def differential_space(self):
"""
Return the vector space of the differential space `\Omega(D)` of the divisor `D`.
OUTPUT:
- a vector space isomorphic to `\Omega(D)`
- an isomorphism from the vector space to the differential space
- the inverse of the isomorphism
EXAMPLES::
sage: K.<x> = FunctionField(GF(5)); R.<t> = K[]
sage: F.<y> = K.extension(t^2 - x^3 - 1)
sage: O = F.maximal_order()
sage: I = O.ideal(x - 2)
sage: P1 = I.divisor().support()[0]
sage: Pinf = F.places_infinite()[0]
sage: D = -3*Pinf + P1
sage: V, from_V, to_V = D.differential_space()
sage: all(to_V(from_V(e)) == e for e in V)
True
"""
F = self._field
W = F.space_of_differentials()
k = F.constant_base_field()
fbasis, coordinates = self._differential_space()
n = len(fbasis)
V = k**n
def from_V(v):
f = sum(v[i] * fbasis[i] for i in range(n))
return differential(F, f)
def to_V(w):
return vector(coordinates(w._f))
from sage.rings.function_field.maps import (
FunctionFieldLinearMap, FunctionFieldPartiallyDefinedLinearMap)
mor_from_V = FunctionFieldLinearMap(Hom(V, W), from_V)
mor_to_V = FunctionFieldPartiallyDefinedLinearMap(Hom(W, V), to_V)
return V, mor_from_V, mor_to_V
def function_space(self):
"""
Return the vector space of the Riemann-Roch space of the divisor.
OUTPUT:
- a vector space, an isomorphism from the vector space
to the Riemann-Roch space, and its inverse.
EXAMPLES::
sage: K.<x> = FunctionField(GF(5)); R.<t> = PolynomialRing(K)
sage: F.<y> = K.extension(t^2-x^3-1)
sage: O = F.maximal_order()
sage: I = O.ideal(x-2)
sage: D = I.divisor()
sage: V, from_V, to_V = D.function_space()
sage: all(to_V(from_V(e)) == e for e in V)
True
"""
F = self._field
k = F.constant_base_field()
basis, coordinates = self._function_space()
n = len(basis)
V = k**n
def from_V(v):
return sum(v[i] * basis[i] for i in range(n))
def to_V(f):
return vector(coordinates(f))
from sage.rings.function_field.maps import (
FunctionFieldLinearMap, FunctionFieldPartiallyDefinedLinearMap)
mor_from_V = FunctionFieldLinearMap(Hom(V, F), from_V)
mor_to_V = FunctionFieldPartiallyDefinedLinearMap(Hom(F, V), to_V)
return V, mor_from_V, mor_to_V