105 lines
2.8 KiB
Python
105 lines
2.8 KiB
Python
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#Class of rational functions on a superelliptic curve C. g = g(x, y) is a polynomial
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#defining the function.
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class superelliptic_function:
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def __init__(self, C, g):
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F = C.base_ring
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Rxy.<x, y> = PolynomialRing(F, 2)
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Fxy = FractionField(Rxy)
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f = C.polynomial
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r = f.degree()
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m = C.exponent
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self.curve = C
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g = reduction(C, g)
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self.function = g
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def __repr__(self):
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return str(self.function)
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def jth_component(self, j):
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g = self.function
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C = self.curve
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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Fx.<x> = FractionField(Rx)
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FxRy.<y> = PolynomialRing(Fx)
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g = FxRy(g)
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return coff(g, j)
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def __add__(self, other):
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C = self.curve
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g1 = self.function
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g2 = other.function
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g = reduction(C, g1 + g2)
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return superelliptic_function(C, g)
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def __sub__(self, other):
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C = self.curve
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g1 = self.function
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g2 = other.function
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g = reduction(C, g1 - g2)
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return superelliptic_function(C, g)
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def __mul__(self, other):
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C = self.curve
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g1 = self.function
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g2 = other.function
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g = reduction(C, g1 * g2)
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return superelliptic_function(C, g)
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def __truediv__(self, other):
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C = self.curve
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g1 = self.function
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g2 = other.function
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g = reduction(C, g1 / g2)
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return superelliptic_function(C, g)
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def __pow__(self, exp):
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C = self.curve
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g = self.function
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return superelliptic_function(C, g^(exp))
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def diffn(self):
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C = self.curve
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f = C.polynomial
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m = C.exponent
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F = C.base_ring
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g = self.function
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Rxy.<x, y> = PolynomialRing(F, 2)
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Fxy = FractionField(Rxy)
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g = Fxy(g)
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A = g.derivative(x)
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B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
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return superelliptic_form(C, A+B)
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def expansion_at_infty(self, i = 0, prec=10):
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C = self.curve
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f = C.polynomial
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m = C.exponent
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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f = Rx(f)
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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RptW.<W> = PolynomialRing(Rt)
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RptWQ = FractionField(RptW)
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Rxy.<x, y> = PolynomialRing(F)
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RxyQ = FractionField(Rxy)
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fct = self.function
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fct = RxyQ(fct)
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r = f.degree()
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delta, a, b = xgcd(m, r)
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b = -b
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M = m/delta
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R = r/delta
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while a<0:
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a += R
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b += M
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g = (x^r*f(x = 1/x))
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gW = RptWQ(g(x = t^M * W^b)) - W^(delta)
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ww = naive_hensel(gW, F, start = root_of_unity(F, delta)^i, prec = prec)
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xx = Rt(1/(t^M*ww^b))
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yy = 1/(t^R*ww^a)
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return Rt(fct(x = Rt(xx), y = Rt(yy)))
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