{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "aedc330d-0f66-48b0-9bab-290cc2d29674",
   "metadata": {},
   "outputs": [],
   "source": [
    "R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')\n",
    "R2.<u, v> = PolynomialRing(QQ, order='negdeglex')\n",
    "R2.<u, v> = PolynomialRing(QQ, order='negdeglex')\n",
    "f = (v^2 - u^3)^4 + u^2 * v^7 + v^9 + u^13\n",
    "g = (v^2 - u^3)^6 + u^2 * v^11 + v^13\n",
    "F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "f8fdf84a-210d-499f-a694-8181041eee1c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "837"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = F.discriminant(z)(x=u, y=v)\n",
    "h.jacobian_ideal().vector_space_dimension()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "9a325ac6-d1c1-4c3f-a9a6-a0cb42024c8a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "586"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4a48f3b9-7ee5-4f3c-afd4-5cb2e3bd26c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(216) * (u + v) * v^6 * (u^6 - u^5*v + u^4*v^2 - u^3*v^3 + u^2*v^4 - u*v^5 + v^6),\n",
       " (108) * u^7 * (2*v^6 - u^7)]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z0 = ZetaFunctions(h)\n",
    "[p.factor() for p in Z0.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "0d5f9c36-da08-415f-9207-7d7c089645bd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1 = F(x=x + 5 * z, y= y + 7 * z)\n",
    "F1.degree(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "d70efb99-6df4-41ac-bc8f-14cd4fd05a8a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "586"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dis = F1.discriminant(z)(x=u, y=v)\n",
    "dis.jacobian_ideal().vector_space_dimension()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f151e667-d7c6-409a-8587-ff34d2b53a32",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 10.4.beta9",
   "language": "sage",
   "name": "sagemath-310"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
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   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
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