763 lines
32 KiB
Plaintext
763 lines
32 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 110,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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],
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"source": [
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"N = 2\n",
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"p = 3\n",
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"RQ = PolynomialRing(QQ, 'X', 2*N)\n",
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"X = RQ.gens()[:N]\n",
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"Y = RQ.gens()[N:]\n",
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"Rpx.<x> = PolynomialRing(GF(p), 1)\n",
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"#RQx.<x> = PolynomialRing(QQ, 1)\n",
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"\n",
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"def witt_pol(lista):\n",
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" n = len(lista)\n",
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" return sum(p^i*lista[i]^(p^(n-i-1)) for i in range(0, n))\n",
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"\n",
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"def witt_sum(n):\n",
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" if n == 0:\n",
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" return X[0] + Y[0]\n",
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" return 1/p^n*(witt_pol(X[:n+1]) + witt_pol(Y[:n+1]) - sum(p^k*witt_sum(k)^(p^(n-k)) for k in range(0, n)))\n",
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"\n",
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"class witt:\n",
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" def __init__(self, coordinates):\n",
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" self.coordinates = coordinates\n",
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" def __repr__(self):\n",
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" lista = [Rpx(a) for a in self.coordinates]\n",
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" return str(lista)\n",
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" def __add__(self, other):\n",
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" lista = []\n",
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" for i in range(0, N):\n",
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" lista+= [witt_sum(i)(self.coordinates + other.coordinates)]\n",
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" return witt(lista)\n",
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" def __rmul__(self, constant):\n",
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" if constant<0:\n",
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" m = (-constant)*(p^N-1)\n",
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" return m*self\n",
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" if constant == 0:\n",
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" return witt(N*[0])\n",
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" return self + (constant - 1)*self\n",
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" def __sub__(self, other):\n",
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" return self + (-1)*other\n",
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"\n",
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"def fraction_pol(g):\n",
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" RxX.<x, X> = PolynomialRing(QQ)\n",
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" g = RxX(g)\n",
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" result = 0\n",
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" for a in g.monomials():\n",
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" c = g.monomial_coefficient(a)\n",
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" c = c%(p^2)\n",
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" dX = a.degree(X)\n",
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" dx = a.degree(x)\n",
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" if dX%p == 0:\n",
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" result += c*x^(dX//p + dx)\n",
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" else:\n",
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" result += c*X^(dX + dx*p)\n",
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" return result\n",
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" \n",
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"def teichmuller(f):\n",
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" Rx.<x> = PolynomialRing(QQ)\n",
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" RxX.<x, X> = PolynomialRing(QQ, 2)\n",
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" f = Rx(f)\n",
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" ff = witt([f, 0])\n",
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" coeffs = f.coefficients(sparse=false)\n",
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" for i in range(0, len(coeffs)):\n",
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" ff -= coeffs[i]*witt([Rx(x^i), 0])\n",
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" f1 = sum(coeffs[i]*RxX(x^(i)) for i in range(0, len(coeffs))) + p*RxX(ff.coordinates[1](x = X))\n",
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" f1 = fraction_pol(f1)\n",
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" #RXp.<Xp> = PolynomialRing(Integers(p^2))\n",
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" #f1 = RXp(f1)\n",
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" return f1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 99,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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],
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"source": [
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"def basis_de_rham_degrees(f, m, p):\n",
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" r = f.degree()\n",
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" delta = GCD(r, m)\n",
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" Rx.<x> = PolynomialRing(QQ)\n",
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" Rxy.<x, y> = PolynomialRing(QQ, 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" basis_holo = holomorphic_differentials_basis(f, m, p)\n",
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" basis = []\n",
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" for k in range(0, len(basis_holo)):\n",
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" basis += [(basis_holo[k], Rx(0))]\n",
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"\n",
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" ## non-holomorphic elts of H^1_dR\n",
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" t = len(basis)\n",
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" degrees0 = {}\n",
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" degrees1 = {}\n",
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" for j in range(1, m):\n",
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" for i in range(1, r):\n",
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" if (r*(m-j) - m*i >= delta): \n",
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" s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n",
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" psi = Rx(cut(s, i))\n",
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" basis += [(Fxy(psi/y^j), Fxy(m*y^(m-j)/x^i))]\n",
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" degrees0[t] = (psi.degree(), j)\n",
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" degrees1[t] = (-i, m-j)\n",
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" t += 1\n",
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" return basis, degrees0, degrees1\n",
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"\n",
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"def de_rham_basis(f, m, p):\n",
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" basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n",
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" return basis\n",
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"\n",
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"def degrees_de_rham0(f, m, p):\n",
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" basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n",
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" return degrees0\n",
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"\n",
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"def degrees_de_rham1(f, m, p):\n",
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" basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n",
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" return degrees1\n",
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"\n",
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"\n",
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"class superelliptic:\n",
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" \n",
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" def __init__(self, f, m, p):\n",
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" Rx.<x> = PolynomialRing(QQ)\n",
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" Rxy.<x, y> = PolynomialRing(QQ, 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.characteristic = p\n",
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" \n",
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" r = Rx(f).degree()\n",
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" delta = GCD(r, m)\n",
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" self.degree_holo = degrees_holomorphic_differentials(f, m, p)\n",
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" self.degree_de_rham0 = degrees_de_rham0(f, m, p)\n",
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" self.degree_de_rham1 = degrees_de_rham1(f, m, p)\n",
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" \n",
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" holo_basis = holomorphic_differentials_basis(f, m, p)\n",
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" holo_basis_converted = []\n",
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" for a in holo_basis:\n",
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" holo_basis_converted += [superelliptic_form(self, a)]\n",
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" \n",
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" self.basis_holomorphic_differentials = holo_basis_converted\n",
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" \n",
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"\n",
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" dr_basis = de_rham_basis(f, m, p)\n",
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" dr_basis_converted = []\n",
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" for (a, b) in dr_basis:\n",
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" dr_basis_converted += [superelliptic_cech(self, superelliptic_form(self, a), superelliptic_function(self, b))]\n",
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" \n",
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" self.basis_de_rham = dr_basis_converted\n",
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" \n",
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" def __repr__(self):\n",
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" f = self.polynomial\n",
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" m = self.exponent\n",
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" p = self.characteristic\n",
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" return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'\n",
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" \n",
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" def is_smooth(self):\n",
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" f = self.polynomial\n",
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" if f.discriminant() == 0:\n",
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" return 0\n",
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" return 1\n",
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" \n",
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" def genus(self):\n",
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" r = self.polynomial.degree()\n",
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" m = self.exponent\n",
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" delta = GCD(r, m)\n",
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" return 1/2*((r-1)*(m-1) - delta + 1)\n",
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" \n",
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"class superelliptic_function:\n",
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" def __init__(self, C, g):\n",
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" p = C.characteristic\n",
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" RxXy.<x, X, y> = PolynomialRing(QQ, 3)\n",
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" FxXy = FractionField(RXy)\n",
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" f = C.polynomial\n",
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" r = f.degree()\n",
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" m = C.exponent\n",
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" \n",
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" self.curve = C\n",
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" self.function = g\n",
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" \n",
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" def __repr__(self):\n",
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" return str(self.function)\n",
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" \n",
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" def jth_component(self, j):\n",
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" g = self.function\n",
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" C = self.curve\n",
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" p = C.characteristic\n",
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" RxX.<x,X> = PolynomialRing(QQ, 2)\n",
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" FxX.<x> = FractionField(RxX)\n",
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" FxXRy.<y> = PolynomialRing(FxX)\n",
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" g = FxXRy(g)\n",
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" return coff(g, j)\n",
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" \n",
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" def __add__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" g = reduction(C, g1 + g2)\n",
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" return superelliptic_function(C, g)\n",
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" \n",
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" def __sub__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" g = reduction(C, g1 - g2)\n",
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" return superelliptic_function(C, g)\n",
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" \n",
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" def __mul__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" g = reduction(C, g1 * g2)\n",
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" return superelliptic_function(C, g)\n",
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" \n",
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" def __truediv__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.function\n",
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" g2 = other.function\n",
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" return superelliptic_function(C, g1 / g2)\n",
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" \n",
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"def diffn(self):\n",
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" C = self.curve\n",
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" f = C.polynomial\n",
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" m = C.exponent\n",
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" p = C.characteristic\n",
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" g = self.function\n",
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" RXy.<X, y> = PolynomialRing(QQ, 2)\n",
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" FXy = FractionField(RXy)\n",
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" g = RXy(g)\n",
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" A = g.derivative(X)*X^(-(p-1))/p\n",
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" t = teichmuller(f)\n",
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" B = g.derivative(y)*t.derivative()/(m*y^(m-1))*X^(-(p-1))/p\n",
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" return superelliptic_form(C, A+B)\n",
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" \n",
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"class superelliptic_form:\n",
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" def __init__(self, C, g):\n",
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" p = C.characteristic\n",
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" Rxy.<x, y> = PolynomialRing(QQ, 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" g = Fxy(reduction_form(C, g))\n",
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" self.form = g\n",
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" self.curve = C \n",
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" \n",
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" def __add__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.form\n",
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" g2 = other.form\n",
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" g = reduction(C, g1 + g2)\n",
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" return superelliptic_form(C, g)\n",
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" \n",
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" def __sub__(self, other):\n",
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" C = self.curve\n",
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" g1 = self.form\n",
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" g2 = other.form\n",
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" g = reduction(C, g1 - g2)\n",
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" return superelliptic_form(C, g)\n",
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" \n",
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" def __repr__(self):\n",
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" g = self.form\n",
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" if len(str(g)) == 1:\n",
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" return str(g) + ' dx'\n",
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" return '('+str(g) + ') dx'\n",
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"\n",
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" def __rmul__(self, constant):\n",
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" C = self.curve\n",
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" omega = self.form\n",
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" return superelliptic_form(C, constant*omega) \n",
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"\n",
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" \n",
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" def coordinates(self):\n",
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" C = self.curve\n",
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" p = C.characteristic\n",
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" m = C.exponent\n",
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" Rx.<x> = PolynomialRing(QQ)\n",
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" Fx = FractionField(Rx)\n",
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" FxRy.<y> = PolynomialRing(Fx)\n",
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" g = C.genus()\n",
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" degrees_holo = C.degree_holo\n",
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" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
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" basis = C.basis_holomorphic_differentials\n",
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" \n",
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" for j in range(1, m):\n",
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" omega_j = Fx(self.jth_component(j))\n",
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" if omega_j != Fx(0):\n",
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" d = degree_of_rational_fctn(omega_j, p)\n",
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" index = degrees_holo_inv[(d, j)]\n",
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" a = coeff_of_rational_fctn(omega_j, p)\n",
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" a1 = coeff_of_rational_fctn(basis[index].jth_component(j), p)\n",
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" elt = self - (a/a1)*basis[index]\n",
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" return elt.coordinates() + a/a1*vector([QQ(i == index) for i in range(0, g)])\n",
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" \n",
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" return vector(g*[0])\n",
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" \n",
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" def jth_component(self, j):\n",
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" g = self.form\n",
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" C = self.curve\n",
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" p = C.characteristic\n",
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" Rx.<x> = PolynomialRing(QQ)\n",
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" Fx = FractionField(Rx)\n",
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" FxRy.<y> = PolynomialRing(Fx)\n",
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" Fxy = FractionField(FxRy)\n",
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" Ryinv.<y_inv> = PolynomialRing(Fx)\n",
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" g = Fxy(g)\n",
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" g = g(y = 1/y_inv)\n",
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" g = Ryinv(g)\n",
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" return coff(g, j)\n",
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" \n",
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" def is_regular_on_U0(self):\n",
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" C = self.curve\n",
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" p = C.characteristic\n",
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" m = C.exponent\n",
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" Rx.<x> = PolynomialRing(QQ)\n",
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" for j in range(1, m):\n",
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" if self.jth_component(j) not in Rx:\n",
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" return 0\n",
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" return 1\n",
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" \n",
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" def is_regular_on_Uinfty(self):\n",
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" C = self.curve\n",
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" p = C.characteristic\n",
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" m = C.exponent\n",
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" f = C.polynomial\n",
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" r = f.degree()\n",
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" delta = GCD(m, r)\n",
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" M = m/delta\n",
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" R = r/delta\n",
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" \n",
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" for j in range(1, m):\n",
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" A = self.jth_component(j)\n",
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" d = degree_of_rational_fctn(A, p)\n",
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" if(-d*M + j*R -(M+1)<0):\n",
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" return 0\n",
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" return 1\n",
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" \n",
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" \n",
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"class superelliptic_cech:\n",
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" def __init__(self, C, omega, fct):\n",
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" self.omega0 = omega\n",
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" self.omega8 = omega - diffn(fct)\n",
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" self.f = fct\n",
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" self.curve = C\n",
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" \n",
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" def __add__(self, other):\n",
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" C = self.curve\n",
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" return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)\n",
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" \n",
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" def __sub__(self, other):\n",
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" C = self.curve\n",
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" return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)\n",
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"\n",
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" def __rmul__(self, constant):\n",
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" C = self.curve\n",
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" w1 = self.omega0.form\n",
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" f1 = self.f.function\n",
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" w2 = superelliptic_form(C, constant*w1)\n",
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" f2 = superelliptic_function(C, constant*f1)\n",
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" return superelliptic_cech(C, w2, f2) \n",
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" \n",
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" def __repr__(self):\n",
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" return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n",
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"\n",
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" def coordinates(self):\n",
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" C = self.curve\n",
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" p = C.characteristic\n",
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" m = C.exponent\n",
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" Rx.<x> = PolynomialRing(QQ)\n",
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" Fx = FractionField(Rx)\n",
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" FxRy.<y> = PolynomialRing(Fx)\n",
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" g = C.genus()\n",
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" degrees_holo = C.degree_holo\n",
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" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
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" degrees0 = C.degree_de_rham0\n",
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" degrees0_inv = {b:a for a, b in degrees0.items()}\n",
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" degrees1 = C.degree_de_rham1\n",
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" degrees1_inv = {b:a for a, b in degrees1.items()}\n",
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" basis = C.basis_de_rham\n",
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" \n",
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" omega = self.omega0\n",
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" fct = self.f\n",
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" \n",
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" if fct.function == Rx(0) and omega.form != Rx(0):\n",
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" for j in range(1, m):\n",
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" omega_j = Fx(omega.jth_component(j))\n",
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" if omega_j != Fx(0):\n",
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" d = degree_of_rational_fctn(omega_j, p)\n",
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" index = degrees_holo_inv[(d, j)]\n",
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" a = coeff_of_rational_fctn(omega_j, p)\n",
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" a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), p)\n",
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" elt = self - (a/a1)*basis[index]\n",
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" return elt.coordinates() + a/a1*vector([QQ(i == index) for i in range(0, 2*g)])\n",
|
|
" \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",
|
|
" \n",
|
|
" if (d, j) in degrees1.values():\n",
|
|
" index = degrees1_inv[(d, j)]\n",
|
|
" a = coeff_of_rational_fctn(fct_j, p)\n",
|
|
" elt = self - (a/m)*basis[index]\n",
|
|
" return elt.coordinates() + a/m*vector([QQ(i == index) for i in range(0, 2*g)])\n",
|
|
" \n",
|
|
" if d<0:\n",
|
|
" a = coeff_of_rational_fctn(fct_j, p)\n",
|
|
" h = superelliptic_function(C, FxRy(a*y^j*x^d))\n",
|
|
" elt = superelliptic_cech(C, self.omega0, self.f - h)\n",
|
|
" return elt.coordinates()\n",
|
|
" \n",
|
|
" if (fct_j != Rx(0)):\n",
|
|
" G = superelliptic_function(C, y^j*x^d)\n",
|
|
" a = coeff_of_rational_fctn(fct_j, p)\n",
|
|
" elt =self - a*superelliptic_cech(C, diffn(G), G)\n",
|
|
" 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",
|
|
" \n",
|
|
"def degree_of_rational_fctn(f, p):\n",
|
|
" Rx.<x> = PolynomialRing(QQ)\n",
|
|
" 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",
|
|
"def coeff_of_rational_fctn(f, p):\n",
|
|
" Rx.<x> = PolynomialRing(QQ)\n",
|
|
" Fx = FractionField(Rx)\n",
|
|
" f = Fx(f)\n",
|
|
" if f == Rx(0):\n",
|
|
" return 0\n",
|
|
" 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",
|
|
" 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",
|
|
" Rx.<x> = PolynomialRing(QQ)\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",
|
|
" return result"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 109,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"-720*X^9 - 15312*X^7 - 107160*X^5 + x^3 - 249984*X^3 - x\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"6*X^7 + 3*X^5 + x^3 + 8*x"
|
|
]
|
|
},
|
|
"execution_count": 109,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"Rx.<x> = PolynomialRing(QQ)\n",
|
|
"f = Rx(x^3 - x)\n",
|
|
"teichmuller(f)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 73,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"ename": "TypeError",
|
|
"evalue": "unsupported operand parent(s) for *: 'Multivariate Polynomial Ring in Xp, y over Rational Field' and 'Univariate Polynomial Ring in Xp over Ring of integers modulo 9'",
|
|
"output_type": "error",
|
|
"traceback": [
|
|
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
|
|
"\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)",
|
|
"\u001b[0;32m/tmp/ipykernel_1111/3447231159.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0mRx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPolynomialRing\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mQQ\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnames\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_first_ngens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuperelliptic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
|
|
"\u001b[0;32m/tmp/ipykernel_1111/3436947063.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, f, m, p)\u001b[0m\n\u001b[1;32m 68\u001b[0m \u001b[0mdr_basis_converted\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 69\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mdr_basis\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 70\u001b[0;31m \u001b[0mdr_basis_converted\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msuperelliptic_form\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msuperelliptic_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 71\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 72\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbasis_de_rham\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdr_basis_converted\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;32m/tmp/ipykernel_1111/3436947063.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, C, omega, fct)\u001b[0m\n\u001b[1;32m 260\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__init__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0momega\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfct\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 261\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0momega0\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0momega\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 262\u001b[0;31m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0momega8\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0momega\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 263\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfct\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 264\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcurve\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;32m/tmp/ipykernel_1111/3436947063.py\u001b[0m in \u001b[0;36mdiffn\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 153\u001b[0m \u001b[0mA\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mderivative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mXp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mXp\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 154\u001b[0m \u001b[0mt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mteichmuller\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 155\u001b[0;31m \u001b[0mB\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mderivative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mderivative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mXp\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 156\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0msuperelliptic_form\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mA\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mB\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 157\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12253)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1514\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_mul_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1515\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mBOTH_ARE_ELEMENT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcl\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1516\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mcoercion_model\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbin_op\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmul\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 1517\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1518\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mlong\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/coerce.pyx\u001b[0m in \u001b[0;36msage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:11751)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1246\u001b[0m \u001b[0;31m# We should really include the underlying error.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1247\u001b[0m \u001b[0;31m# This causes so much headache.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1248\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mbin_op_exception\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mop\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 1249\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1250\u001b[0m \u001b[0mcpdef\u001b[0m \u001b[0mcanonical_coercion\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;31mTypeError\u001b[0m: unsupported operand parent(s) for *: 'Multivariate Polynomial Ring in Xp, y over Rational Field' and 'Univariate Polynomial Ring in Xp over Ring of integers modulo 9'"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"Rx.<x> = PolynomialRing(QQ)\n",
|
|
"C = superelliptic(x^3 - x, 2, 3)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 58,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[((1/y) dx, 0, (1/y) dx), ((x/y) dx, 2/x*y, ((-1)/(x*y)) dx)]"
|
|
]
|
|
},
|
|
"execution_count": 58,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"C.basis_de_rham"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 52,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
],
|
|
"source": [
|
|
"g = basis_de_rham_degrees(x^3 - x, 2, 3)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 54,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
],
|
|
"source": [
|
|
"RXpy.<Xp, y> = PolynomialRing(QQ, 2)\n",
|
|
"FXpy = FractionField(RXpy)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 55,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"2*y/Xp^3"
|
|
]
|
|
},
|
|
"execution_count": 55,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"g(x = Xp^p, y = y)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 68,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
],
|
|
"source": [
|
|
"t = teichmuller(x^3 - x)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 71,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"6*Xp^6 + 6*Xp^4 + 6*Xp^2"
|
|
]
|
|
},
|
|
"execution_count": 71,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"t.derivative()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 102,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
],
|
|
"source": [
|
|
"RxX.<x, X> = PolynomialRing(QQ)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 103,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
],
|
|
"source": [
|
|
"g = RxX(x*X + 2*x*X^3)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 82,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"3"
|
|
]
|
|
},
|
|
"execution_count": 82,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"g.monomials()[0].degree(X)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 106,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"<class 'sage.rings.rational.Rational'>"
|
|
]
|
|
},
|
|
"execution_count": 106,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"type(g.monomial_coefficient(X))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 0,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
],
|
|
"source": [
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "SageMath 9.5",
|
|
"language": "sagemath",
|
|
"metadata": {
|
|
"cocalc": {
|
|
"description": "Open-source mathematical software system",
|
|
"priority": 10,
|
|
"url": "https://www.sagemath.org/"
|
|
}
|
|
},
|
|
"name": "sage-9.5",
|
|
"resource_dir": "/ext/jupyter/kernels/sage-9.5"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.9.9"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
} |