189 lines
6.0 KiB
Python
189 lines
6.0 KiB
Python
class superelliptic_witt:
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def __init__(self, t, f):
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''' Define Witt function on C of the form [t] + V(f). '''
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self.curve = t.curve
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C = t.curve
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p = C.characteristic
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self.t = t #superelliptic_function
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self.f = f #superelliptic_function
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def __repr__(self):
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f = self.f
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t = self.t
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if f.function == 0:
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return "[" + str(t) + "]"
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if t.function == 0:
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return "V(" + str(f) + ")"
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return "[" + str(t) + "] + V(" + str(f) + ")"
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def __neg__(self):
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f = self.f
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t = self.t
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return superelliptic_witt(-t, -f)
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def __add__(self, other):
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C = self.curve
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second_coor = 0*C.x
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X = self.t
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Y = other.t
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F = C.base_ring
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p = F.characteristic()
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for i in range(1, p):
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second_coor -= binomial_prim(p, i)*X^i*Y^(p-i)
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return superelliptic_witt(self.t + other.t, self.f + other.f + second_coor)
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def __sub__(self, other):
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return self + (-other)
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def __rmul__(self, other):
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p = self.curve.characteristic
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if other in ZZ:
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if other == 0:
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return superelliptic_witt(0*C.x, 0*C.x)
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if other > 0:
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return self + (other-1)*self
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if other < 0:
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return (-other)*(-self)
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if other in QQ:
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other_integer = Integers(p^2)(other)
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return other_integer*self
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def __mul__(self, other):
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C = self.curve
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p = C.characteristic
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if isinstance(other, superelliptic_witt):
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t1 = self.t
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f1 = self.f
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t2 = other.t
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f2 = other.f
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return superelliptic_witt(t1*t2, t1^p*f2 + t2^p*f1)
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if isinstance(other, superelliptic_drw_form):
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h1 = other.h1
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h2 = other.h2
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omega = other.omega
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t = self.t
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f = self.f
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aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
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return superelliptic_drw_form(t*h1, aux_form, t^p*h2)
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def __eq__(self, other):
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return self.t == other.t and self.f == other.f
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def diffn(self, dy_w = 0):
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if dy_w == 0:
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dy_w = self.curve.dy_w()
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C = self.curve
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t = self.t
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f = self.f
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fC = C.polynomial
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F = C.base_ring
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Rxy.<x, y> = PolynomialRing(F, 2)
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if t.function == 0:
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return superelliptic_drw_form(0*C.x, 0*C.dx, f)
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t_polynomial = t.function
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num = t_polynomial.numerator()
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den = t_polynomial.denominator()
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num_t_fct = superelliptic_function(C, num)
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den_t_fct = superelliptic_function(C, den)
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inv_den_t_fct = superelliptic_function(C, 1/den)
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if den != 1:
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# d([N/D] + V(f)) = [1/D]*d([N]) - [N]*[D^(-2)]*d([D]) + dV(f)
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return ((den_t_fct)^(-1)).teichmuller()*num_t_fct.teichmuller().diffn() - ((den_t_fct)^(-2)).teichmuller()*num_t_fct.teichmuller()*den_t_fct.teichmuller().diffn() + superelliptic_drw_form(0*C.x, 0*C.dx, f)
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t_polynomial = Rxy(t_polynomial)
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M = t_polynomial.monomials()[0]
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a = t_polynomial.monomial_coefficient(M)
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#[P] = [aM] + Q, where Q = ([P] - [aM])
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aM_fct = superelliptic_function(C, a*M)
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Q = self - aM_fct.teichmuller()
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exp_x = M.exponents()[0][0]
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exp_y = M.exponents()[0][1]
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return Q.diffn() + exp_x*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x) + exp_y*(aM_fct/C.y).teichmuller()*dy_w
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def binomial_prim(p, i):
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return binomial(p, i)/p
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def reduce_rational_fct(fct, p):
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Rxy.<x, y> = PolynomialRing(QQ)
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Fxy = FractionField(Rxy)
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fct = Fxy(fct)
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num = Rxy(fct.numerator())
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den = Rxy(fct.denominator())
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num1 = Rxy(0)
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for m in num.monomials():
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a = num.monomial_coefficient(m)
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num1 += (a%p^2)*m
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den1 = Rxy(0)
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for m in den.monomials():
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a = den.monomial_coefficient(m)
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den1 += (a%p^2)*m
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return num1/den1
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def teichmuller(fct):
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C = fct.curve
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return superelliptic_witt(fct, 0*C.x)
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superelliptic_function.teichmuller = teichmuller
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#dy = [f(x)]'/2*y dx
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#[f1 + M] = [f1] + [M] + V(cos)
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#d[f1 + M] = d[f1] + d[M] + dV(f1*M)
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#M = b x^a
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#d[M] = a*[b x^(a-1)]
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def auxilliary_derivative(P):
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'''Return "derivative" of P, where P depends only on x. In other words d[P(x)].'''
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P0 = P.t.function
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P1 = P.f.function
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C = P.curve
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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P0 = Rx(P0.numerator())
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P1 = Rx(P1.numerator())
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if P0 == 0:
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return superelliptic_drw_form(0*C.x, 0*C.dx, P.f)
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M = P0.monomials()[0]
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a = P0.monomial_coefficient(M)
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#[P] = [aM] + Q, where Q = ([P] - [aM])
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aM_fct = superelliptic_function(C, a*M)
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Q = P - aM_fct.teichmuller()
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exp = M.exponents()[0]
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return auxilliary_derivative(Q) + exp*superelliptic_drw_form(aM_fct/C.x, 0*C.dx, 0*C.x)
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def mult_by_p(elt):
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C = elt.curve
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F = C.base_ring
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p = F.characteristic()
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if isinstance(elt, superelliptic_form):
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fct = elt.form
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Fxy, Rxy, x, y = C.fct_field
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omega = superelliptic_form(C, fct^p * x^(p-1))
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result = superelliptic_drw_form(0*C.x, elt, 0*C.x)
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return result
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if isinstance(elt, superelliptic_function):
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return superelliptic_witt(0*C.x, elt^p)
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if isinstance(elt, superelliptic_cech):
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om0 = elt.omega0
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f = elt.f
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return superelliptic_drw_cech(mult_by_p(om0), mult_by_p(f))
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def verschiebung(elt):
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C = elt.curve
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if isinstance(elt, superelliptic_function):
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return superelliptic_witt(0*C.x, elt)
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if isinstance(elt, superelliptic_form):
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return superelliptic_drw_form(0*C.x, elt, 0*C.x)
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superelliptic_form.verschiebung = verschiebung
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superelliptic_function.verschiebung = verschiebung
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def dy_w(C):
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'''Return d[y].'''
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fC = C.polynomial
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fC = superelliptic_function(C, fC)
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fC = fC.teichmuller()
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dy_w = 1/2* ((C.y)^(-1)).teichmuller()*auxilliary_derivative(fC)
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return dy_w
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superelliptic.dy_w = dy_w |