559 lines
16 KiB
Plaintext
559 lines
16 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": false
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},
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"source": [
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"# Theory\n",
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"Let $C : y^m = f(x)$. Then:\n",
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"\n",
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" - the basis of $H^0(C, \\Omega_{C/k})$ is given by:\n",
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" $$x^{i-1} dx/y^j,$$\n",
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" where $1 \\le i \\le r-1$, $1 \\le j \\le m-1$, $-mi + rj \\ge \\delta$ and $\\delta := GCD(m, r)$, $r := \\deg f$.\n",
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" \n",
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" - the above forms along with\n",
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" $$\\lambda_{ij} = \\left[ \\left( \\frac{\\psi_{ij} \\, dx}{m x^{i+1} y^{m - j}},\n",
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" \\frac{-\\phi_{ij} \\, dx}{m x^{i+1} y^{m - j}}, \\frac{y^j}{x^i} \\right) \\right]$$\n",
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" (where $s_{ij} = jx f'(x) - mi f(x)$, \n",
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" $\\psi_{ij}(x) = s_{ij}^{\\ge i+1}$,\n",
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" $\\phi_{ij}(x) = s_{ij}^{< i+1}$)\n",
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"form a basis of $H^1_{dR}(C/K)$."
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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": 2,
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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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"# The program computes the basis of holomorphic differentials of y^m = f(x) in char p.\n",
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"# The coefficient j means that we compute the j-th eigenpart, i.e.\n",
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"# forms y^j * f(x) dx. Output is [f(x), 0]\n",
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"\n",
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"def baza_holo(m, f, j, p):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" f = R(f)\n",
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" r = f.degree()\n",
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" delta = GCD(m, r)\n",
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" baza = {}\n",
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" k = 0\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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" baza[k] = [x^(i-1), R(0), j]\n",
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" k = k+1\n",
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" return baza"
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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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"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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"# The program computes the basis of de Rham cohomology of y^m = f(x) in char p.\n",
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"# We treat them as pairs [omega, f], where omega is regular on the affine part\n",
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"# and omega - df is regular on the second atlas.\n",
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"# The coefficient j means that we compute the j-th eigenpart, i.e.\n",
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"# [f(x) dx/y^j, y^(m-j)*g(x)]. Output is [f(x), g(x)]\n",
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"\n",
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"def baza_dr(m, f, j, p):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" f = R(f) \n",
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" r = f.degree()\n",
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" delta = GCD(m, r)\n",
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" baza = {}\n",
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" holo = baza_holo(m, f, j, p)\n",
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" for k in range(0, len(holo)):\n",
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" baza[k] = holo[k]\n",
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" \n",
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" k = len(baza)\n",
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" \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 = R(m-j)*R(x)*R(f.derivative()) - R(m)*R(i)*f\n",
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" psi = R(obciecie(s, i, p))\n",
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" baza[k] = [psi, R(m)/x^i, j]\n",
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" k = k+1\n",
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" return baza"
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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": 4,
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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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"#auxiliary programs\n",
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"def stopnie_bazy_holo(m, f, j, p):\n",
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" baza = baza_holo(m, f, j, p)\n",
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" stopnie = {}\n",
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" for k in range(0, len(baza)):\n",
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" stopnie[k] = baza[k][0].degree()\n",
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" return stopnie\n",
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"\n",
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"def stopnie_bazy_dr(m, f, j, p):\n",
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" baza = baza_dr(m, f, j, p)\n",
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" stopnie = {}\n",
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" for k in range(0, len(baza)):\n",
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" stopnie[k] = baza[k][0].degree()\n",
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" return stopnie\n",
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"\n",
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"def stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p):\n",
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" baza = baza_dr(m, f, j, p)\n",
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" stopnie = {}\n",
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" for k in range(0, len(baza)):\n",
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" if baza[k][1] != 0:\n",
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" stopnie[k] = baza[k][1].denominator().degree()\n",
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" return stopnie\n",
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"\n",
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"def obciecie(f, i, p):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" f = R(f)\n",
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" coeff = f.coefficients(sparse = false)\n",
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" return sum(x^(j-i-1) * coeff[j] for j in range(i+1, f.degree() + 1))\n",
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"\n",
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"\n",
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"#Any element [f dx, g] is represented as a combination of the basis vectors.\n",
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"\n",
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"def zapis_w_bazie_dr(elt, m, f, p):\n",
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" j = elt[2]\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" RR = FractionField(R)\n",
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" f = R(f)\n",
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" r = f.degree()\n",
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" delta = GCD(m, r)\n",
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" baza = baza_dr(m, f, j, p)\n",
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" wymiar = len(baza)\n",
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" zapis = vector([GF(p)(0) for i in baza])\n",
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" stopnie = stopnie_bazy_dr(m, f, j, p)\n",
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" inv_stopnie = {v: k for k, v in stopnie.items()}\n",
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" stopnie_holo = stopnie_bazy_holo(m, f, j, p)\n",
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" inv_stopnie_holo = {v: k for k, v in stopnie_holo.items()} \n",
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" \n",
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" ## zmiana\n",
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" if elt[0]== 0 and elt[1] == 0:\n",
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" return zapis\n",
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" \n",
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" if elt[1] == 0:\n",
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" elt[0] = R(elt[0])\n",
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" d = elt[0].degree()\n",
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" a = elt[0].coefficients(sparse = false)[d]\n",
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" k = inv_stopnie_holo[d] #ktory element bazy jest stopnia d? ten o indeksie k\n",
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" \n",
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" a1 = baza[k][0].coefficients(sparse = false)[d]\n",
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" elt1 = [R(0),R(0),j]\n",
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" elt1[0] = elt[0] - a/a1 * baza[k][0]\n",
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" elt1[1] = R(0)\n",
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" return zapis_w_bazie_dr(elt1, m, f, p) + vector([a/a1*GF(p)(i == k) for i in range(0, len(baza))])\n",
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"\n",
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" g = elt[1]\n",
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" a = wspolczynnik_wiodacy(g)\n",
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" d = -stopien_roznica(g)\n",
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" Rr = r/delta\n",
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" Mm = m/delta\n",
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" \n",
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" stopnie2 = stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p)\n",
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" inv_stopnie2 = {v: k for k, v in stopnie2.items()}\n",
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" if (d not in stopnie2.values()):\n",
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" if d> 0:\n",
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" j1 = m-j\n",
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" elt1 = [elt[0], RR(elt[1]) - a*1/R(x^d), j]\n",
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" else:\n",
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" d1 = -d\n",
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" j1 = m-j\n",
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" elt1 = [elt[0] - a*(j1*x^(d1) * f.derivative()/m + d1*f*x^(d1 - 1)), RR(elt[1]) - a*R(x^(d1)), j]\n",
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" return zapis_w_bazie_dr(elt1, m, f, p)\n",
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" \n",
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" k = inv_stopnie2[d]\n",
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" b = wspolczynnik_wiodacy(baza[k][1])\n",
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" elt1 = [R(0), R(0), j]\n",
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" elt1[0] = elt[0] - a/b*baza[k][0]\n",
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" elt1[1] = elt[1] - a/b*baza[k][1]\n",
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" return zapis_w_bazie_dr(elt1, m, f, p) + vector([a*GF(p)(i == k) for i in range(0, len(baza))])\n",
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" \n",
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" \n",
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"def zapis_w_bazie_holo(elt, m, f, p):\n",
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" j = elt[2]\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" f = R(f) \n",
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" r = f.degree()\n",
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" delta = GCD(m, r)\n",
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" baza = baza_holo(m, f, j, p)\n",
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" wymiar = len(baza)\n",
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" zapis = vector([GF(p)(0) for i in baza])\n",
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" stopnie = stopnie_bazy_holo(m, f, j, p)\n",
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" inv_stopnie = {v: k for k, v in stopnie.items()}\n",
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" \n",
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" if elt[0] == 0:\n",
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" return zapis\n",
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" \n",
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" d = elt[0].degree()\n",
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" a = elt[0].coefficients(sparse = false)[d]\n",
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" \n",
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" k = inv_stopnie[d] #ktory element bazy jest stopnia d? ten o indeksie k\n",
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" \n",
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" a1 = baza[k][0].coefficients(sparse = false)[d]\n",
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" elt1 = [R(0),R(0), j]\n",
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" elt1[0] = elt[0] - a/a1 * baza[k][0]\n",
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" \n",
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" return zapis_w_bazie_holo(elt1, m, f, p) + vector([a/a1 * GF(p)(i == k) for i in range(0, len(baza))])\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": false
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},
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"source": [
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"We have: $V(\\omega, f) = (C(\\omega), 0)$ and $F(\\omega, f) = (0, f^p)$, where C denotes the Cartier operator. Moreover:\n",
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"\n",
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"let $t = multord_m(p)$, $M := (p^t - 1)/m$. Then: $y^{p^t - 1} = f(x)^M$ and $1/y = f(x)^M/y^{p^t}$. Thus:\n",
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"\n",
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"\n",
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"$$ C(P(x) \\, dx / y^j) = C(P(x) \\, f(x)^{M \\cdot j} \\, dx /y^{p^t \\cdot j}) = \\frac{1}{y^{p^{t - 1} \\cdot j}} C(P(x) \\, f(x)^{M \\cdot j} \\, dx) = \\frac{1}{y^{(p^{t - 1} \\cdot j) \\, mod \\, m}} \\cdot \\frac{1}{f(x)^{[p^{t - 1} \\cdot j/m]}} \\cdot C(P(x) \\, f(x)^{M \\cdot j} \\, dx)$$\n"
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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": 5,
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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 czesc_wielomianu(p, h):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" h = R(h)\n",
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" wynik = R(0)\n",
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" for i in range(0, h.degree()+1):\n",
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" if (i%p) == p-1:\n",
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" potega = Integer((i-(p-1))/p)\n",
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" wynik = wynik + Integer(h[i]) * x^(potega) \n",
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" return wynik\n",
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"\n",
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"def cartier_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" f = R(f)\n",
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" r = f.degree()\n",
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" delta = GCD(m, r)\n",
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" rzad = Integers(m)(p).multiplicative_order()\n",
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" M = Integer((p^(rzad)-1)/m)\n",
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" W = R(elt[0])\n",
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" h = R(W*f^(M*j))\n",
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" B = floor(p^(rzad-1)*j/m)\n",
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" g = czesc_wielomianu(p, h)/f^B\n",
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" jj = (p^(rzad-1)*j)%m\n",
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" #jj = Integers(m)(j/p)\n",
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" return [g, 0, jj] #jest to w czesci indeksowanej jj\n",
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"\n",
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"def macierz_cartier_dr(p, m, f, j):\n",
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" baza = baza_dr(m, f, j, p)\n",
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" A = matrix(GF(p), len(baza), len(baza))\n",
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" for k in range(0, len(baza)):\n",
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" cart = cartier_dr(p, m, f, baza[k], j)\n",
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" v = zapis_w_bazie_dr(cart, m, f, p)\n",
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" A[k, :] = matrix(v)\n",
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" return A.transpose()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": false
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},
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"source": [
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"$F((\\omega, P(x) \\cdot y^j)) = (0, P(x)^p \\cdot y^{p \\cdot j}) = (0, P(x)^p \\cdot f(x)^{[p \\cdot j/m]} \\cdot y^{(p \\cdot j) \\, mod \\, m})$"
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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": 6,
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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 frobenius_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" RR = FractionField(R)\n",
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" f = R(f)\n",
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" j1 = m-j\n",
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" M = floor(j1*p/m)\n",
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" return [0, f^M * RR(elt[1])^p, (j1*p)%m] #eigenspace = j1*p mod m\n",
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"\n",
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"def macierz_frob_dr(p, m, f, j):\n",
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" baza = baza_dr(m, f, j, p)\n",
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" A = matrix(GF(p), len(baza), len(baza))\n",
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" for k in range(0, len(baza)):\n",
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" frob = frobenius_dr(p, m, f, baza[k], j)\n",
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" v = zapis_w_bazie_dr(frob, m, f, p)\n",
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" A[k, :] = matrix(v)\n",
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" return A.transpose()\n",
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"\n",
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"def wspolczynnik_wiodacy(f):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" RR = FractionField(R)\n",
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" f = RR(f)\n",
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" f1 = f.numerator()\n",
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" f2 = f.denominator()\n",
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" d1 = f1.degree()\n",
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" d2 = f2.degree()\n",
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" a1 = f1.coefficients(sparse = false)[d1]\n",
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" a2 = f2.coefficients(sparse = false)[d2]\n",
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" return(a1/a2)\n",
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"\n",
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"def stopien_roznica(f):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" RR = FractionField(R)\n",
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" f = RR(f)\n",
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" f1 = f.numerator()\n",
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" f2 = f.denominator()\n",
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" d1 = f1.degree()\n",
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" d2 = f2.degree()\n",
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" return(d1 - d2)\n",
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"\n",
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"def czy_w_de_rhamie(elt, m, f, j, p):\n",
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" j1 = m - j\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" RR = FractionField(R)\n",
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" f = R(f)\n",
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" elt = [RR(elt[0]), RR(elt[1])]\n",
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" auxiliary = elt[0] - j1/m*elt[1]*f.derivative() - f*elt[1].derivative()\n",
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" deg = stopien_roznica(auxiliary)\n",
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" \n",
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" r = f.degree()\n",
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" delta = GCD(r, m)\n",
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" Rr = r/delta\n",
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" Mm = m/delta\n",
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" return(j*Rr - deg*Mm >= 0)\n",
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"\n",
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"def full_cartier(m, f, p):\n",
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" R.<x> = PolynomialRing(GF(p))\n",
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" f = R(f)\n",
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" r = f.degree()\n",
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" delta = GCD(m, r)\n",
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" g = 1/2*((m-1)*(r-1) - delta)\n",
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" print(g)\n",
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" \n",
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" wymiary = [0]+[len(baza_holo(m, f, j, p)) for j in range(1, m)]\n",
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" print(wymiary)\n",
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" for j1 in range(1, m):\n",
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" for j2 in range(1, m):\n",
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" print(j1, j2)\n",
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" print(matrix(GF(p), wymiary[j1], wymiary[j2]))\n",
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" lista = [[matrix(GF(p), wymiary[j1], wymiary[j2]) for j1 in range(0, m)] for j2 in range(0, m)]\n",
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" rzad = Integers(m)(p).multiplicative_order()\n",
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" \n",
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" for j in range(1, m):\n",
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" jj = (p^(rzad-1)*j)%m\n",
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" print(j, jj)\n",
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" print('wymiary', macierz_cartier_dr(p, m, f, j).dimensions(), wymiary[j], wymiary[jj])\n",
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" lista[j][jj] = macierz_cartier_dr(p, m, f, j)\n",
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" return lista \n",
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" return block_matrix(lista)"
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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": 243,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"5/2\n",
|
|
"[0, 0, 1, 2]\n",
|
|
"1 1\n",
|
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"[]\n",
|
|
"1 2\n",
|
|
"[]\n",
|
|
"1 3\n",
|
|
"[]\n",
|
|
"2 1\n",
|
|
"[]\n",
|
|
"2 2\n",
|
|
"[0]\n",
|
|
"2 3\n",
|
|
"[0 0]\n",
|
|
"3 1\n",
|
|
"[]\n",
|
|
"3 2\n",
|
|
"[0]\n",
|
|
"[0]\n",
|
|
"3 3\n",
|
|
"[0 0]\n",
|
|
"[0 0]\n",
|
|
"1 1\n",
|
|
"wymiary (2, 2) 0 0\n",
|
|
"2 2\n",
|
|
"wymiary (2, 2) 1 1\n",
|
|
"3 3\n",
|
|
"wymiary (2, 2) 2 2\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"m = 4\n",
|
|
"p = 5\n",
|
|
"f = x^3 + x+2\n",
|
|
"lista = full_cartier(m, f, p)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 241,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[\n",
|
|
" [2 3] [0]\n",
|
|
"[], [], [0 0], [0]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"execution_count": 241,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"lista[2]"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 247,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[0 0]\n",
|
|
"[0 0]"
|
|
]
|
|
},
|
|
"execution_count": 247,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"macierz_cartier_dr(p, m, f, 1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 249,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"{0: [3*x, 4/x, 0], 1: [4, 4/x^2, 0]}"
|
|
]
|
|
},
|
|
"execution_count": 249,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"baza_dr(m, f, 0, p)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 8,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"{0: [1, 0, 3], 1: [0, 2/x, 3]}"
|
|
]
|
|
},
|
|
"execution_count": 8,
|
|
"metadata": {
|
|
},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"p = 5\n",
|
|
"R.<x> = PolynomialRing(GF(p))\n",
|
|
"f = x^3 + x + 2\n",
|
|
"m = 7\n",
|
|
"baza_dr(m, f, 3, p)"
|
|
]
|
|
},
|
|
{
|
|
"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": "3.7.3"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
} |