po powrocie obliczania bazy do srodka klasy
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parent
4277b74355
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91df146a40
@ -2,81 +2,13 @@
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 8,
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"execution_count": 44,
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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_holomorphic_differentials_degree(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(GF(p))\n",
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" Rxy.<x, y> = PolynomialRing(GF(p), 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" #########basis of holomorphic differentials and de Rham\n",
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" \n",
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" basis_holo = []\n",
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" degrees0 = {}\n",
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" k = 0\n",
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" \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*j - m*i >= delta):\n",
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" basis_holo += [Fxy(x^(i-1)/y^j)]\n",
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" degrees0[k] = (i-1, j)\n",
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" k = k+1\n",
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" \n",
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" return(basis_holo, degrees0)\n",
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"\n",
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"def holomorphic_differentials_basis(f, m, p):\n",
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" basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)\n",
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" return basis_holo\n",
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" \n",
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"def degrees_holomorphic_differentials(f, m, p):\n",
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" basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)\n",
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" return degrees0\n",
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" \n",
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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(GF(p))\n",
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" Rxy.<x, y> = PolynomialRing(GF(p), 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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@ -89,30 +21,84 @@
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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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" \n",
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" def basis_holomorphic_differentials_degree(self):\n",
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" f = self.polynomial\n",
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" m = self.exponent\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(GF(p))\n",
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" Rxy.<x, y> = PolynomialRing(GF(p), 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" #########basis of holomorphic differentials and de Rham\n",
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"\n",
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" basis_holo = []\n",
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" degrees0 = {}\n",
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" k = 0\n",
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"\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*j - m*i >= delta):\n",
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" basis_holo += [superelliptic_form(self, Fxy(x^(i-1)/y^j))]\n",
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" degrees0[k] = (i-1, j)\n",
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" k = k+1\n",
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"\n",
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" return(basis_holo, degrees0)\n",
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"\n",
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" def holomorphic_differentials_basis(self):\n",
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" basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()\n",
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" return basis_holo\n",
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" \n",
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" def degrees_holomorphic_differentials(self):\n",
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" basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()\n",
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" return degrees0\n",
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"\n",
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" def basis_de_rham_degrees(self):\n",
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" f = self.polynomial\n",
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" m = self.exponent\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(GF(p))\n",
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" Rxy.<x, y> = PolynomialRing(GF(p), 2)\n",
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" Fxy = FractionField(Rxy)\n",
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" basis_holo = self.holomorphic_differentials_basis()\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 += [superelliptic_cech(self, basis_holo[k], superelliptic_function(self, 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 += [superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, 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(self):\n",
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" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
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" return basis\n",
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"\n",
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" def degrees_de_rham0(self):\n",
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" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
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" return degrees0\n",
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"\n",
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" def degrees_de_rham1(self):\n",
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" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
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" return degrees1 \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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" return 1/2*((r-1)*(m-1) - delta + 1)\n",
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" \n",
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" def verschiebung_matrix(self):\n",
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" basis = self.basis_de_rham\n",
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" basis = self.de_rham_basis()\n",
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" g = self.genus()\n",
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" p = self.characteristic\n",
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" M = matrix(GF(p), 2*g, 2*g)\n",
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" return M\n",
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" \n",
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" def frobenius_matrix(self):\n",
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" basis = self.basis_de_rham\n",
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" basis = self.de_rham_basis()\n",
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" g = self.genus()\n",
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" p = self.characteristic\n",
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" M = matrix(GF(p), 2*g, 2*g)\n",
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@ -150,7 +136,7 @@
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" return M\n",
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"\n",
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" def cartier_matrix(self):\n",
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" basis = self.basis_holomorphic_differentials\n",
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" basis = self.basis_holomorphic_differentials()\n",
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" g = self.genus()\n",
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" p = self.characteristic\n",
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" M = matrix(GF(p), g, g)\n",
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" v = w.cartier().coordinates()\n",
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" M[i, :] = v\n",
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" return M \n",
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" \n",
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"\n",
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"# def p_rank(self):\n",
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"# return self.cartier_matrix().rank()\n",
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" \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 = 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.basis_holomorphic_differentials\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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" 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 = 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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" degrees0 = C.degree_de_rham0\n",
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" degrees0 = C.degrees_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 = C.degrees_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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" basis = C.de_rham_basis()\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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@ -567,7 +553,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"execution_count": 45,
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"metadata": {
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"collapsed": false
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},
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@ -659,19 +645,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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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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"from datetime import datetime"
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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": 3,
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"execution_count": 46,
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"metadata": {
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"collapsed": false
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},
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@ -687,7 +661,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"execution_count": 47,
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"metadata": {
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"collapsed": false
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},
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@ -695,17 +669,17 @@
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{
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"data": {
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"text/plain": [
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"1/y"
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"[1]"
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]
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},
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"execution_count": 7,
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"execution_count": 47,
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"metadata": {
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},
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"output_type": "execute_result"
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}
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],
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"source": [
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"C.basis_holomorphic_differentials[0].form"
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"C.final_type()"
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]
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},
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{
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