{
 "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",
    " - the above forms along with\n",
    " $$\\lambda_{ij} = \\left[ \\left( \\frac{\\psi_{ij} \\, dx}{m x^{i+1} y^{m - j}},\n",
    "  \\frac{-\\phi_{ij} \\, dx}{m x^{i+1} y^{m - j}}, \\frac{y^j}{x^i} \\right) \\right]$$\n",
    "  (where $s_{ij} = jx f'(x) - mi f(x)$, \n",
    " $\\psi_{ij}(x) = s_{ij}^{\\ge i+1}$,\n",
    " $\\phi_{ij}(x) = s_{ij}^{< i+1}$)\n",
    "form a basis of $H^1_{dR}(C/K)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": 6,
   "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",
    "# [f(x) dx/y^j, y^(m-j)*g(x)]. 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": 140,
   "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, 'czy w dr', czy_w_de_rhamie(elt, m, f, j, p))\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    RR = FractionField(R)\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",
    "    a = wspolczynnik_wiodacy(g)\n",
    "    d = -stopien_roznica(g)\n",
    "    Rr = r/delta\n",
    "    Mm = m/delta\n",
    "    \n",
    "    stopnie2 = stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p)\n",
    "    inv_stopnie2 = {v: k for k, v in stopnie2.items()}      \n",
    "    print('d', d, 'a', a, 'dostepne:', stopnie2, 'czy w stopnie 2', d not in stopnie2)\n",
    "    if (d not in stopnie2.values()):\n",
    "        print('p3')\n",
    "        if d>= 0:\n",
    "            print('p3a')\n",
    "            j1 = m-j\n",
    "            print('?', -j1*Rr+d*Mm)\n",
    "            elt1 = [elt[0], RR(elt[1]) - a*1/R(x^d)]\n",
    "        else:\n",
    "            print('p3b')\n",
    "            d1 = -d\n",
    "            j1 = m-j\n",
    "            elt1 = [elt[0] - a*(j1*x^(d1) * f.derivative()/m + d1*f*x^(d1 - 1)), RR(elt[1]) - a*R(x^(d1))]\n",
    "        return zapis_w_bazie_dr(elt1, m, f, j, p)\n",
    "    \n",
    "    print('p4')\n",
    "    k = inv_stopnie2[d]\n",
    "    b = wspolczynnik_wiodacy(baza[k][1])\n",
    "    print('k', k, 'a', a)\n",
    "    elt1 = [R(0), R(0)]\n",
    "    elt1[0] = elt[0] - a/b*baza[k][0]\n",
    "    elt1[1] = elt[1] - a/b*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": 131,
   "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": 132,
   "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",
    "    j1 = m-j\n",
    "    M = floor(j1*p/m)\n",
    "    return [0, f^M * RR(elt[1])^p] #eigenspace = j1*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()\n",
    "\n",
    "def wspolczynnik_wiodacy(f):\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    RR = FractionField(R)\n",
    "    f = RR(f)\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 stopien_roznica(f):\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    RR = FractionField(R)\n",
    "    f = RR(f)\n",
    "    f1 = f.numerator()\n",
    "    f2 = f.denominator()\n",
    "    d1 = f1.degree()\n",
    "    d2 = f2.degree()\n",
    "    return(d1 - d2)\n",
    "\n",
    "def czy_w_de_rhamie(elt, m, f, j, p):\n",
    "    j1 = m - j\n",
    "    R.<x> = PolynomialRing(GF(p))\n",
    "    RR = FractionField(R)\n",
    "    f = R(f)\n",
    "    elt = [RR(elt[0]), RR(elt[1])]\n",
    "    auxiliary = elt[0] - j1/m*elt[1]*f.derivative() - f*elt[1].derivative()\n",
    "    deg = stopien_roznica(auxiliary)\n",
    "    \n",
    "    r = f.degree()\n",
    "    delta = GCD(r, m)\n",
    "    Rr = r/delta\n",
    "    Mm = m/delta\n",
    "    return(j*Rr - deg*Mm >= 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{0: [1, 0], 1: [x, 2/x]}\n",
      "[0, 0] czy w dr True\n",
      "p1\n",
      "[0, (2*x^6 + 4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5] czy w dr True\n",
      "d -1 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "[2*x + 4, (4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5] czy w dr True\n",
      "d 1 a 4 dostepne: {1: 1} czy w stopnie 2 False\n",
      "p4\n",
      "k 1 a 4\n",
      "[4, (2*x^3 + 2*x^2 + 2*x + 3)/x^5] czy w dr True\n",
      "d 2 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 1\n",
      "[4, (2*x^2 + 2*x + 3)/x^5] czy w dr True\n",
      "d 3 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 3\n",
      "[4, (2*x + 3)/x^5] czy w dr True\n",
      "d 4 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 5\n",
      "[4, 3/x^5] czy w dr True\n",
      "d 5 a 3 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 7\n",
      "[4, 0] czy w dr True\n",
      "p2\n",
      "[0, 0] czy w dr True\n",
      "p1\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[0 4]\n",
       "[0 4]"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 2\n",
    "j = 1\n",
    "p = 5\n",
    "f = x^3 + x+3\n",
    "print(baza_dr(m, f, j, p))\n",
    "macierz_frob_dr(p, m, f, j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "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": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "frobenius_dr(p, m, f, [x, 2/x], j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "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] czy w dr True\n",
      "d -1 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "[2*x + 4, (4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5] czy w dr True\n",
      "d 1 a 4 dostepne: {1: 1} czy w stopnie 2 False\n",
      "p4\n",
      "k 1 a 4\n",
      "[4, (2*x^3 + 2*x^2 + 2*x + 3)/x^5] czy w dr True\n",
      "d 2 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 1\n",
      "[4, (2*x^2 + 2*x + 3)/x^5] czy w dr True\n",
      "d 3 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 3\n",
      "[4, (2*x + 3)/x^5] czy w dr True\n",
      "d 4 a 2 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 5\n",
      "[4, 3/x^5] czy w dr True\n",
      "d 5 a 3 dostepne: {1: 1} czy w stopnie 2 True\n",
      "p3\n",
      "? 7\n",
      "[4, 0] czy w dr True\n",
      "p2\n",
      "[0, 0] czy w dr True\n",
      "p1\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(4, 4)"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": 137,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4*x^4 + 4*x^3 + 4*x^2 + 4*x + 4"
      ]
     },
     "execution_count": 137,
     "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": 138,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{0: [0, 4/x], 1: [1, 4/x^2]}\n",
      "{0: 1, 1: 2}\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 4\n",
    "j = 1\n",
    "p = 5\n",
    "f = x^3 + x+2\n",
    "print(baza_dr(m, f, j, p))\n",
    "print(stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p))\n",
    "t = stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p)\n",
    "1 in t.values()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, (4*x^9 + 2*x^7 + 4*x^6 + 2*x^5 + 3*x^4 + 2*x^3 + 4*x^2 + 3*x + 2)/x^5] czy w dr True\n",
      "d -4 a 4 dostepne: {0: 1, 1: 2} czy w stopnie 2 True\n",
      "p3\n",
      "p3b\n",
      "[x^4 + 3*x^3, (2*x^7 + 4*x^6 + 2*x^5 + 3*x^4 + 2*x^3 + 4*x^2 + 3*x + 2)/x^5] czy w dr True\n",
      "d -2 a 2 dostepne: {0: 1, 1: 2} czy w stopnie 2 True\n",
      "p3\n",
      "p3b\n",
      "[3*x^3 + 2*x^2 + 2*x, (4*x^6 + 2*x^5 + 3*x^4 + 2*x^3 + 4*x^2 + 3*x + 2)/x^5] czy w dr True\n",
      "d -1 a 4 dostepne: {0: 1, 1: 2} czy w stopnie 2 True\n",
      "p3\n",
      "p3b\n",
      "[2*x^2 + 2, (2*x^5 + 3*x^4 + 2*x^3 + 4*x^2 + 3*x + 2)/x^5] czy w dr True\n",
      "d 0 a 2 dostepne: {0: 1, 1: 2} czy w stopnie 2 False\n",
      "p3\n",
      "p3a\n",
      "? -9\n",
      "[2*x^2 + 2, (3*x^4 + 2*x^3 + 4*x^2 + 3*x + 2)/x^5] czy w dr False\n",
      "d 1 a 3 dostepne: {0: 1, 1: 2} czy w stopnie 2 False\n",
      "p4\n",
      "k 0 a 3\n",
      "[2*x^2 + 2, (2*x^3 + 4*x^2 + 3*x + 2)/x^5] czy w dr False\n",
      "d 2 a 2 dostepne: {0: 1, 1: 2} czy w stopnie 2 True\n",
      "p4\n",
      "k 1 a 2\n",
      "[2*x^2 + 4, (4*x^2 + 3*x + 2)/x^5] czy w dr False\n",
      "d 3 a 4 dostepne: {0: 1, 1: 2} czy w stopnie 2 True\n",
      "p3\n",
      "p3a\n",
      "? 3\n",
      "[2*x^2 + 4, (3*x + 2)/x^5] czy w dr False\n",
      "d 4 a 3 dostepne: {0: 1, 1: 2} czy w stopnie 2 True\n",
      "p3\n",
      "p3a\n",
      "? 7\n",
      "[2*x^2 + 4, 2/x^5] czy w dr False\n",
      "d 5 a 2 dostepne: {0: 1, 1: 2} czy w stopnie 2 True\n",
      "p3\n",
      "p3a\n",
      "? 11\n",
      "[2*x^2 + 4, 0] czy w dr False\n",
      "p2\n"
     ]
    },
    {
     "ename": "KeyError",
     "evalue": "2",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-141-86a3d5ca17ee>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mmacierz_frob_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\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[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[0m\n\u001b[0m",
      "\u001b[0;32m<ipython-input-132-3ef6f7d47fcd>\u001b[0m in \u001b[0;36mmacierz_frob_dr\u001b[0;34m(p, m, f, j)\u001b[0m\n\u001b[1;32m     12\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\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[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbaza\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     13\u001b[0m         \u001b[0mfrob\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfrobenius_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\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[0mbaza\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 14\u001b[0;31m         \u001b[0mv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfrob\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[1;32m     15\u001b[0m         \u001b[0mA\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmatrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     16\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mA\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtranspose\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<ipython-input-140-9ce96c4e8631>\u001b[0m in \u001b[0;36mzapis_w_bazie_dr\u001b[0;34m(elt, m, f, j, p)\u001b[0m\n\u001b[1;32m     85\u001b[0m             \u001b[0mj1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mj\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[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[0;34m(\u001b[0m\u001b[0mj1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mf\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[0mm\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0md1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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[0mRR\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;36m1\u001b[0m\u001b[0;34m)\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[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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---> 87\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0melt1\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[1;32m     88\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     89\u001b[0m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'p4'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<ipython-input-140-9ce96c4e8631>\u001b[0m in \u001b[0;36mzapis_w_bazie_dr\u001b[0;34m(elt, m, f, j, p)\u001b[0m\n\u001b[1;32m     85\u001b[0m             \u001b[0mj1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mj\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[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[0;34m(\u001b[0m\u001b[0mj1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mf\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[0mm\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0md1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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[0mRR\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;36m1\u001b[0m\u001b[0;34m)\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[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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---> 87\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0melt1\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[1;32m     88\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     89\u001b[0m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'p4'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
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      "\u001b[0;32m<ipython-input-140-9ce96c4e8631>\u001b[0m in \u001b[0;36mzapis_w_bazie_dr\u001b[0;34m(elt, m, f, j, p)\u001b[0m\n\u001b[1;32m     85\u001b[0m             \u001b[0mj1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mj\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[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[0;34m(\u001b[0m\u001b[0mj1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mf\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[0mm\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0md1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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[0mRR\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;36m1\u001b[0m\u001b[0;34m)\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[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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---> 87\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0melt1\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[1;32m     88\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     89\u001b[0m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'p4'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<ipython-input-140-9ce96c4e8631>\u001b[0m in \u001b[0;36mzapis_w_bazie_dr\u001b[0;34m(elt, m, f, j, p)\u001b[0m\n\u001b[1;32m     85\u001b[0m             \u001b[0mj1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mj\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[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[0;34m(\u001b[0m\u001b[0mj1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mf\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[0mm\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0md1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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[0mRR\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;36m1\u001b[0m\u001b[0;34m)\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[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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---> 87\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0melt1\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[1;32m     88\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     89\u001b[0m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'p4'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<ipython-input-140-9ce96c4e8631>\u001b[0m in \u001b[0;36mzapis_w_bazie_dr\u001b[0;34m(elt, m, f, j, p)\u001b[0m\n\u001b[1;32m     85\u001b[0m             \u001b[0mj1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mj\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[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[0;34m(\u001b[0m\u001b[0mj1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mf\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[0mm\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0md1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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[0mRR\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;36m1\u001b[0m\u001b[0;34m)\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[0mR\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md1\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---> 87\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0melt1\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[1;32m     88\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     89\u001b[0m     \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'p4'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<ipython-input-140-9ce96c4e8631>\u001b[0m in \u001b[0;36mzapis_w_bazie_dr\u001b[0;34m(elt, m, f, j, p)\u001b[0m\n\u001b[1;32m     56\u001b[0m         \u001b[0md\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[0mdegree\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     57\u001b[0m         \u001b[0ma\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[0mcoefficients\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msparse\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfalse\u001b[0m\u001b[0;34m)\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[0;32m---> 58\u001b[0;31m         \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0minv_stopnie_holo\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;31m#ktory element bazy jest stopnia d? ten o indeksie k\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     59\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     60\u001b[0m         \u001b[0ma1\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[0mcoefficients\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msparse\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfalse\u001b[0m\u001b[0;34m)\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[0;31mKeyError\u001b[0m: 2"
     ]
    }
   ],
   "source": [
    "macierz_frob_dr(p, m, f, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wspolczynnik_wiodacy(2/x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "degree() takes exactly one argument (0 given)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-97-2687f469fb6b>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \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[0;34m.\u001b[0m\u001b[0mdegree\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;31mTypeError\u001b[0m: degree() takes exactly one argument (0 given)"
     ]
    }
   ],
   "source": [
    "(2/x).degree()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
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   "name": "sagemath"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
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