def divisor_of_function(self, r):
"""
Return the divisor of a function on a curve.
INPUT: r is a rational function on X
OUTPUT:
- ``list`` - The divisor of r represented as a list of
coefficients and points. (TODO: This will change to a more
structural output in the future.)
EXAMPLES::
sage: F = GF(5)
sage: P2 = AffineSpace(2, F, names = 'xy')
sage: R = P2.coordinate_ring()
sage: x, y = R.gens()
sage: f = y^2 - x^9 - x
sage: C = Curve(f)
sage: K = FractionField(R)
sage: r = 1/x
sage: C.divisor_of_function(r) # todo: not implemented (broken)
[[-1, (0, 0, 1)]]
sage: r = 1/x^3
sage: C.divisor_of_function(r) # todo: not implemented (broken)
[[-3, (0, 0, 1)]]
"""
F = self.base_ring()
f = self.defining_polynomial()
pts = self.places_on_curve()
numpts = len(pts)
R = f.parent()
x, y = R.gens()
R0 = PolynomialRing(F, 3, names=[str(x), str(y), "t"])
vars0 = R0.gens()
t = vars0[2]
divf = []
for pt0 in pts:
if pt0[2] != F(0):
lcs = self.local_coordinates(pt0, 5)
yt = lcs[1]
xt = lcs[0]
ldg = degree_lowest_rational_function(r(xt, yt), t)
if ldg[0] != 0:
divf.append([ldg[0], pt0])
return divf
def divisor_of_function(self, r):
"""
Return the divisor of a function on a curve.
INPUT: r is a rational function on X
OUTPUT:
- ``list`` - The divisor of r represented as a list of
coefficients and points. (TODO: This will change to a more
structural output in the future.)
EXAMPLES::
sage: F = GF(5)
sage: P2 = AffineSpace(2, F, names = 'xy')
sage: R = P2.coordinate_ring()
sage: x, y = R.gens()
sage: f = y^2 - x^9 - x
sage: C = Curve(f)
sage: K = FractionField(R)
sage: r = 1/x
sage: C.divisor_of_function(r) # todo: not implemented (broken)
[[-1, (0, 0, 1)]]
sage: r = 1/x^3
sage: C.divisor_of_function(r) # todo: not implemented (broken)
[[-3, (0, 0, 1)]]
"""
F = self.base_ring()
f = self.defining_polynomial()
pts = self.places_on_curve()
numpts = len(pts)
R = f.parent()
x,y = R.gens()
R0 = PolynomialRing(F,3,names = [str(x),str(y),"t"])
vars0 = R0.gens()
t = vars0[2]
divf = []
for pt0 in pts:
if pt0[2] != F(0):
lcs = self.local_coordinates(pt0,5)
yt = lcs[1]
xt = lcs[0]
ldg = degree_lowest_rational_function(r(xt,yt),t)
if ldg[0] != 0:
divf.append([ldg[0],pt0])
return divf
def divisor_of_function(self, r):
"""
Return the divisor of a function on a curve.
INPUT: r is a rational function on X
OUTPUT:
- ``list`` - The divisor of r represented as a list of
coefficients and points. (TODO: This will change to a more
structural output in the future.)
EXAMPLES::
sage: FF = FiniteField(5)
sage: P2 = ProjectiveSpace(2, FF, names = ['x','y','z'])
sage: R = P2.coordinate_ring()
sage: x, y, z = R.gens()
sage: f = y^2*z^7 - x^9 - x*z^8
sage: C = Curve(f)
sage: K = FractionField(R)
sage: r = 1/x
sage: C.divisor_of_function(r) # todo: not implemented !!!!
[[-1, (0, 0, 1)]]
sage: r = 1/x^3
sage: C.divisor_of_function(r) # todo: not implemented !!!!
[[-3, (0, 0, 1)]]
"""
F = self.base_ring()
f = self.defining_polynomial()
x, y, z = f.parent().gens()
pnts = self.rational_points()
divf = []
for P in pnts:
if P[2] != F(0):
# What is the '5' in this line and the 'r()' in the next???
lcs = self.local_coordinates(P,5)
ldg = degree_lowest_rational_function(r(lcs[0],lcs[1]),z)
if ldg[0] != 0:
divf.append([ldg[0],P])
return divf
def divisor_of_function(self, r):
"""
Return the divisor of a function on a curve.
INPUT: r is a rational function on X
OUTPUT:
- ``list`` - The divisor of r represented as a list of
coefficients and points. (TODO: This will change to a more
structural output in the future.)
EXAMPLES::
sage: FF = FiniteField(5)
sage: P2 = ProjectiveSpace(2, FF, names = ['x','y','z'])
sage: R = P2.coordinate_ring()
sage: x, y, z = R.gens()
sage: f = y^2*z^7 - x^9 - x*z^8
sage: C = Curve(f)
sage: K = FractionField(R)
sage: r = 1/x
sage: C.divisor_of_function(r) # todo: not implemented !!!!
[[-1, (0, 0, 1)]]
sage: r = 1/x^3
sage: C.divisor_of_function(r) # todo: not implemented !!!!
[[-3, (0, 0, 1)]]
"""
F = self.base_ring()
f = self.defining_polynomial()
x, y, z = f.parent().gens()
pnts = self.rational_points()
divf = []
for P in pnts:
if P[2] != F(0):
# What is the '5' in this line and the 'r()' in the next???
lcs = self.local_coordinates(P, 5)
ldg = degree_lowest_rational_function(r(lcs[0], lcs[1]), z)
if ldg[0] != 0:
divf.append([ldg[0], P])
return divf
