def compute_separable_model(self):
r"""
Compute a separable model of the curve (if necessary).
OUTPUT: ``None``
This function only has to be called only once. It then decides whether
or not the function field of the curve is given as a separable extension
of the base field or not. If it is not separable then we compute a
separable model, which is a tripel `(Y_1,\phi, q)` where
- `Y_1` is a smooth projective curve over the same constant base field
`k` as the curve `Y` itself, and which is given by a separable extension,
- `\phi` is a field homomorphism from the function field of `Y_1` into
the function field of `Y` corresponding to a purely inseparable extension,
- `q` is the degree of the extension given by `\phi`, i.e. the degree of
inseparability of the map `Y\to Y_1` given by `\phi`.
Note that `q` is a power of the characteristic of `k`.
"""
F = self.function_field()
p = F.characteristic()
if p == 0:
self._is_separable = True
return
F0 = F.base_field()
if F is F0:
self._is_separable = True
return
G = F.polynomial()
q = ZZ(1)
while q < G.degree():
if all([G[i].is_zero() for i in range(G.degree()+1) if not (p*q).divides(i)]):
q = q*p
else:
break
# now q is the degree of inseparability
if q.is_one():
self._is_separable = True
return
# now q>1 and hence F/F_0 is inseparable
self._is_separable = False
self._degree_of_inseparability = q
R1 = PolynomialRing(F0, F.variable_name()+'_s')
G1 = R1([G[q*i] for i in range((G.degree()/q).floor()+1)])
F1 = F0.extension(G1, R1.variable_name())
self._separable_model = SmoothProjectiveCurve(F1)
self._phi = F1.hom([F.gen()**q])
self._degree_of_inseparability = q
