rozniczkowanie form drw zrobione (?)
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File diff suppressed because one or more lines are too long
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@ -1,9 +1,9 @@
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p = 2
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m = 1
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p = 3
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m = 2
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F = GF(p)
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Rx.<x> = PolynomialRing(F)
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f = x
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f = x^3 - x
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C = superelliptic(f, m)
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xx = C.x
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AS = as_cover(C, [xx^5, xx^5 + xx^3])
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print(AS.magical_element())
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a = superelliptic_drw_form(C.one, 0*C.dx, 0*C.x)
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b = a+a+a+a+a+a+a+a+a
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print(b)
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@ -0,0 +1,30 @@
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def patch(C):
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if C.exponent != 2:
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raise ValueError("Not implemented yet!")
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Fxy, Rxy, x, y = C.fct_field
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F = C.base_ring
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Rx.<x> = PolynomialRing(F)
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f = C.polynomial
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g = C.genus()
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f_star = Rx(x^(2*g+2)*f(1/x))
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return superelliptic(f_star, 2)
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def second_patch(argument):
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C = argument.curve
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C1 = patch(C)
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Fxy, Rxy, x, y = C.fct_field
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g = C.genus()
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if isinstance(argument, superelliptic_function):
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fct = Fxy(argument.function)
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fct1 = Fxy(fct.subs({x : 1/x, y : y/x^(g+1)}))
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return superelliptic_function(C1, fct1)
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if isinstance(argument, superelliptic_form):
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fct = Fxy(argument.form)
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fct1 = Fxy(fct.subs({x : 1/x, y : y/x^(g+1)}))
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fct1 *= -x^(-2)
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return superelliptic_form(C1, fct1)
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def lift_form_to_drw(omega):
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A, B = regular_form(omega)
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A, B = A.change_ring(QQ), B.change_ring(QQ)
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print("%s dx + %s dy"%(A, B))
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@ -1,39 +1,105 @@
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class superelliptic_witt:
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def __init__(self, C, t, f0, f1):
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''' Define Witt function on C of the form [t] + f0([x], [y]) + V(f1). '''
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self.curve = C
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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.f0 = reduce_rational_fct(f0, p) #polynomial/rational function over Z/p^2
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self.f1 = f1 #superelliptic_function
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self.f = f #superelliptic_function
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def __repr__(self):
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f0 = self.f0
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f1 = self.f1
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f = self.f
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t = self.t
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return "[" + str(t) + "] + " + str(f0).replace("x", "[x]").replace("y", "[y]") + " + V(" + str(f1) + ")"
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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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return superelliptic_witt(C, self.t + other.t, self.f0 + other.f0, self.f1 + other.f1)
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def teichmuller_representation(self):
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'''Represents as [f] + V(g), i.e. f0 = 0.'''
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C = self.curve
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Fxy, Rxy, x, y = self.fct_field
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F = C.base_ring
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function = Rxy(self.f0)
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if self.f0 == 0:
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return self
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M = Rxy(function.monomials()[0])
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a = F(function.monomial_coefficient(M))
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fct1 = fct - superelliptic_function(C, a*M)
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function1 = fct1.function
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return teichmuller(fct1) + superelliptic_witt(C, (a.lift())^p*M.change_ring(QQ), superelliptic_function(C, function1^2*a*M + function1*(a*M)^2))
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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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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 antiteichmuller_representation(self):
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'''Represents as f([x], [y]) + V(g), i.e. teichmuller part is zero.'''
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return 0
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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)^(-1)
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return other_integer*self
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def __mul__(self, other):
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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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p = self.curve.characteristic
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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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p = other.curve.characteristic
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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):
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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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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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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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@ -53,58 +119,127 @@ def reduce_rational_fct(fct, p):
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def teichmuller(fct):
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C = fct.curve
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Fxy, Rxy, x, y = C.fct_field
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F = C.base_ring
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function = Rxy(fct.function)
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if function == 0:
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return superelliptic_witt(C, 0, 0*C.x)
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M = Rxy(function.monomials()[0])
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a = F(function.monomial_coefficient(M))
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fct1 = fct - superelliptic_function(C, a*M)
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function1 = fct1.function
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return teichmuller(fct1) + superelliptic_witt(C, (a.lift())^p*M.change_ring(QQ), superelliptic_function(C, function1^2*a*M + function1*(a*M)^2))
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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. '''
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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)
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P1 = Rx(P1)
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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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class superelliptic_drw_form:
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def __init__(self, C, omega_x, omega_y, omega, h):
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'''Form [omega_x] d[x] + [omega_y] d[y] + V(omega) + dV([h])'''
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self.curve = C
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self.omega_x = omega_x
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self.omega_y = omega_y
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def __init__(self, h1, omega, h2):
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'''Form [h1] d[x] + V(omega) + dV([h])'''
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self.curve = h1.curve
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self.h1 = h1
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self.omega = omega
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self.h = h
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self.h2 = h2
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def __eq__(self, other):
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eq1 = (self.omega1 == self.omega1)
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eq1 = (self.h1 == self.h1)
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try:
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H = (self.h - other.h).pthroot()
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H = (self.h2 - other.h2).pth_root()
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except:
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return False
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eq2 = (self.omega2 - other.omega2).cartier() - H.diffn()
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eq2 = (self.omega - other.omega).cartier() - H.diffn()
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if eq1 and eq2:
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return True
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return False
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def __repr__(self):
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C = self.curve
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omega_x = self.omega_x
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omega_y = self.omega_y
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h = self.h
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return str(omega_x) + "] dx + [" + str(omega_x.form) + "] d[x] " + "+ V(" + str(omega2) + ") + dV([" + str(h) +"])"
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h1 = self.h1
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omega = self.omega
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h2 = self.h2
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result = ""
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if h1.function != 0:
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result += "[" + str(h1) + "] d[x]"
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if h1.function !=0 and omega.form != 0:
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result += " + "
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if omega.form != 0:
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result += "V(" + str(omega) + ")"
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if h2.function !=0 and omega.form != 0:
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result += " + "
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if h2.function != 0:
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result += "dV([" + str(h2) +"])"
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if result == "":
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result += "0"
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return result
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def __rmul__(self, other):
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h1 = self.h1
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h2 = self.h2
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omega = self.omega
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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_drw_form(0*C.x, 0*C.dx, 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)^(-1)
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return other_integer*self
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t = other.t
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f = other.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 __neg__(self):
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C = self.curve
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h1 = self.h1
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h2 = self.h2
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omega = self.omega
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return superelliptic_drw_form(-h1, -omega, -h2)
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def __add__(self, other):
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C = self.curve
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h1 = self.h1
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h2 = self.h2
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omega = self.omega
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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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aux = (teichmuller(h1) + teichmuller(H1))*superelliptic_drw_form(C.one, 0*C.dx, 0*C.x)
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h1_aux = aux.h1
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h2_aux = aux.h2
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omega_aux = aux.omega
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return superelliptic_drw_form(h1_aux, omega + OMEGA + omega_aux, h2 + H2 + h2_aux)
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def frobenius(self):
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C = self.curve
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h1 = self.h1
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h2 = self.h2
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p = C.characteristic
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return h1^p*C.x^(p-1)*C.dx + h2.diffn()
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def mult_by_p(omega):
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C = omega.curve
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fct = omega.form
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Fxy, Rxy, x, y = C.fct_field
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omega2 = superelliptic_form(C, fct^p * x^(p-1))
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result = superelliptic_drw_form(C, 0*C.dx, omega2, 0*C.x)
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omega = superelliptic_form(C, fct^p * x^(p-1))
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result = superelliptic_drw_form(C, 0*C.dx, omega, 0*C.x)
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return result
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def basis_W2Omega(C):
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basis = C.holomorphic_differentials_basis()
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result = []
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for omega in basis:
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result += [mult_by_p(omega)]
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image_of_cartier = []
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return result
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@ -21,6 +21,11 @@ class superelliptic_form:
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g = reduction(C, g1 - g2)
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return superelliptic_form(C, g)
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def __neg__(self):
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C = self.curve
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g = self.form
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return superelliptic_form(C, -g)
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def __repr__(self):
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g = self.form
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if len(str(g)) == 1:
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@ -32,6 +37,9 @@ class superelliptic_form:
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omega = self.form
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return superelliptic_form(C, constant*omega)
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def __eq__(self, other):
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return self.form == other.form
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def cartier(self):
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'''Computes Cartier operator on the form. Idea: y^m = f(x) -> y^(p^r - 1) = f(x)^M, where r = ord_p(m),
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M = (p^r - 1)/m. Thus h(x)/y^j dx = h(x) f(x)^(M*j)/y^(p^r * j) dx. Thus C(h(x)/y^j dx) = 1/y^(p^(r-1)*j) C(h(x) f(x)^(M*j) dx).'''
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@ -38,6 +38,11 @@ class superelliptic_function:
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g = reduction(C, g1 + g2)
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return superelliptic_function(C, g)
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def __neg__(self):
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C = self.curve
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g = self.function
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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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