DeRhamComputation/crystalline_p2.ipynb

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    "N = 2\n",
    "p = 3\n",
    "RQ = PolynomialRing(QQ, 'X', 2*N)\n",
    "X = RQ.gens()[:N]\n",
    "Y = RQ.gens()[N:]\n",
    "Rpx.<x> = PolynomialRing(GF(p), 1)\n",
    "#RQx.<x> = PolynomialRing(QQ, 1)\n",
    "\n",
    "def witt_pol(lista):\n",
    "    n = len(lista)\n",
    "    return sum(p^i*lista[i]^(p^(n-i-1)) for i in range(0, n))\n",
    "\n",
    "def witt_sum(n):\n",
    "    if n == 0:\n",
    "        return X[0] + Y[0]\n",
    "    return 1/p^n*(witt_pol(X[:n+1]) + witt_pol(Y[:n+1]) - sum(p^k*witt_sum(k)^(p^(n-k)) for k in range(0, n)))\n",
    "\n",
    "class witt:\n",
    "    def __init__(self, coordinates):\n",
    "        self.coordinates = coordinates\n",
    "    def __repr__(self):\n",
    "        lista = [Rpx(a) for a in self.coordinates]\n",
    "        return str(lista)\n",
    "    def __add__(self, other):\n",
    "        lista = []\n",
    "        for i in range(0, N):\n",
    "            lista+= [witt_sum(i)(self.coordinates + other.coordinates)]\n",
    "        return witt(lista)\n",
    "    def __rmul__(self, constant):\n",
    "        if constant<0:\n",
    "            m = (-constant)*(p^N-1)\n",
    "            return m*self\n",
    "        if constant == 0:\n",
    "            return witt(N*[0])\n",
    "        return self + (constant - 1)*self\n",
    "    def __sub__(self, other):\n",
    "        return self + (-1)*other\n",
    "\n",
    "def teichmuller(f):\n",
    "    Rx.<x> = PolynomialRing(QQ)\n",
    "    RXp.<Xp> = PolynomialRing(QQ)\n",
    "    f = Rx(f)\n",
    "    ff = witt([f, 0])\n",
    "    coeffs = f.coefficients(sparse=false)\n",
    "    for i in range(0, len(coeffs)):\n",
    "        ff -= coeffs[i]*witt([Rx(x^i), 0])\n",
    "    print(ff)\n",
    "    f1 = sum(coeffs[i]*RXp(Xp^(3*i)) for i in range(0, len(coeffs))) + p*RXp(ff.coordinates[1](x = Xp))\n",
    "    RXp.<Xp> = PolynomialRing(Integers(p^2))\n",
    "    f1 = RXp(f1)\n",
    "    return f1"
   ]
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, -x^7 + x^5]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Xp^9 + 6*Xp^7 + 3*Xp^5 + 8*Xp^3"
      ]
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     "execution_count": 29,
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   "source": [
    "Rx.<x> = PolynomialRing(QQ)\n",
    "f = Rx(x^3 - x)\n",
    "teichmuller(f)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
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   "source": [
    "f = Rx(x^3 - x)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
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   "outputs": [
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     "data": {
      "text/plain": [
       "[0, -1, 0, 1]"
      ]
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     "execution_count": 13,
     "metadata": {
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     "output_type": "execute_result"
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   ],
   "source": [
    "f.coefficients(sparse=false)"
   ]
  },
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   "execution_count": 0,
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