teichmuller dziala

crystalline_p2.ipynb

@@ -1,5 +1,134 @@
 {
  "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
+   "source": [
+    "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"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {
+    "collapsed": false
+   },
+   "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"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {
+     },
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Rx.<x> = PolynomialRing(QQ)\n",
+    "f = Rx(x^3 - x)\n",
+    "teichmuller(f)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
+   "source": [
+    "f = Rx(x^3 - x)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[0, -1, 0, 1]"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {
+     },
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "f.coefficients(sparse=false)"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 0,
@@ -25,6 +154,18 @@
    },
    "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": "3.9.9"
   }
  },
  "nbformat": 4,