{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "72d5a30c-1a59-4a55-9680-fdfa7979625b",
   "metadata": {},
   "outputs": [],
   "source": [
    "# load('zetas_triples_general.py')\n",
    "R.<x, y, z> = PolynomialRing(QQ, order='neglex')\n",
    "S.<u, v> = PolynomialRing(QQ, order='neglex')\n",
    "f = (v - u^2)^2 + u * v^3\n",
    "g = f^2 + v^6\n",
    "p = v - u^2\n",
    "F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)\n",
    "h = F.discriminant(z)(x=u, y=v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "57f4e587-fd6a-4aa2-8423-78e8b5f0d2dd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.jacobian_ideal().vector_space_dimension()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "93b0a2d7-b561-4251-a991-34e8c69fbf89",
   "metadata": {},
   "outputs": [],
   "source": [
    "# (F + 1 * p(u=x, v=y) * z^2).jacobian_ideal().vector_space_dimension()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "a932f83b-81ad-47d2-8c2e-b944239fc364",
   "metadata": {},
   "outputs": [],
   "source": [
    "# load('zetas_triples_general.py')\n",
    "R.<x, y, z> = PolynomialRing(QQ, order='neglex')\n",
    "S.<u, v> = PolynomialRing(QQ, order='neglex')\n",
    "f = v^2 - u^7 + u^5 * v\n",
    "f2 = v^2\n",
    "g = f^2 + (v + u^2)^6\n",
    "g1 = 3 / 2 * v * f\n",
    "g3 = v^3\n",
    "F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)\n",
    "h = F.discriminant(z)(x=u, y=v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "65fb99b3-89fb-4fb7-a0d4-667050dd5cab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F.jacobian_ideal().vector_space_dimension()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "8cc358aa-03d1-4b9c-b4ff-87b0324bd55b",
   "metadata": {},
   "outputs": [],
   "source": [
    "t = -3\n",
    "F1 = F - 3 * f2(u=x, v=y) * t^2 * z + 2 * g1(u=x, v=y) * t +  2 * g3(u=x, v=y) * t^3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "d21a93b0-b0c4-4cd2-b648-f4bca8f78376",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1.jacobian_ideal().vector_space_dimension()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "9974eefe-db38-4da1-b438-eb56f6effbe4",
   "metadata": {},
   "outputs": [],
   "source": [
    "def nwt(p):\n",
    "    return p(u=u^2, v=u^7 * (v + 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "05705617-d0f6-464a-a9a0-4cefb5bce4d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "u^14 * (2*v + v^2 + u^3 + u^3*v)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nwt(f).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "4eb24e43-3a10-4db9-8502-8c62289a3fe7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "u^24 * (1 + 6*u^3 + 6*u^3*v + 4*u^4*v^2 + 4*u^4*v^3 + u^4*v^4 + 15*u^6 + 30*u^6*v + 15*u^6*v^2 + 4*u^7*v + 6*u^7*v^2 + 2*u^7*v^3 + 20*u^9 + 60*u^9*v + 60*u^9*v^2 + 20*u^9*v^3 + u^10 + 2*u^10*v + u^10*v^2 + 15*u^12 + 60*u^12*v + 90*u^12*v^2 + 60*u^12*v^3 + 15*u^12*v^4 + 6*u^15 + 30*u^15*v + 60*u^15*v^2 + 60*u^15*v^3 + 30*u^15*v^4 + 6*u^15*v^5 + u^18 + 6*u^18*v + 15*u^18*v^2 + 20*u^18*v^3 + 15*u^18*v^4 + 6*u^18*v^5 + u^18*v^6)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nwt(g).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "28776340-947d-4ed6-a492-79f9341b9f18",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1 + v)^2 * u^14"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nwt(f2).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "358b6044-7de1-4bfe-9e4f-283adfc93aca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3/2) * (1 + v) * u^21 * (2*v + v^2 + u^3 + u^3*v)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nwt(g1).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "cda085bf-987d-44c2-8dbd-ae21afaa3251",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1 + v)^3 * u^21"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nwt(g3).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "9684ce50-7ff0-4b4c-9580-497c3e3b3964",
   "metadata": {},
   "outputs": [],
   "source": [
    "h1 = F1.discriminant(z)(x=u, y=v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "220532f1-226a-4898-9ff8-5f0344a7de16",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-27) * (v^2 - u^7) * (3965*v^4 + 2735*u^7*v^2 + 2048*u^14)]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z1 = ZetaFunctions(h1)\n",
    "[q.factor() for q in Z1.get_polyfaces_dictionary(compact=True) if q.number_of_terms() > 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "17c661cd-7293-41a1-b9ec-2e71269ae582",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(108) * u^21 * (3*v^2 - 2*u^3)]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h1a = S(h1(u=u^2, v=u^7 * (1 + v)) / u^21)\n",
    "Z1a = ZetaFunctions(h1a)\n",
    "[q.factor() for q in Z1a.get_polyfaces_dictionary(compact=True) if q.number_of_terms() > 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "aa670fe5-7aa2-4b88-9e85-4351e528b2a1",
   "metadata": {},
   "outputs": [],
   "source": [
    "Rt.<x, y, z, t> = QQ[]\n",
    "Ft = Rt(F - 3 * f2(u=x, v=y) * t^2 * z + 2 * g1(u=x, v=y) * t +  2 * g3(u=x, v=y) * t^3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "198a37ca-5a9d-4fe4-8949-e9ca06d48119",
   "metadata": {},
   "outputs": [],
   "source": [
    "J = Rt.ideal([Ft.derivative(a) for a in Rt.gens()[:-1]] + [Ft])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "77792a73-db8e-43a5-93b7-dacb75f92c4f",
   "metadata": {},
   "outputs": [],
   "source": [
    "Gt = Ft(z=z + y * t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "18ec2301-7665-436a-9a9c-e1070d99dfd1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*y*z^2"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Gt.derivative(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0fa40cb1-67c3-40c4-bf4b-60dd202979d2",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 10.4.rc1",
   "language": "sage",
   "name": "sagemath-devel"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}