{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "232ac4e6-80fb-4842-bab2-97c82a517a6d",
   "metadata": {},
   "outputs": [],
   "source": [
    "S0 = PolynomialRing(QQ, 'a', 12)\n",
    "S0.inject_variables(verbose=False)\n",
    "R.<x, y> = S0[]\n",
    "def fct(p):\n",
    "    RF = PolynomialRing(QQ, S0.gens() + R.gens())\n",
    "    p1 = sum(RF(a0) * RF(b0) for a0, b0 in p)\n",
    "    fc = p1.factor()\n",
    "    unidad = fc.unit()\n",
    "    fcts = [(R(p0), m) for p0, m in fc]\n",
    "    return (unidad, fcts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "ec7af5b3-8106-4d16-999f-09da392c48f5",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = (y^4 - x^6 + a0 * x^2 * y^3) * (x^4 - y^6 + a1 * x^3 * y^2)\n",
    "g = (y^6 + x^7 + a2 * x^6 * y + a4 * x^5 * y^2 + a6 * x^4 * y^3 + a8 * x^3 * y^4 + a10 * x * y^6) * (x^6 + y^7 + a3 * x * y^6 + a5 * x^2 * y^5 + a7 * x^3 * y^4 + a9 * x^4 * y^3 + a11 * x^6 * y)\n",
    "p = f^3 - g^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "15b0b7dd-286d-41cd-b6a8-afb1243bbf1b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1, [(x, 19), (x^7 + 2*y^6, 1)]),\n",
       " (-1, [(y, 19), (y^7 + 2*x^6, 1)]),\n",
       " (-1,\n",
       "  [(y, 6),\n",
       "   (x, 6),\n",
       "   (2*x^13 + 2*a2*x^12*y + 2*a4*x^11*y^2 + 2*a6*x^10*y^3 + 2*a8*x^9*y^4 + (-3*a0)*x^8*y^5 + 2*a10*x^7*y^6 + 2*a11*x^6*y^7 + (-3*a1)*x^5*y^8 + 2*a9*x^4*y^9 + 2*a7*x^3*y^10 + 2*a5*x^2*y^11 + 2*a3*x*y^12 + 2*y^13,\n",
       "    1)])]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = ZetaFunctions(p)\n",
    "u = [fct(q) for q in Z.get_polyfaces_dictionary(compact=True) if q.number_of_terms() > 1]\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "b9a91f1d-2d4e-46fc-b2fb-b19e8ac3f097",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2*a2 - 26)*x^12*y + (2*a4 - 156)*x^11*y^2 + (2*a6 - 572)*x^10*y^3 + (2*a8 - 1430)*x^9*y^4 + (-3*a0 - 2574)*x^8*y^5 + (2*a10 - 3432)*x^7*y^6 + (2*a11 - 3432)*x^6*y^7 + (-3*a1 - 2574)*x^5*y^8 + (2*a9 - 1430)*x^4*y^9 + (2*a7 - 572)*x^3*y^10 + (2*a5 - 156)*x^2*y^11 + (2*a3 - 26)*x*y^12"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = 2*x^13 + 2*a2*x^12*y + 2*a4*x^11*y^2 + 2*a6*x^10*y^3 + 2*a8*x^9*y^4 + (-3*a0)*x^8*y^5 + 2*a10*x^7*y^6 + 2*a11*x^6*y^7 + (-3*a1)*x^5*y^8 + 2*a9*x^4*y^9 + 2*a7*x^3*y^10 + 2*a5*x^2*y^11 + 2*a3*x*y^12 + 2*y^13 - 2 * (x + y)^13\n",
    "q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "a573d7e3-c568-4210-8b03-1a2333c432cd",
   "metadata": {},
   "outputs": [],
   "source": [
    "J = S0.ideal(q.coefficients())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "id": "4869bc3a-a30c-4f74-b33f-cd9a621fe2ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1716*x^6*y + 715*x^4*y^3 + 286*x^3*y^4 + 78*x^2*y^5 + 13*x*y^6 + y^7 + x^6) * (x^7 + 13*x^6*y + 78*x^5*y^2 + 286*x^4*y^3 + 715*x^3*y^4 + 1716*x*y^6 + y^6)"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "7d0e90b3-1955-4a84-8c8b-137765241570",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1) * x^19 * (x^7 + 2*y^6),\n",
       " (-1) * y^19 * (y^7 + 2*x^6),\n",
       " (-2) * y^6 * x^6 * (x + y)^13]"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = f.map_coefficients(lambda u: u.reduce(J)).change_ring(QQ)\n",
    "G = g.map_coefficients(lambda u: u.reduce(J)).change_ring(QQ)\n",
    "P = F^3 - G^2\n",
    "Z = ZetaFunctions(P)\n",
    "u = [q.factor() for q in Z.get_polyfaces_dictionary(compact=True) if q.number_of_terms() > 1]\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "7b07cd90-991b-4c1a-899d-4eb51c3f1c5a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Defining x, y\n"
     ]
    }
   ],
   "source": [
    "RQ = R.change_ring(QQ)\n",
    "RQ.inject_variables()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "id": "b97e77ca-e99e-4050-847e-bb8287a94a8e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-2) * (y^13 - 3784272*x)]"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1 = RQ(P(y=x*(y - 1)) / x^25)\n",
    "Z = ZetaFunctions(P1)\n",
    "u = [q.factor() for q in Z.get_polyfaces_dictionary(compact=True) if q.number_of_terms() > 1]\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d16f3d74-fb25-4332-8049-f360eeb9da92",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "SageMath 10.5.beta8",
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   "codemirror_mode": {
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