC
OUTPUT: a positive real number
EXAMPLES::
sage: from sage.dynamics.arithmetic_dynamics.endPN_minimal_model import get_bound_dynamical
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: f = DynamicalSystem([50*x^2 + 795*x*y + 2120*y^2, 265*x^2 + 106*y^2])
sage: get_bound_dynamical(f.dynatomic_polynomial(1), f)
35.5546923182219
"""
def coshdelta(z):
#The cosh of the hyperbolic distance from z = t+uj to j
return (z.norm() + 1)/(2*z.imag())
if F.base_ring() != ComplexField(prec=prec):
if emb is None:
compF = F.change_ring(ComplexField(prec=prec))
else:
compF = F.change_ring(emb)
else:
compF = F
n = F.degree()
z0F, thetaF = covariant_z0(compF, prec=prec, emb=emb)
cosh_delta = coshdelta(z0F)
d = f.degree()
hF = e**f.global_height(prec=prec)
#get precomputed constants C,k
if m == 1:
C = 4*d+2;k=2
else:
Ck_values = {(False, 2, 2): (322, 6), (False, 2, 3): (385034, 14),\
(False, 2, 4): (4088003923454, 30), (False, 3, 2): (18044, 8),\
(False, 4, 2): (1761410, 10), (False, 5, 2): (269283820, 12),\
(True, 2, 2): (43, 4), (True, 2, 3): (106459, 12),\
(True, 2, 4): (39216735905, 24), (True, 3, 2): (1604, 6),\
(True, 4, 2): (114675, 8), (True, 5, 2): (14158456, 10)}
try:
C,k = Ck_values[(dynatomic,d,m)]
except KeyError:
raise ValueError("constants not computed for this (m,d) pair")
if n == 2 and d == 2:
#bound with epsilonF = 1
bound = 2*((2*C*(hF**k))/(thetaF))
else:
bound = cosh(epsinv(F, (2**(n-1))*C*(hF**k)/thetaF, prec=prec))
return bound