def Deltoid(cls, par_type='rational'):
r"""
Return a deltoid motion.
"""
if par_type == 'rational':
FF = FunctionField(QQ, 't')
t = FF.gen()
C = {
_sage_const_0 : vector((_sage_const_0 , _sage_const_0 )),
_sage_const_1 : vector((_sage_const_1 , _sage_const_0 )),
_sage_const_2 : vector((_sage_const_4 *(t**_sage_const_2 - _sage_const_2
)/(t**_sage_const_2 + _sage_const_4 ), _sage_const_12 *t/(t**_sage_const_2 + _sage_const_4 ))),
_sage_const_3 : vector(((t**_sage_const_4 - _sage_const_13 *t**_sage_const_2 + _sage_const_4
)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 ),
_sage_const_6 *(t**_sage_const_3 - _sage_const_2 *t)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 )))
}
G = FlexRiGraph([[0, 1], [1, 2], [2, 3], [0, 3]])
return GraphMotion.ParametricMotion(G, C, 'rational', sampling_type='tan', check=False)
elif par_type == 'symbolic':
t = var('t')
C = {
_sage_const_0 : vector((_sage_const_0 , _sage_const_0 )),
_sage_const_1 : vector((_sage_const_1 , _sage_const_0 )),
_sage_const_2 : vector((_sage_const_4 *(t**_sage_const_2 - _sage_const_2
)/(t**_sage_const_2 + _sage_const_4 ), _sage_const_12 *t/(t**_sage_const_2 + _sage_const_4 ))),
_sage_const_3 : vector(((t**_sage_const_4 - _sage_const_13 *t**_sage_const_2 + _sage_const_4
)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 ),
_sage_const_6 *(t**_sage_const_3 - _sage_const_2 *t)/(t**_sage_const_4 + _sage_const_5 *t**_sage_const_2 + _sage_const_4 )))
}
G = FlexRiGraph([[0, 1], [1, 2], [2, 3], [0, 3]])
return ParametricGraphMotion.ParametricMotion(G, C, 'symbolic', sampling_type='tan', check=False)
else:
raise exceptions.ValueError('Deltoid with par_type ' + str(par_type) + ' is not supported.')
def _test(self):
from sage.all import QQ, PolynomialRing, GaussValuation, FunctionField
R = PolynomialRing(QQ, 'x')
x = R.gen()
v = GaussValuation(R, QQ.valuation(2))
K = FunctionField(QQ, 'x')
x = K.gen()
v = K.valuation(v)
K.valuation((v, K.hom(x/2), K.hom(2*x)))
def _test(self):
from sage.all import FunctionField, QQ, PolynomialRing
K = FunctionField(QQ, 'x')
x = K.gen()
R = PolynomialRing(K, 'y')
y = R.gen()
L = K.extension(y**3 - 1 / x**3 * y + 2 / x**4, 'y')
v = K.valuation(x)
v.extensions(L)
def time_is_semistable(self):
K = FunctionField(QQ, 'x')
x = K.gen()
R = PolynomialRing(K, 'T')
T = R.gen()
f = 64 * x**3 * T - 64 * x**3 + 36 * x**2 * T**2 + 208 * x**2 * T + 192 * x**2 + 9 * x * T**3 + 72 * x * T**2 + 240 * x * T + 64 * x + T**4 + 9 * T**3 + 52 * T**2 + 48 * T
L = K.extension(f, 'y')
Y = SmoothProjectiveCurve(L)
v = QQ.valuation(13)
M = SemistableModel(Y, v)
return M.is_semistable()
def _test(self):
from sage.all import GF, FunctionField, PolynomialRing
k = GF(4)
a = k.gen()
R = PolynomialRing(k, 'b')
b = R.gen()
l = k.extension(b**2 + b + a, 'b')
K = FunctionField(l, 'x')
x = K.gen()
R = PolynomialRing(K, 't')
t = R.gen()
F = t * x
F.factor(proof=False)
def _test(self):
from sage.all import PolynomialRing, QQ, NumberField, GaussValuation, FunctionField
R = PolynomialRing(QQ, 'x')
x = R.gen()
K = NumberField(x**6 + 126 * x**3 + 126, 'pi')
v = K.valuation(2)
R = PolynomialRing(K, 'x')
x = R.gen()
v = GaussValuation(R, v).augmentation(x, QQ(2) / 3)
F = FunctionField(K, 'x')
x = F.gen()
v = F.valuation(v)
S = PolynomialRing(F, 'y')
y = S.gen()
w0 = GaussValuation(S, v)
G = y**2 - x**3 - 3
w1 = w0.mac_lane_step(G)[0]
w1.mac_lane_step(G)
def make_function_field(K):
r"""
Return the function field isomorphic to this field, an isomorphism,
and its inverse.
INPUT:
- ``K`` -- a field
OUTPUT: A triple `(F,\phi,\psi)`, where `F` is a rational function field,
`\phi:K\to F` is a field isomorphism and `\psi` the inverse of `\phi`.
It is assumed that `K` is either the fraction field of a polynomial ring
over a finite field `k`, or a finite simple extension of such a field.
In the first case, `F=k_1(x)` is a rational function field over a finite
field `k_1`, where `k_1` as an *absolute* finite field isomorphic to `k`.
In the second case, `F` is a finite simple extension of a rational function
field as in the first case.
.. NOTE::
this command seems to be partly superflous by now, because the residue
of a valuation is already of type "function field" whenever this makes sense.
However, even if `K` is a function field over a finite field, it is not
guaranteed that the constant base field is a 'true' finite field, and then
it is important to change that.
"""
from mclf.curves.smooth_projective_curves import make_finite_field
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.categories.function_fields import FunctionFields
from sage.rings.function_field.function_field import is_FunctionField
if is_FunctionField(K):
k = K.constant_base_field()
if k.is_finite() and hasattr(k, "base_field"):
# k seems to be finite, but not a true finite field
# we construct a true finite field k1 isomorphic to k
# and F isomorphic to K with constant base field k1
k1, phi, psi = make_finite_field(k)
if hasattr(K, "polynomial"):
# K is an extension of a rational function field
K0 = K.rational_function_field()
F0, phi0, psi0 = make_function_field(K0)
f = K.polynomial().change_ring(phi0)
F = F0.extension(f)
return F, K.hom(F.gen(), phi0), F.hom(K.gen(), psi0)
else:
F = FunctionField(k1, K.variable_name())
return F, K.hom(F.gen(), phi), F.hom(K.gen(), psi)
else:
return K, K.Hom(K).identity(), K.Hom(K).identity()
if hasattr(K, "modulus") or hasattr(K, "polynomial"):
# we hope that K is a simple finite extension of a field which is
# isomorphic to a rational function field
K_base = K.base_field()
F_base, phi_base, psi_base = make_function_field(K_base)
if hasattr(K, "modulus"):
G = K.modulus()
else:
G = K.polynomial()
R = G.parent()
R_new = PolynomialRing(F_base, R.variable_name())
G_new = R_new([phi_base(c) for c in G.list()])
assert G_new.is_irreducible(), "G must be irreducible!"
# F = F_base.extension(G_new, R.variable_name())
F = F_base.extension(G_new, 'y')
# phi0 = R.hom(F.gen(), F)
# to construct phi:K=K_0[x]/(G) --> F=F_0[y]/(G),
# we first 'map' from K to K_0[x]
phi = K.hom(R.gen(), R, check=False)
# then from K_0[x] to F_0[y]
psi = R.hom(phi_base, R_new)
# then from F_0[y] to F = F_0[y]/(G)
phi = phi.post_compose(psi.post_compose(R_new.hom(F.gen(), F)))
psi = F.hom(K.gen(), psi_base)
return F, phi, psi
else:
# we hope that K is isomorphic to a rational function field over a
# finite field
if K in FunctionFields():
# K is already a function field
k = K.constant_base_field()
k_new, phi_base, psi_base = make_finite_field(k)
F = FunctionField(k_new, K.variable_name())
phi = K.hom(F.gen(), phi_base)
psi = F.hom(K.gen(), psi_base)
return F, phi, psi
elif hasattr(K, "function_field"):
F1 = K.function_field()
phi1 = F1.coerce_map_from(K)
psi1 = F1.hom(K.gen())
F, phi2, psi2 = make_function_field(F1)
phi = phi1.post_compose(phi2)
psi = psi2.post_compose(psi1)
return F, phi, psi
else:
raise NotImplementedError
# Utility functions:
#
# - pp as generator of function field over QQ
# - substitution in ratonal function, star involution
# - lists of monic and non-monic polys and forms over any base F
# - roots of a polynomial or binary form
# - affine linear combinations
from sage.all import (FunctionField, QQ, PolynomialRing, ProjectiveSpace,
VectorSpace, infinity)
from collections import Counter
Qp = FunctionField(QQ, 'p')
pp = Qp.gen()
def subs(f, p):
"""Substitute p for the variable of the rational function f.
"""
if f in QQ:
return f
n = f.numerator()(p)
d = f.denominator()(p)
if d:
return n / d
else:
return infinity
def star(r):
return r if r in QQ else subs(r, 1 / pp)
def make_function_field(K):
r"""
Return the function field isomorphic to this field, an isomorphism,
and its inverse.
INPUT:
- ``K`` -- a field
OUTPUT: A triple `(F,\phi,\psi)`, where `F` is a rational function field,
`\phi:K\to F` is a field isomorphism and `\psi` the inverse of `\phi`.
It is assumed that `K` is either the fraction field of a polynomial ring
over a finite field `k`, or a finite simple extension of such a field.
In the first case, `F=k_1(x)` is a rational function field over a finite
field `k_1`, where `k_1` as an *absolute* finite field isomorphic to `k`.
In the second case, `F` is a finite simple extension of a rational function
field as in the first case.
"""
from mclf.curves.smooth_projective_curves import make_finite_field
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.categories.function_fields import FunctionFields
if hasattr(K, "modulus") or hasattr(K, "polynomial"):
# we hope that K is a simple finite extension of a field which is
# isomorphic to a rational function field
K_base = K.base_field()
F_base, phi_base, psi_base = make_function_field(K_base)
if hasattr(K, "modulus"):
G = K.modulus()
else:
G = K.polynomial()
R = G.parent()
R_new = PolynomialRing(F_base, R.variable_name())
G_new = R_new([phi_base(c) for c in G.list()])
assert G_new.is_irreducible(), "G must be irreducible!"
# F = F_base.extension(G_new, R.variable_name())
F = F_base.extension(G_new, 'y')
# phi0 = R.hom(F.gen(), F)
# to construct phi:K=K_0[x]/(G) --> F=F_0[y]/(G),
# we first 'map' from K to K_0[x]
phi = K.hom(R.gen(), R, check=False)
# then from K_0[x] to F_0[y]
psi = R.hom(phi_base, R_new)
# then from F_0[y] to F = F_0[y]/(G)
phi = phi.post_compose(psi.post_compose(R_new.hom(F.gen(), F)))
psi = F.hom(K.gen(), psi_base)
return F, phi, psi
else:
# we hope that K is isomorphic to a rational function field over a
# finite field
if K in FunctionFields():
# K is already a function field
k = K.constant_base_field()
k_new, phi_base, psi_base = make_finite_field(k)
F = FunctionField(k_new, K.variable_name())
phi = K.hom(F.gen(), phi_base)
psi = F.hom(K.gen(), psi_base)
return F, phi, psi
elif hasattr(K, "function_field"):
F1 = K.function_field()
phi1 = F1.coerce_map_from(K)
psi1 = F1.hom(K.gen())
F, phi2, psi2 = make_function_field(F1)
phi = phi1.post_compose(phi2)
psi = psi2.post_compose(psi1)
return F, phi, psi
else:
raise NotImplementedError