DeRhamComputation/superelliptic.ipynb

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2021-08-19 22:35:11 +02:00
{
"cells": [
{
"cell_type": "code",
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"execution_count": 55,
2022-03-07 13:48:30 +01:00
"metadata": {
"collapsed": false
},
"outputs": [
],
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"source": [
"class superelliptic:\n",
" \n",
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" def __init__(self, f, m):\n",
" Rx = f.parent()\n",
" x = Rx.gen()\n",
" F = Rx.base() \n",
" Rx.<x> = PolynomialRing(F)\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" self.polynomial = Rx(f)\n",
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" self.exponent = m\n",
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" self.base_ring = F\n",
" self.characteristic = F.characteristic()\n",
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" \n",
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" r = Rx(f).degree()\n",
" delta = GCD(r, m)\n",
" \n",
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" def __repr__(self):\n",
" f = self.polynomial\n",
" m = self.exponent\n",
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" F = self.base_ring\n",
" return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over ' + str(F)\n",
"\n",
" \n",
" def basis_holomorphic_differentials_degree(self):\n",
" f = self.polynomial\n",
" m = self.exponent\n",
" r = f.degree()\n",
" delta = GCD(r, m)\n",
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" F = self.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" #########basis of holomorphic differentials and de Rham\n",
"\n",
" basis_holo = []\n",
" degrees0 = {}\n",
" k = 0\n",
"\n",
" for j in range(1, m):\n",
" for i in range(1, r):\n",
" if (r*j - m*i >= delta):\n",
" basis_holo += [superelliptic_form(self, Fxy(x^(i-1)/y^j))]\n",
" degrees0[k] = (i-1, j)\n",
" k = k+1\n",
"\n",
" return(basis_holo, degrees0)\n",
"\n",
" def holomorphic_differentials_basis(self):\n",
" basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()\n",
" return basis_holo\n",
" \n",
" def degrees_holomorphic_differentials(self):\n",
" basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()\n",
" return degrees0\n",
"\n",
" def basis_de_rham_degrees(self):\n",
" f = self.polynomial\n",
" m = self.exponent\n",
" r = f.degree()\n",
" delta = GCD(r, m)\n",
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" F = self.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" basis_holo = self.holomorphic_differentials_basis()\n",
" basis = []\n",
" for k in range(0, len(basis_holo)):\n",
" basis += [superelliptic_cech(self, basis_holo[k], superelliptic_function(self, 0))]\n",
"\n",
" ## non-holomorphic elts of H^1_dR\n",
" t = len(basis)\n",
" degrees0 = {}\n",
" degrees1 = {}\n",
" for j in range(1, m):\n",
" for i in range(1, r):\n",
" if (r*(m-j) - m*i >= delta): \n",
" s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n",
" psi = Rx(cut(s, i))\n",
" basis += [superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))]\n",
" degrees0[t] = (psi.degree(), j)\n",
" degrees1[t] = (-i, m-j)\n",
" t += 1\n",
" return basis, degrees0, degrees1\n",
"\n",
" def de_rham_basis(self):\n",
" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
" return basis\n",
"\n",
" def degrees_de_rham0(self):\n",
" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
" return degrees0\n",
"\n",
" def degrees_de_rham1(self):\n",
" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
" return degrees1 \n",
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" \n",
" def is_smooth(self):\n",
" f = self.polynomial\n",
" if f.discriminant() == 0:\n",
" return 0\n",
" return 1\n",
" \n",
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" def genus(self):\n",
" r = self.polynomial.degree()\n",
" m = self.exponent\n",
" delta = GCD(r, m)\n",
" return 1/2*((r-1)*(m-1) - delta + 1)\n",
" \n",
" def verschiebung_matrix(self):\n",
" basis = self.de_rham_basis()\n",
" g = self.genus()\n",
" p = self.characteristic\n",
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" F = self.base_ring\n",
" M = matrix(F, 2*g, 2*g)\n",
" for i in range(0, len(basis)):\n",
" w = basis[i]\n",
" v = w.verschiebung().coordinates()\n",
" M[i, :] = v\n",
" return M\n",
" \n",
" def frobenius_matrix(self):\n",
" basis = self.de_rham_basis()\n",
" g = self.genus()\n",
" p = self.characteristic\n",
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" F = self.base_ring\n",
" M = matrix(F, 2*g, 2*g)\n",
" \n",
" for i in range(0, len(basis)):\n",
" w = basis[i]\n",
" v = w.frobenius().coordinates()\n",
" M[i, :] = v\n",
" return M\n",
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"\n",
" def cartier_matrix(self):\n",
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" basis = self.holomorphic_differentials_basis()\n",
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" g = self.genus()\n",
" p = self.characteristic\n",
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" F = self.base_ring\n",
" M = matrix(F, g, g)\n",
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" for i in range(0, len(basis)):\n",
" w = basis[i]\n",
" v = w.cartier().coordinates()\n",
" M[i, :] = v\n",
" return M \n",
"\n",
"# def p_rank(self):\n",
"# return self.cartier_matrix().rank()\n",
" \n",
" def a_number(self):\n",
" g = C.genus()\n",
" return g - self.cartier_matrix().rank()\n",
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" \n",
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" def final_type(self, test = 0):\n",
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" Fr = self.frobenius_matrix()\n",
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" V = self.verschiebung_matrix()\n",
" p = self.characteristic\n",
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" return flag(Fr, V, p, test)\n",
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" \n",
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"def reduction(C, g):\n",
" p = C.characteristic\n",
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" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" f = C.polynomial\n",
" r = f.degree()\n",
" m = C.exponent\n",
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" g = Fxy(g)\n",
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" g1 = g.numerator()\n",
" g2 = g.denominator()\n",
" \n",
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" Rx.<x> = PolynomialRing(F)\n",
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" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx) \n",
" (A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))\n",
" g = FxRy(g1*B/A)\n",
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" \n",
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" while(g.degree(Rxy(y)) >= m):\n",
" d = g.degree(Rxy(y))\n",
" G = coff(g, d)\n",
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" i = floor(d/m)\n",
" g = g - G*y^d + f^i * y^(d%m) *G\n",
" \n",
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" return(FxRy(g))\n",
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"\n",
"def reduction_form(C, g):\n",
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" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" f = C.polynomial\n",
" r = f.degree()\n",
" m = C.exponent\n",
" g = reduction(C, g)\n",
"\n",
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" g1 = Rxy(0)\n",
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" Rx.<x> = PolynomialRing(F)\n",
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" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
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" \n",
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" g = FxRy(g)\n",
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" for j in range(0, m):\n",
" if j==0:\n",
" G = coff(g, 0)\n",
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" g1 += FxRy(G)\n",
" else:\n",
" G = coff(g, j)\n",
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" g1 += Fxy(y^(j-m)*f*G)\n",
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" return(g1)\n",
" \n",
"class superelliptic_function:\n",
" def __init__(self, C, g):\n",
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" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" f = C.polynomial\n",
" r = f.degree()\n",
" m = C.exponent\n",
" \n",
" self.curve = C\n",
" g = reduction(C, g)\n",
" self.function = g\n",
" \n",
" def __repr__(self):\n",
" return str(self.function)\n",
" \n",
" def jth_component(self, j):\n",
" g = self.function\n",
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" C = self.curve\n",
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" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
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" Fx.<x> = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = FxRy(g)\n",
" return coff(g, j)\n",
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" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 + g2)\n",
" return superelliptic_function(C, g)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 - g2)\n",
" return superelliptic_function(C, g)\n",
" \n",
" def __mul__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 * g2)\n",
" return superelliptic_function(C, g)\n",
" \n",
" def __truediv__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 / g2)\n",
" return superelliptic_function(C, g)\n",
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"\n",
" def diffn(self):\n",
" C = self.curve\n",
" f = C.polynomial\n",
" m = C.exponent\n",
" F = C.base_ring\n",
" g = self.function\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" g = Fxy(g)\n",
" A = g.derivative(x)\n",
" B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))\n",
" return superelliptic_form(C, A+B)\n",
"\n",
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" \n",
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" def expansion_at_infty(self, i = 0, prec=10):\n",
" C = self.curve\n",
" f = C.polynomial\n",
" m = C.exponent\n",
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" f = Rx(f)\n",
" Rt.<t> = LaurentSeriesRing(F, default_prec=prec)\n",
" RptW.<W> = PolynomialRing(Rt)\n",
" RptWQ = FractionField(RptW)\n",
" Rxy.<x, y> = PolynomialRing(F)\n",
" RxyQ = FractionField(Rxy)\n",
" fct = self.function\n",
" fct = RxyQ(fct)\n",
" r = f.degree()\n",
" delta, a, b = xgcd(m, r)\n",
" b = -b\n",
" M = m/delta\n",
" R = r/delta\n",
" while a<0:\n",
" a += R\n",
" b += M\n",
" \n",
" g = (x^r*f(x = 1/x))\n",
" gW = RptWQ(g(x = t^M * W^b)) - W^(delta)\n",
" ww = naive_hensel(gW, F, start = root_of_unity(F, delta), prec = prec)\n",
" xx = Rt(1/(t^M*ww^b))\n",
" yy = 1/(t^R*ww^a)\n",
" return Rt(fct(x = Rt(xx), y = Rt(yy)))\n",
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" \n",
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"def naive_hensel(fct, F, start = 1, prec=10):\n",
" Rt.<t> = LaurentSeriesRing(F, default_prec=prec)\n",
" RtQ = FractionField(Rt)\n",
" RptW.<W> = PolynomialRing(RtQ)\n",
" fct = RptW(fct)\n",
" alpha = (fct.derivative())(W = start)\n",
" w0 = Rt(1)\n",
" i = 1\n",
" while(i < prec):\n",
" a = 0\n",
" while(Rt(fct(W = w0)).valuation()) <= i:\n",
" w0 = w0 - fct(W = w0)/alpha\n",
" i += 1\n",
" return w0\n",
"\n",
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"class superelliptic_form:\n",
" def __init__(self, C, g):\n",
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" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
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" Fxy = FractionField(Rxy)\n",
" g = Fxy(reduction_form(C, g))\n",
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" self.form = g\n",
" self.curve = C \n",
" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" g1 = self.form\n",
" g2 = other.form\n",
" g = reduction(C, g1 + g2)\n",
" return superelliptic_form(C, g)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" g1 = self.form\n",
" g2 = other.form\n",
" g = reduction(C, g1 - g2)\n",
" return superelliptic_form(C, g)\n",
" \n",
" def __repr__(self):\n",
" g = self.form\n",
" if len(str(g)) == 1:\n",
" return str(g) + ' dx'\n",
" return '('+str(g) + ') dx'\n",
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"\n",
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" def __rmul__(self, constant):\n",
" C = self.curve\n",
" omega = self.form\n",
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" return superelliptic_form(C, constant*omega) \n",
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" \n",
" def cartier(self):\n",
" C = self.curve\n",
" m = C.exponent\n",
" p = C.characteristic\n",
" f = C.polynomial\n",
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" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" Fxy = FractionField(FxRy)\n",
" result = superelliptic_form(C, FxRy(0))\n",
" mult_order = Integers(m)(p).multiplicative_order()\n",
" M = Integer((p^(mult_order)-1)/m)\n",
" \n",
" for j in range(1, m):\n",
" fct_j = self.jth_component(j)\n",
" h = Rx(fct_j*f^(M*j))\n",
" j1 = (p^(mult_order-1)*j)%m\n",
" B = floor(p^(mult_order-1)*j/m)\n",
" result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))\n",
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" return result \n",
" \n",
" def coordinates(self):\n",
" C = self.curve\n",
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" F = C.base_ring\n",
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" m = C.exponent\n",
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" Rx.<x> = PolynomialRing(F)\n",
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" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = C.genus()\n",
" degrees_holo = C.degrees_holomorphic_differentials()\n",
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" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
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" basis = C.holomorphic_differentials_basis()\n",
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" \n",
" for j in range(1, m):\n",
" omega_j = Fx(self.jth_component(j))\n",
" if omega_j != Fx(0):\n",
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" d = degree_of_rational_fctn(omega_j, F)\n",
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" index = degrees_holo_inv[(d, j)]\n",
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" a = coeff_of_rational_fctn(omega_j, F)\n",
" a1 = coeff_of_rational_fctn(basis[index].jth_component(j), F)\n",
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" elt = self - (a/a1)*basis[index]\n",
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" return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, g)])\n",
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" \n",
" return vector(g*[0])\n",
" \n",
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" def jth_component(self, j):\n",
" g = self.form\n",
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" C = self.curve\n",
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" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
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" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" Fxy = FractionField(FxRy)\n",
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" Ryinv.<y_inv> = PolynomialRing(Fx)\n",
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" g = Fxy(g)\n",
" g = g(y = 1/y_inv)\n",
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" g = Ryinv(g)\n",
" return coff(g, j)\n",
" \n",
" def is_regular_on_U0(self):\n",
" C = self.curve\n",
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" F = C.base_ring\n",
" m = C.exponent\n",
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" Rx.<x> = PolynomialRing(F)\n",
" for j in range(1, m):\n",
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" if self.jth_component(j) not in Rx:\n",
" return 0\n",
" return 1\n",
" \n",
" def is_regular_on_Uinfty(self):\n",
" C = self.curve\n",
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" F = C.base_ring\n",
" m = C.exponent\n",
" f = C.polynomial\n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" M = m/delta\n",
" R = r/delta\n",
" \n",
" for j in range(1, m):\n",
" A = self.jth_component(j)\n",
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" d = degree_of_rational_fctn(A, F)\n",
" if(-d*M + j*R -(M+1)<0):\n",
" return 0\n",
" return 1\n",
2022-10-11 21:17:22 +02:00
"\n",
" def expansion_at_infty(self, i = 0, prec=10):\n",
" g = self.form\n",
" C = self.curve\n",
" g = superelliptic_function(C, g)\n",
" g = g.expansion_at_infty(i = i, prec=prec)\n",
" x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)\n",
" dx_series = x_series.derivative()\n",
" return g*dx_series\n",
" \n",
"class superelliptic_cech:\n",
2021-08-21 18:40:44 +02:00
" def __init__(self, C, omega, fct):\n",
" self.omega0 = omega\n",
2022-10-11 21:17:22 +02:00
" self.omega8 = omega - fct.diffn()\n",
" self.f = fct\n",
2021-08-21 18:40:44 +02:00
" self.curve = C\n",
" \n",
2021-08-21 19:52:03 +02:00
" def __add__(self, other):\n",
" C = self.curve\n",
" return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)\n",
2021-09-13 12:11:08 +02:00
"\n",
" def __rmul__(self, constant):\n",
2021-08-22 21:48:21 +02:00
" C = self.curve\n",
" w1 = self.omega0.form\n",
" f1 = self.f.function\n",
" w2 = superelliptic_form(C, constant*w1)\n",
" f2 = superelliptic_function(C, constant*f1)\n",
2021-09-13 12:11:08 +02:00
" return superelliptic_cech(C, w2, f2) \n",
2021-08-22 21:48:21 +02:00
" \n",
2021-08-21 19:52:03 +02:00
" def __repr__(self):\n",
" return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n",
2021-08-22 21:48:21 +02:00
" \n",
" def verschiebung(self):\n",
2021-08-22 21:48:21 +02:00
" C = self.curve\n",
" omega = self.omega0\n",
2022-10-10 20:44:49 +02:00
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))\n",
" \n",
" def frobenius(self):\n",
" C = self.curve\n",
" fct = self.f.function\n",
" p = C.characteristic\n",
2022-10-10 20:44:49 +02:00
" Rx.<x> = PolynomialRing(F)\n",
" return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))\n",
2021-08-22 21:48:21 +02:00
"\n",
" def coordinates(self):\n",
" C = self.curve\n",
2022-10-10 20:44:49 +02:00
" F = C.base_ring\n",
2021-08-22 21:48:21 +02:00
" m = C.exponent\n",
2022-10-10 20:44:49 +02:00
" Rx.<x> = PolynomialRing(F)\n",
2021-08-22 21:48:21 +02:00
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = C.genus()\n",
" degrees_holo = C.degrees_holomorphic_differentials()\n",
" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
" degrees0 = C.degrees_de_rham0()\n",
" degrees0_inv = {b:a for a, b in degrees0.items()}\n",
" degrees1 = C.degrees_de_rham1()\n",
2021-08-22 21:48:21 +02:00
" degrees1_inv = {b:a for a, b in degrees1.items()}\n",
" basis = C.de_rham_basis()\n",
2021-08-22 21:48:21 +02:00
" \n",
" omega = self.omega0\n",
" fct = self.f\n",
" \n",
" if fct.function == Rx(0) and omega.form != Rx(0):\n",
2021-08-22 21:48:21 +02:00
" for j in range(1, m):\n",
" omega_j = Fx(omega.jth_component(j))\n",
" if omega_j != Fx(0):\n",
2022-10-10 20:44:49 +02:00
" d = degree_of_rational_fctn(omega_j, F)\n",
" index = degrees_holo_inv[(d, j)]\n",
2022-10-10 20:44:49 +02:00
" a = coeff_of_rational_fctn(omega_j, F)\n",
" a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), F)\n",
2021-09-13 12:11:08 +02:00
" elt = self - (a/a1)*basis[index]\n",
2022-10-10 20:44:49 +02:00
" return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, 2*g)])\n",
2021-08-22 21:48:21 +02:00
" \n",
" for j in range(1, m):\n",
" fct_j = Fx(fct.jth_component(j))\n",
" if (fct_j != Rx(0)):\n",
" d = degree_of_rational_fctn(fct_j, p)\n",
2021-08-22 21:48:21 +02:00
" \n",
" if (d, j) in degrees1.values():\n",
" index = degrees1_inv[(d, j)]\n",
2022-10-10 20:44:49 +02:00
" a = coeff_of_rational_fctn(fct_j, F)\n",
2021-09-13 12:11:08 +02:00
" elt = self - (a/m)*basis[index]\n",
2022-10-10 20:44:49 +02:00
" return elt.coordinates() + a/m*vector([F(i == index) for i in range(0, 2*g)])\n",
2021-08-22 21:48:21 +02:00
" \n",
" if d<0:\n",
2022-10-10 20:44:49 +02:00
" a = coeff_of_rational_fctn(fct_j, F)\n",
" h = superelliptic_function(C, FxRy(a*y^j*x^d))\n",
" elt = superelliptic_cech(C, self.omega0, self.f - h)\n",
2021-08-22 21:48:21 +02:00
" return elt.coordinates()\n",
" \n",
" if (fct_j != Rx(0)):\n",
" G = superelliptic_function(C, y^j*x^d)\n",
2022-10-10 20:44:49 +02:00
" a = coeff_of_rational_fctn(fct_j, F)\n",
2021-09-13 12:11:08 +02:00
" elt =self - a*superelliptic_cech(C, diffn(G), G)\n",
2021-08-22 21:48:21 +02:00
" return elt.coordinates()\n",
"\n",
" return vector(2*g*[0])\n",
" \n",
" def is_cocycle(self):\n",
" w0 = self.omega0\n",
" w8 = self.omega8\n",
" fct = self.f\n",
" if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():\n",
" return('w0 & w8')\n",
" if not w0.is_regular_on_U0():\n",
" return('w0')\n",
" if not w8.is_regular_on_Uinfty():\n",
" return('w8')\n",
" if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():\n",
" return 1\n",
" return 0\n",
2021-08-21 19:52:03 +02:00
" \n",
2022-10-10 20:44:49 +02:00
"def degree_of_rational_fctn(f, F):\n",
" Rx.<x> = PolynomialRing(F)\n",
2021-08-21 20:50:10 +02:00
" Fx = FractionField(Rx)\n",
" f = Fx(f)\n",
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" return(d1 - d2)\n",
"\n",
2022-10-10 20:44:49 +02:00
"def coeff_of_rational_fctn(f, F):\n",
" Rx.<x> = PolynomialRing(F)\n",
2021-08-22 21:48:21 +02:00
" Fx = FractionField(Rx)\n",
" f = Fx(f)\n",
" if f == Rx(0):\n",
" return 0\n",
2021-08-22 21:48:21 +02:00
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" a1 = f1.coefficients(sparse = false)[d1]\n",
" a2 = f2.coefficients(sparse = false)[d2]\n",
" return(a1/a2)\n",
"\n",
"def coff(f, d):\n",
" lista = f.coefficients(sparse = false)\n",
" if len(lista) <= d:\n",
" return 0\n",
" return lista[d]\n",
"\n",
"def cut(f, i):\n",
2021-08-21 18:40:44 +02:00
" R = f.parent()\n",
" coeff = f.coefficients(sparse = false)\n",
" return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))\n",
"\n",
"def polynomial_part(p, h):\n",
2022-10-10 20:44:49 +02:00
" F = GF(p)\n",
" Rx.<x> = PolynomialRing(F)\n",
" h = Rx(h)\n",
" result = Rx(0)\n",
" for i in range(0, h.degree()+1):\n",
" if (i%p) == p-1:\n",
" power = Integer((i-(p-1))/p)\n",
" result += Integer(h[i]) * x^(power) \n",
2022-10-11 21:17:22 +02:00
" return result\n",
"\n",
"#Find delta-th root of unity in field F\n",
"def root_of_unity(F, delta):\n",
" Rx.<x> = PolynomialRing(F)\n",
" cyclotomic = x^(delta) - 1\n",
" for root, a in cyclotomic.roots():\n",
" powers = [root^d for d in delta.divisors() if d!= delta]\n",
" if 1 not in powers:\n",
" return root\n",
" \n",
"def preimage(U, V, M): #preimage of subspace U under M\n",
" basis_preimage = M.right_kernel().basis()\n",
" imageU = U.intersection(M.transpose().image())\n",
" basis = imageU.basis()\n",
" for v in basis:\n",
" w = M.solve_right(v)\n",
" basis_preimage = basis_preimage + [w]\n",
" return V.subspace(basis_preimage)\n",
"\n",
"def image(U, V, M):\n",
" basis = U.basis()\n",
" basis_image = []\n",
" for v in basis:\n",
" basis_image += [M*v]\n",
" return V.subspace(basis_image)\n",
"\n",
2021-10-08 18:47:24 +02:00
"def flag(F, V, p, test = 0):\n",
" dim = F.dimensions()[0]\n",
" space = VectorSpace(GF(p), dim)\n",
" flag_subspaces = (dim+1)*[0]\n",
" flag_used = (dim+1)*[0]\n",
2021-10-08 18:47:24 +02:00
" final_type = (dim+1)*['?']\n",
" \n",
" flag_subspaces[dim] = space\n",
" flag_used[dim] = 1\n",
" \n",
" \n",
" while 1 in flag_used:\n",
" index = flag_used.index(1)\n",
" flag_used[index] = 0\n",
" U = flag_subspaces[index]\n",
" U_im = image(U, space, V)\n",
" d_im = U_im.dimension()\n",
" final_type[index] = d_im\n",
" U_pre = preimage(U, space, F)\n",
" d_pre = U_pre.dimension()\n",
" \n",
" if flag_subspaces[d_im] == 0:\n",
" flag_subspaces[d_im] = U_im\n",
" flag_used[d_im] = 1\n",
" \n",
" if flag_subspaces[d_pre] == 0:\n",
" flag_subspaces[d_pre] = U_pre\n",
" flag_used[d_pre] = 1\n",
" \n",
2021-10-08 18:47:24 +02:00
" if test == 1:\n",
" print('(', final_type, ')')\n",
" \n",
" for i in range(0, dim+1):\n",
2021-10-08 18:47:24 +02:00
" if final_type[i] == '?' and final_type[dim - i] != '?':\n",
" i1 = dim - i\n",
" final_type[i] = final_type[i1] - i1 + dim/2\n",
" \n",
" final_type[0] = 0\n",
2021-10-08 18:47:24 +02:00
" \n",
" for i in range(1, dim+1):\n",
" if final_type[i] == '?':\n",
" prev = final_type[i-1]\n",
" if prev != '?' and prev in final_type[i+1:]:\n",
" final_type[i] = prev\n",
" \n",
" for i in range(1, dim+1):\n",
2021-10-08 18:47:24 +02:00
" if final_type[i] == '?':\n",
2021-09-11 16:02:09 +02:00
" final_type[i] = min(final_type[i-1] + 1, dim/2)\n",
" \n",
" if is_final(final_type, dim/2):\n",
" return final_type[1:dim/2 + 1]\n",
2021-10-08 18:47:24 +02:00
" print('error:', final_type[1:dim/2 + 1])\n",
" \n",
"def is_final(final_type, dim):\n",
" n = len(final_type)\n",
" if final_type[0] != 0:\n",
" return 0\n",
" \n",
" if final_type[n-1] != dim:\n",
" return 0\n",
" \n",
" for i in range(1, n):\n",
2021-09-11 16:02:09 +02:00
" if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:\n",
" return 0\n",
" return 1"
]
},
{
"cell_type": "code",
2022-10-11 21:17:22 +02:00
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 56,
2022-03-07 13:48:30 +01:00
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
2022-10-10 20:44:49 +02:00
"Rx.<x> = PolynomialRing(GF(5))\n",
"f = x^7 + x + 1\n",
"C = superelliptic(f, 2)"
]
},
{
"cell_type": "code",
2022-10-11 21:17:22 +02:00
"execution_count": 58,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"omega = C.de_rham_basis()[3]"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"3*t^2 + 3*t^14 + t^16 + 3*t^26 + 3*t^28 + 3*t^30 + 3*t^38 + t^40 + 3*t^50 + 3*t^62 + t^72 + t^74 + 2*t^76 + 4*t^84 + 2*t^86 + 4*t^88 + 4*t^90 + 2*t^96 + 4*t^98 + t^100 + O(t^114)"
]
},
"execution_count": 51,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"omega.omega8().exp"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"F = QQ\n",
"Rt.<t> = LaurentSeriesRing(F, default_prec=20)\n",
"RtQ = FractionField(Rt)\n",
"RptW.<W> = PolynomialRing(RtQ)\n",
"fct = W^2 - (1+t)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"A = naive_hensel(fct, F, start = 1, prec=10)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"WARNING: Some output was deleted.\n"
]
}
],
"source": [
"A^2 - (1+t)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"FF = GF(5^6)"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[(4, 1),\n",
" (1, 1),\n",
" (3*z6^5 + 4*z6^4 + 3*z6^2 + 2*z6 + 2, 1),\n",
" (3*z6^5 + 4*z6^4 + 3*z6^2 + 2*z6 + 1, 1),\n",
" (2*z6^5 + z6^4 + 2*z6^2 + 3*z6 + 4, 1),\n",
" (2*z6^5 + z6^4 + 2*z6^2 + 3*z6 + 3, 1)]"
]
},
"execution_count": 56,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Rx.<x> = PolynomialRing(FF)\n",
"(x^6 - 1).roots()"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
2022-10-11 21:17:22 +02:00
"3*z2 + 3"
]
},
2022-10-11 21:17:22 +02:00
"execution_count": 57,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
2022-10-11 21:17:22 +02:00
"root_of_unity(GF(5^2), 3)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"x^2 + x + 1"
]
},
"execution_count": 52,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Rx.<x> = PolynomialRing(GF(5^3))\n",
"Rx(x^2 + x + 1).factor()"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"a = GF(5^3).random_element()"
]
},
2021-08-19 22:35:11 +02:00
{
"cell_type": "code",
2022-03-07 13:48:30 +01:00
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
2022-10-11 21:17:22 +02:00
"a"
2022-03-07 13:48:30 +01:00
]
2021-08-19 22:35:11 +02:00
}
],
"metadata": {
"kernelspec": {
2022-03-07 13:48:30 +01:00
"display_name": "SageMath 9.5",
"language": "sagemath",
"metadata": {
"cocalc": {
"description": "Open-source mathematical software system",
"priority": 1,
2022-03-07 13:48:30 +01:00
"url": "https://www.sagemath.org/"
}
},
"name": "sage-9.5",
"resource_dir": "/ext/jupyter/kernels/sage-9.5"
2021-08-19 22:35:11 +02:00
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
2022-03-07 13:48:30 +01:00
"version": "3.9.9"
2021-08-19 22:35:11 +02:00
}
},
"nbformat": 4,
2022-03-07 13:48:30 +01:00
"nbformat_minor": 4
}