DeRhamComputation/deRhamComputation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Theory\n",
    "Let $C : y^m = f(x)$. Then:\n",
    "\n",
    " - the basis of $H^0(C, \\Omega_{C/k})$ is given by:\n",
    " $$x^{i-1} dx/y^j,$$\n",
    " 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",
    " \n",
    " - "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The program computes the basis of holomorphic differentials of y^m = f(x) in char p.\n",
    "# The coefficient j means that we compute the j-th eigenpart, i.e.\n",
    "# forms y^j * f(x) dx. Output is [f(x), 0]\n",
    "\n",
    "def baza_holo(m, f, j, p):\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    f = R(f)\n",
    "    r = f.degree()\n",
    "    delta = GCD(m, r)\n",
    "    baza = {}\n",
    "    k = 0\n",
    "    for i in range(1, r):\n",
    "        if (r*j - m*i >= delta):\n",
    "            baza[k] = [x^(i-1), R(0)]\n",
    "            k = k+1\n",
    "    return baza"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The program computes the basis of de Rham cohomology of y^m = f(x) in char p.\n",
    "# We treat them as pairs [omega, f], where omega is regular on the affine part\n",
    "# and omega - df is regular on the second atlas.\n",
    "# The coefficient j means that we compute the j-th eigenpart, i.e.\n",
    "# [y^j * f(x) dx, g(x)/y^j]. Output is [f(x), g(x)]\n",
    "\n",
    "def baza_dr(m, f, j, p):\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    f = R(f)    \n",
    "    r = f.degree()\n",
    "    delta = GCD(m, r)\n",
    "    baza = {}\n",
    "    holo = baza_holo(m, f, j, p)\n",
    "    for k in range(0, len(holo)):\n",
    "        baza[k] = holo[k]\n",
    "    \n",
    "    k = len(baza)\n",
    "    \n",
    "    for i in range(1, r):\n",
    "        if (r*(m-j) - m*i >= delta):\n",
    "            s = R(m-j)*R(x)*R(f.derivative()) - R(m)*R(i)*f\n",
    "            psi = R(obciecie(s, i, p))\n",
    "            baza[k] = [psi, R(m)/x^i]\n",
    "            k = k+1\n",
    "    return baza"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [],
   "source": [
    "#auxiliary programs\n",
    "def stopnie_bazy_holo(m, f, j, p):\n",
    "    baza = baza_holo(m, f, j, p)\n",
    "    stopnie = {}\n",
    "    for k in range(0, len(baza)):\n",
    "        stopnie[k] = baza[k][0].degree()\n",
    "    return stopnie\n",
    "\n",
    "def stopnie_bazy_dr(m, f, j, p):\n",
    "    baza = baza_dr(m, f, j, p)\n",
    "    stopnie = {}\n",
    "    for k in range(0, len(baza)):\n",
    "        stopnie[k] = baza[k][0].degree()\n",
    "    return stopnie\n",
    "\n",
    "def stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p):\n",
    "    baza = baza_dr(m, f, j, p)\n",
    "    stopnie = {}\n",
    "    for k in range(0, len(baza)):\n",
    "        if baza[k][1] != 0:\n",
    "            stopnie[k] = baza[k][1].denominator().degree()\n",
    "    return stopnie\n",
    "\n",
    "def obciecie(f, i, p):\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    f = R(f)\n",
    "    coeff = f.coefficients(sparse = false)\n",
    "    return sum(x^(j-i-1) * coeff[j] for j in range(i+1, f.degree() + 1))\n",
    "\n",
    "\n",
    "#Any element [f dx, g] is represented as a combination of the basis vectors.\n",
    "\n",
    "def zapis_w_bazie_dr(elt, m, f, j, p):\n",
    "    print(elt)\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    f = R(f)\n",
    "    r = f.degree()\n",
    "    delta = GCD(m, r)\n",
    "    baza = baza_dr(m, f, j, p)\n",
    "    wymiar = len(baza)\n",
    "    zapis = vector([GF(p)(0) for i in baza])\n",
    "    stopnie = stopnie_bazy_dr(m, f, j, p)\n",
    "    inv_stopnie = {v: k for k, v in stopnie.items()}\n",
    "    stopnie_holo = stopnie_bazy_holo(m, f, j, p)\n",
    "    inv_stopnie_holo = {v: k for k, v in stopnie_holo.items()}    \n",
    "    \n",
    "    ## zmiana\n",
    "    if elt[0]== 0 and elt[1] == 0:\n",
    "        print('p1')\n",
    "        return zapis\n",
    "    \n",
    "    if elt[1] == 0:\n",
    "        print('p2')\n",
    "        elt[0] = R(elt[0])\n",
    "        d = elt[0].degree()\n",
    "        a = elt[0].coefficients(sparse = false)[d]\n",
    "        k = inv_stopnie_holo[d] #ktory element bazy jest stopnia d? ten o indeksie k\n",
    "    \n",
    "        a1 = baza[k][0].coefficients(sparse = false)[d]\n",
    "        elt1 = [R(0),R(0)]\n",
    "        elt1[0] = elt[0] - a/a1 * baza[k][0]\n",
    "        elt1[1] = R(0)\n",
    "        return zapis_w_bazie_dr(elt1, m, f, j, p) + vector([a/a1*GF(p)(i == k) for i in range(0, len(baza))])\n",
    "\n",
    "    g = elt[1]\n",
    "    g1 = R(elt[1].numerator())\n",
    "    g2 = R(elt[1].denominator())\n",
    "    d1 = g1.degree()\n",
    "    d2 = g2.degree()\n",
    "    a1 = g1.coefficients(sparse = false)[d1]\n",
    "    a2 = g2.coefficients(sparse = false)[d2]\n",
    "    a = a1/a2\n",
    "    d = d2 - d1\n",
    "    \n",
    "    if (d*m - (m-j)*r >= 0):\n",
    "        print('p3')\n",
    "        elt1 = [R(0), R(0)]\n",
    "        elt1[0] = elt[0]\n",
    "        return zapis_w_bazie_dr(elt1, m, f, j, p)\n",
    "    \n",
    "    print('p4')\n",
    "    stopnie2 = stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p)\n",
    "    inv_stopnie2 = {v: k for k, v in stopnie2.items()}  \n",
    "    k = inv_stopnie2[d]\n",
    "    elt1 = [R(0), R(0)]\n",
    "    elt1[0] = elt[0] - a*baza[k][0]\n",
    "    elt1[1] = elt[1] - a*baza[k][1]\n",
    "    return zapis_w_bazie_dr(elt1, m, f, j, p) + vector([a*GF(p)(i == k) for i in range(0, len(baza))])\n",
    "    \n",
    "    \n",
    "def zapis_w_bazie_holo(elt, m, f, j, p):\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    f = R(f)    \n",
    "    r = f.degree()\n",
    "    delta = GCD(m, r)\n",
    "    baza = baza_holo(m, f, j, p)\n",
    "    wymiar = len(baza)\n",
    "    zapis = vector([GF(p)(0) for i in baza])\n",
    "    stopnie = stopnie_bazy_holo(m, f, j, p)\n",
    "    inv_stopnie = {v: k for k, v in stopnie.items()}\n",
    "    \n",
    "    if elt[0] == 0:\n",
    "        return zapis\n",
    "    \n",
    "    d = elt[0].degree()\n",
    "    a = elt[0].coefficients(sparse = false)[d]\n",
    "    \n",
    "    k = inv_stopnie[d] #ktory element bazy jest stopnia d? ten o indeksie k\n",
    "    \n",
    "    a1 = baza[k][0].coefficients(sparse = false)[d]\n",
    "    elt1 = [R(0),R(0)]\n",
    "    elt1[0] = elt[0] - a/a1 * baza[k][0]\n",
    "    \n",
    "    return zapis_w_bazie_holo(elt1, m, f, j, p) + vector([a/a1 * GF(p)(i == k) for i in range(0, len(baza))])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have: $V(\\omega, f) = (C(\\omega), 0)$ and $F(\\omega, f) = (0, f^p)$, where C denotes the Cartier operator. Moreover:\n",
    "\n",
    "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",
    "\n",
    "\n",
    "$$ C(P(x) \\, dx / y^j) = C(P(x) \\, f(x)^{M \\cdot j} \\, dx /y^{p^t \\cdot j}) = \\frac{1}{y^{p^{t - 1} \\cdot j}} C(P(x) \\, f(x)^{M \\cdot j} \\, dx) = \\frac{1}{y^{(p^{t - 1} \\cdot j) \\, mod \\, m}} \\cdot \\frac{1}{f(x)^{[p^{t - 1} \\cdot j/m]}} \\cdot C(P(x) \\, f(x)^{M \\cdot j} \\, dx)$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [],
   "source": [
    "def czesc_wielomianu(p, h):\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    h = R(h)\n",
    "    wynik = R(0)\n",
    "    for i in range(0, h.degree()+1):\n",
    "        if (i%p) == p-1:\n",
    "            potega = Integer((i-(p-1))/p)\n",
    "            wynik = wynik + Integer(h[i]) * x^(potega)    \n",
    "    return wynik\n",
    "\n",
    "def cartier_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    f = R(f)\n",
    "    r = f.degree()\n",
    "    delta = GCD(m, r)\n",
    "    rzad = Integers(m)(p).multiplicative_order()\n",
    "    M = Integer((p^(rzad)-1)/m)\n",
    "    W = R(elt[0])\n",
    "    h = R(W*f^(M*j))\n",
    "    B = floor(p^(rzad-1)*j/m)\n",
    "    g = czesc_wielomianu(p, h)/f^B\n",
    "    jj = (p^(rzad-1)*j)%m\n",
    "    #jj = Integers(m)(j/p)\n",
    "    return [g, 0] #jest to w czesci indeksowanej jj\n",
    "\n",
    "def macierz_cartier_dr(p, m, f, j):\n",
    "    baza = baza_dr(m, f, j, p)\n",
    "    A = matrix(GF(p), len(baza), len(baza))\n",
    "    for k in range(0, len(baza)):\n",
    "        cart = cartier_dr(p, m, f, baza[k], j)\n",
    "        v = zapis_w_bazie_dr(cart, m, f, j, p)\n",
    "        A[k, :] = matrix(v)\n",
    "    return A.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$F((\\omega, P(x) \\cdot y^j)) = (0, P(x)^p \\cdot y^{p \\cdot j}) = (0, P(x)^p \\cdot f(x)^{[p \\cdot j/m]} \\cdot y^{(p \\cdot j) \\, mod \\, m})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [],
   "source": [
    "def frobenius_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    RR = FractionField(R)\n",
    "    f = R(f)\n",
    "    M = floor(j*p/m)\n",
    "    return [0, f^M * RR(elt[1])^p] #eigenspace = j*p mod m\n",
    "\n",
    "def macierz_frob_dr(p, m, f, j):\n",
    "    baza = baza_dr(m, f, j, p)\n",
    "    A = matrix(GF(p), len(baza), len(baza))\n",
    "    for k in range(0, len(baza)):\n",
    "        frob = frobenius_dr(p, m, f, baza[k], j)\n",
    "        v = zapis_w_bazie_dr(frob, m, f, j, p)\n",
    "        A[k, :] = matrix(v)\n",
    "    return A.transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: [1, 0], 1: [x, 2/x]}"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 2\n",
    "j = 1\n",
    "p = 5\n",
    "f = x^3 + x+3\n",
    "baza_dr(m, f, j, p)\n",
    "#macierz_frob_dr(p, m, f, j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, (2*x^6 + 4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5]"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "frobenius_dr(p, m, f, [x, 2/x], j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, (2*x^6 + 4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5]\n",
      "p4\n"
     ]
    },
    {
     "ename": "KeyError",
     "evalue": "-1",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-97-b5994066838e>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\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[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m6\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\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;36m4\u001b[0m\u001b[0;34m)\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[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[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\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;36m2\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;36m3\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[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\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[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\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<ipython-input-92-50e115077f5c>\u001b[0m in \u001b[0;36mzapis_w_bazie_dr\u001b[0;34m(elt, m, f, j, p)\u001b[0m\n\u001b[1;32m     82\u001b[0m     \u001b[0mstopnie2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mstopnie_drugiej_wspolrzednej_bazy_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     83\u001b[0m     \u001b[0minv_stopnie2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0mv\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mstopnie2\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mitems\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[0;32m---> 84\u001b[0;31m     \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0minv_stopnie2\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0md\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     85\u001b[0m     \u001b[0melt1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\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     86\u001b[0m     \u001b[0melt1\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0melt\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mbaza\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\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;31mKeyError\u001b[0m: -1"
     ]
    }
   ],
   "source": [
    "zapis_w_bazie_dr([0, (2*x^6 + 4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5], m, f, j, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4*x^4 + 4*x^3 + 4*x^2 + 4*x + 4"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = sum(i*x^i for i in range(0, p^2))\n",
    "czesc_wielomianu(p, h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: [x^2, 0], 1: [x, 0], 2: [1, 0]}"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 5\n",
    "j = 1\n",
    "p = 5\n",
    "f = x^4 + x+2\n",
    "baza_dr(m, f, j, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0 0 0]\n",
       "[0 0 0]\n",
       "[0 0 0]"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "macierz_frob_dr(p, m, f, j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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