DeRhamComputation/deRhamComputation.ipynb

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    "# 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",
    " - "
   ]
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   "execution_count": 5,
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   "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"
   ]
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   "cell_type": "code",
   "execution_count": 6,
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   "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"
   ]
  },
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   "execution_count": 7,
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   "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",
    "    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",
    "        return zapis\n",
    "    \n",
    "    if elt[1] == 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",
    "        elt1 = [R(0), R(0)]\n",
    "        elt1[0] = elt[0]\n",
    "        return zapis_w_bazie_dr(elt1, m, f, j, p)\n",
    "    \n",
    "    \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"
   ]
  },
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    {
     "data": {
      "text/plain": [
       "{0: [x, 2/x], 1: [2, 2/x^2]}"
      ]
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     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "baza_dr(2, x^3 + 1, 0, 3)"
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   "execution_count": null,
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   "source": []
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