From bb63124a4c1d93bc4aa9203313a03c92e618d633 Mon Sep 17 00:00:00 2001
From: jgarnek <jgarnek@amu.edu.pl>
Date: Thu, 5 May 2022 11:09:44 +0000
Subject: [PATCH] dodalem_klase fraction polynomial - x oraz X razem

---
 crystalline_p2.ipynb | 119 ++++++++++++++++++++++++++++++++++++-------
 1 file changed, 101 insertions(+), 18 deletions(-)

diff --git a/crystalline_p2.ipynb b/crystalline_p2.ipynb
index 790c82d..4479b59 100644
--- a/crystalline_p2.ipynb
+++ b/crystalline_p2.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 61,
+   "execution_count": 83,
    "metadata": {
     "collapsed": false
    },
@@ -47,15 +47,31 @@
     "    def __sub__(self, other):\n",
     "        return self + (-1)*other\n",
     "\n",
+    "class fraction_pol:\n",
+    "    def __init__(self, g):\n",
+    "        RxX.<x, X> = PolynomialRing(QQ)\n",
+    "        g = RxX(g)\n",
+    "        result = 0\n",
+    "        for a in g.monomials():\n",
+    "            c = g.coefficient(a)\n",
+    "            c = c%9\n",
+    "            dX = a.degree(X)\n",
+    "            dx = a.degree(x)\n",
+    "            if dX%p == 0:\n",
+    "                result += c*x^(dX//p + dx)\n",
+    "            else:\n",
+    "                result += c*X^(dX + dx*p)\n",
+    "        self.function = result\n",
+    "    \n",
     "def teichmuller(f):\n",
     "    Rx.<x> = PolynomialRing(QQ)\n",
-    "    RXp.<Xp> = PolynomialRing(QQ)\n",
+    "    RxX.<x, X> = PolynomialRing(QQ, 2)\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",
-    "    f1 = sum(coeffs[i]*RXp(Xp^(3*i)) for i in range(0, len(coeffs))) + p*RXp(ff.coordinates[1](x = Xp))\n",
+    "    f1 = sum(coeffs[i]*RxX(x^(i)) for i in range(0, len(coeffs))) + p*RxX(ff.coordinates[1](x = X))\n",
     "    #RXp.<Xp> = PolynomialRing(Integers(p^2))\n",
     "    #f1 = RXp(f1)\n",
     "    return f1"
@@ -63,7 +79,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 72,
+   "execution_count": 74,
    "metadata": {
     "collapsed": false
    },
@@ -94,9 +110,6 @@
     "                degrees0[t] = (psi.degree(), j)\n",
     "                degrees1[t] = (-i, m-j)\n",
     "                t += 1\n",
-    "    RXpy.<Xp, y> = PolynomialRing(QQ, 2)\n",
-    "    FXpy = FractionField(RXpy)\n",
-    "    basis = [(a[0](x = Xp^p, y = y), a[1](x = Xp^p, y = y)) for a in basis]\n",
     "    return basis, degrees0, degrees1\n",
     "\n",
     "def de_rham_basis(f, m, p):\n",
@@ -164,8 +177,8 @@
     "class superelliptic_function:\n",
     "    def __init__(self, C, g):\n",
     "        p = C.characteristic\n",
-    "        RXpy.<Xp, y> = PolynomialRing(QQ, 2)\n",
-    "        FXpy = FractionField(RXpy)\n",
+    "        RxXy.<x, X, y> = PolynomialRing(QQ, 3)\n",
+    "        FxXy = FractionField(RXy)\n",
     "        f = C.polynomial\n",
     "        r = f.degree()\n",
     "        m = C.exponent\n",
@@ -180,10 +193,10 @@
     "        g = self.function\n",
     "        C = self.curve\n",
     "        p = C.characteristic\n",
-    "        RXp.<Xp> = PolynomialRing(GF(p))\n",
-    "        FXp.<x> = FractionField(RXp)\n",
-    "        FXpRy.<y> = PolynomialRing(FXp)\n",
-    "        g = FXpRy(g)\n",
+    "        RxX.<x,X> = PolynomialRing(QQ, 2)\n",
+    "        FxX.<x> = FractionField(RxX)\n",
+    "        FxXRy.<y> = PolynomialRing(FxX)\n",
+    "        g = FxXRy(g)\n",
     "        return coff(g, j)\n",
     "    \n",
     "    def __add__(self, other):\n",
@@ -219,12 +232,12 @@
     "    m = C.exponent\n",
     "    p = C.characteristic\n",
     "    g = self.function\n",
-    "    RXpy.<Xp, y> = PolynomialRing(QQ, 2)\n",
-    "    FXpy = FractionField(RXpy)\n",
-    "    g = RXpy(g)\n",
-    "    A = g.derivative(Xp)*Xp^(-(p-1))/p\n",
+    "    RXy.<X, y> = PolynomialRing(QQ, 2)\n",
+    "    FXy = FractionField(RXy)\n",
+    "    g = RXy(g)\n",
+    "    A = g.derivative(X)*X^(-(p-1))/p\n",
     "    t = teichmuller(f)\n",
-    "    B = g.derivative(y)*t.derivative()/(m*y^(m-1))*Xp^(-(p-1))/p\n",
+    "    B = g.derivative(y)*t.derivative()/(m*y^(m-1))*X^(-(p-1))/p\n",
     "    return superelliptic_form(C, A+B)\n",
     "        \n",
     "class superelliptic_form:\n",
@@ -662,6 +675,76 @@
     "t.differentiate(Xp)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 75,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
+   "source": [
+    "RxX.<x, X> = PolynomialRing(QQ)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 76,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+   ],
+   "source": [
+    "g = RxX(x*X + 2*x*X^3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 82,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "3"
+      ]
+     },
+     "execution_count": 82,
+     "metadata": {
+     },
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "g.monomials()[0].degree(X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 78,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "2"
+      ]
+     },
+     "execution_count": 78,
+     "metadata": {
+     },
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "g.coefficient(x*X^3)"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 0,