{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "4778e32e-f561-4e4d-b63a-d4d60f91abde",
   "metadata": {},
   "outputs": [],
   "source": [
    "R.<ss, a1, a2, alpha1, alpha2, alpha, nu1, nu2, t, p1, p2, cd, rt> = QQ[]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f042e81b-abf8-4e09-bc67-a19d778033e6",
   "metadata": {},
   "outputs": [],
   "source": [
    "e1, e2, e3 = [v.change_ring(R) for v in (QQ^3).gens()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0733b364-bbf6-486c-a5d1-e72c7a5d48ca",
   "metadata": {},
   "outputs": [],
   "source": [
    "V1 = vector([0, 0, 3])\n",
    "V0 = vector([a1, a2, 1])\n",
    "nu = vector([nu1, nu2, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b81e1db9-883e-4a03-a38c-1ca296c9d25b",
   "metadata": {},
   "outputs": [],
   "source": [
    "Q1 = vector([2, 0, a1]) / alpha1\n",
    "Q2 = vector([0, 2, a2]) / alpha2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f97fbf75-4da5-45db-8f66-cdccdd052326",
   "metadata": {},
   "outputs": [],
   "source": [
    "def cono3(u, v, w, vert, nu0):\n",
    "    m = Matrix([u, v, w]).det()\n",
    "    den = 1\n",
    "    for u0 in (u, v, w):\n",
    "        den *= (u0 * vert) * ss + u0 * nu0\n",
    "    return m / den"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "04072945-1ada-4d66-b44e-95318f073634",
   "metadata": {},
   "outputs": [],
   "source": [
    "sb = {ss: (2 * t - 1) / 3}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "d9834ff4-e09f-4cf9-abd0-9b1eea6ca033",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{2}\\right) \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (a_{1} a_{2} t + a_{2} \\nu_{1} + a_{1} \\nu_{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{2}\\right) \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (a_{1} a_{2} t + a_{2} \\nu_{1} + a_{1} \\nu_{2})$"
      ],
      "text/plain": [
       "(1/2) * nu2^-1 * nu1^-1 * (a2*t + nu2)^-1 * (a1*t + nu1)^-1 * (a1*a2*t + a2*nu1 + a1*nu2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vertical = cono3(e1, e2, e3, V1, nu) - cono3(Q1, Q2, e3, V1, nu)\n",
    "show(vertical.subs(sb).factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "fe5aa4f3-db61-4a38-a1d1-dd100d819937",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{3}{8}\\right) \\cdot 2^{2} \\cdot (t + 1)^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{3}{8}\\right) \\cdot 2^{2} \\cdot (t + 1)^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1}$"
      ],
      "text/plain": [
       "(3/8) * 2^2 * (t + 1)^-1 * (a2*t + nu2)^-1 * (a1*t + nu1)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "horizontal = cono3(Q1, Q2, e3, V0, nu)\n",
    "horizontal = (horizontal.numerator() / 1) / (horizontal.denominator() / 1)\n",
    "show(horizontal.subs(sb).factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "a3ea1fac-c5e0-4eed-8074-482879c11b70",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{2}\\right) \\cdot (t + 1)^{-1} \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (a_{1} a_{2} t^{2} + a_{1} a_{2} t + a_{2} \\nu_{1} t + a_{1} \\nu_{2} t + a_{2} \\nu_{1} + a_{1} \\nu_{2} + 3 \\nu_{1} \\nu_{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{2}\\right) \\cdot (t + 1)^{-1} \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (a_{1} a_{2} t^{2} + a_{1} a_{2} t + a_{2} \\nu_{1} t + a_{1} \\nu_{2} t + a_{2} \\nu_{1} + a_{1} \\nu_{2} + 3 \\nu_{1} \\nu_{2})$"
      ],
      "text/plain": [
       "(1/2) * (t + 1)^-1 * nu2^-1 * nu1^-1 * (a2*t + nu2)^-1 * (a1*t + nu1)^-1 * (a1*a2*t^2 + a1*a2*t + a2*nu1*t + a1*nu2*t + a2*nu1 + a1*nu2 + 3*nu1*nu2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show((vertical + horizontal).subs(sb).factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "29605b7d-37f7-4f22-80bd-1694134cd9b7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4/(alpha1*alpha2), 2*a2/(alpha1*alpha2), (-2*a1)/(alpha1*alpha2)]"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix([Q1, Q2]).minors(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "83121ff8-547c-424b-b3ce-860a80f0541e",
   "metadata": {},
   "outputs": [],
   "source": [
    "mgamma = 2 * cd / alpha1 / alpha2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "b3963226-7397-4cb5-bdff-5382f74df8df",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3*a1/alpha1, 3*a1/alpha1, 3*a2/alpha2, 3*a2/alpha2, (a1 + 2*nu1)/alpha1)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q1 * V1, Q1 * V0, Q2 * V1, Q2 * V0, Q1 * nu"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "36201bc7-a7e2-4c9e-a8e9-5865cb9686f9",
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = ((Q1 * V1) * ss + Q1 * nu)\n",
    "f2 = ((Q2 * V1) * ss + Q2 * nu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "d77f48f5-2236-4cb6-a211-14f3623539c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot (t + 1)^{-1} \\cdot (2 t - 1) \\cdot \\mathit{cd}^{2} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot (t + 1)^{-1} \\cdot (2 t - 1) \\cdot \\mathit{cd}^{2} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1}$"
      ],
      "text/plain": [
       "(1/4) * (t + 1)^-1 * (2*t - 1) * cd^2 * (a2*t + nu2)^-1 * (a1*t + nu1)^-1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "arista = mgamma * ss / (ss + 1) / f1 / f2 * cd \n",
    "show(arista.subs(sb).factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "c18631c1-f18a-48a2-8e9a-727d347ef89a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot (t + 1)^{-1} \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (2 \\nu_{1} \\nu_{2} t \\mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \\nu_{1} t - 2 a_{1} \\nu_{2} t - 2 a_{2} \\nu_{1} - 2 a_{1} \\nu_{2} - 6 \\nu_{1} \\nu_{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot (t + 1)^{-1} \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (2 \\nu_{1} \\nu_{2} t \\mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \\nu_{1} t - 2 a_{1} \\nu_{2} t - 2 a_{2} \\nu_{1} - 2 a_{1} \\nu_{2} - 6 \\nu_{1} \\nu_{2})$"
      ],
      "text/plain": [
       "(-1/4) * (t + 1)^-1 * nu2^-1 * nu1^-1 * (a2*t + nu2)^-1 * (a1*t + nu1)^-1 * (2*nu1*nu2*t*cd^2 - 2*a1*a2*t^2 - nu1*nu2*cd^2 - 2*a1*a2*t - 2*a2*nu1*t - 2*a1*nu2*t - 2*a2*nu1 - 2*a1*nu2 - 6*nu1*nu2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "prueba = horizontal - arista + vertical\n",
    "# prueba.numerator().variables() + prueba.denominator().variables()\n",
    "show(prueba.subs(sb).factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "3dca988e-ab5a-45df-ad41-ef3f0a72dd68",
   "metadata": {},
   "outputs": [],
   "source": [
    "def compara(aa1, aa2, nnu1, nnu2):\n",
    "    R.<x, y, z> = QQ[]\n",
    "    F = z^3 + x^aa1 * y^aa2 * z\n",
    "    zt = ZetaFunctions(F).topological_zeta(local=True, weights=(nnu1, nnu2, 1))\n",
    "    S = zt.parent()\n",
    "    s = S.gen(0)\n",
    "    aalpha1 = aa1.gcd(2)\n",
    "    aalpha2 = aa2.gcd(2)\n",
    "    ccd = aalpha1.gcd(aalpha2)\n",
    "    pr = prueba(nu1=nnu1, nu2=nnu2, a1=aa1, a2=aa2, cd=ccd, ss=s)\n",
    "    return (zt, pr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "id": "073d82a2-a980-49b8-af90-6eceb62f3e8d",
   "metadata": {},
   "outputs": [],
   "source": [
    "zt, pr = compara(5, 7, 2, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "8dcbe7a7-d101-4467-816b-be27da69ec4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/18) * (s + 3/5)^-1 * (s + 13/21)^-1 * (s + 1)^-1 * (s^2 + 62/35*s + 39/35)"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "859d400a-7531-4d67-9f2b-93d504e67a4c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/18) * (s + 3/5)^-1 * (s + 13/21)^-1 * (s + 1)^-1 * (s^2 + 62/35*s + 39/35)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pr.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "17be1a78-85e6-43d1-9195-2eb8087e8014",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt - pr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "71b3a501-34ea-4088-8e1e-cbb2443d8bf3",
   "metadata": {},
   "outputs": [],
   "source": [
    "zt, pr = compara(8, 4, 2, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "4e819dc3-3ec2-4f56-beab-f1f3305810a6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/30) * (s + 1/2)^-1 * (s + 1)^-1 * (s + 7/6)^-1 * (s^2 + 3/2*s + 7/4)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "id": "5b253517-325f-46db-99ca-0ccfeba8ddd0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/30) * (s + 1/2)^-1 * (s + 1)^-1 * (s + 7/6)^-1 * (s^2 + 3/2*s + 7/4)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pr.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "36c06394-c632-47af-9763-4087cd66b85c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt - pr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "id": "b33a603c-96ba-4fe6-8db1-270536077a49",
   "metadata": {},
   "outputs": [],
   "source": [
    "zt, pr = compara(5, 0, 2, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "3a9accb9-47b2-4e8e-ad24-555933f516a5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/10) * (s + 3/5)^-1 * (s + 1)^-1 * (s + 3)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "id": "9f299f5e-36e3-46e4-a0cd-8c08d107d229",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/10) * (s + 3/5)^-1 * (s + 1)^-1 * (s + 3)"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pr.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "94a24032-ab8a-4d90-a88a-8695ca4255e1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt - pr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "96473553-7d90-4537-83f8-3a1c7e16a4df",
   "metadata": {},
   "outputs": [],
   "source": [
    "zt, pr = compara(6, 0, 2, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "id": "bdec0869-286f-46e4-a615-93779957bd70",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1/18) * (s + 5/9)^-1 * (s + 1)^-1 * (s - 5)"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "f0bdb839-c6e7-4928-bb94-dbed2ab5dca5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1/18) * (s + 5/9)^-1 * (s + 1)^-1 * (s - 5)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pr.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "bb8c2b27-5112-412f-b77c-21e6b854f8f6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt - pr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "id": "124c73b9-0e00-4b54-b361-834aae7093d3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot (t + 1)^{-1} \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (2 \\nu_{1} \\nu_{2} t \\mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \\nu_{1} t - 2 a_{1} \\nu_{2} t - 2 a_{2} \\nu_{1} - 2 a_{1} \\nu_{2} - 6 \\nu_{1} \\nu_{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot (t + 1)^{-1} \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{2} t + \\nu_{2})^{-1} \\cdot (a_{1} t + \\nu_{1})^{-1} \\cdot (2 \\nu_{1} \\nu_{2} t \\mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \\nu_{1} t - 2 a_{1} \\nu_{2} t - 2 a_{2} \\nu_{1} - 2 a_{1} \\nu_{2} - 6 \\nu_{1} \\nu_{2})$"
      ],
      "text/plain": [
       "(-1/4) * (t + 1)^-1 * nu2^-1 * nu1^-1 * (a2*t + nu2)^-1 * (a1*t + nu1)^-1 * (2*nu1*nu2*t*cd^2 - 2*a1*a2*t^2 - nu1*nu2*cd^2 - 2*a1*a2*t - 2*a2*nu1*t - 2*a1*nu2*t - 2*a2*nu1 - 2*a1*nu2 - 6*nu1*nu2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "prt = prueba.subs(sb)\n",
    "show(prt.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "655a992c-c841-4ebd-b641-fbfa107b1a2f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (2 \\nu_{1} \\nu_{2} t \\mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \\nu_{1} t - 2 a_{1} \\nu_{2} t - 2 a_{2} \\nu_{1} - 2 a_{1} \\nu_{2} - 6 \\nu_{1} \\nu_{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (2 \\nu_{1} \\nu_{2} t \\mathit{cd}^{2} - 2 a_{1} a_{2} t^{2} - \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 2 a_{1} a_{2} t - 2 a_{2} \\nu_{1} t - 2 a_{1} \\nu_{2} t - 2 a_{2} \\nu_{1} - 2 a_{1} \\nu_{2} - 6 \\nu_{1} \\nu_{2})$"
      ],
      "text/plain": [
       "(-1/4) * nu2^-1 * nu1^-1 * (2*nu1*nu2*t*cd^2 - 2*a1*a2*t^2 - nu1*nu2*cd^2 - 2*a1*a2*t - 2*a2*nu1*t - 2*a1*nu2*t - 2*a2*nu1 - 2*a1*nu2 - 6*nu1*nu2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "prt1 = prt * (a2 * t + nu2) * (a1 * t + nu1) * (t + 1)\n",
    "show(prt1.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "348ed4c3-3e5b-437d-b2ce-80cf913b2ab0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{3}{4}\\right) \\cdot (a_{2} - \\nu_{2})^{-1} \\cdot (-a_{1} + \\nu_{1})^{-1} \\cdot (\\mathit{cd}^{2} + 2)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{3}{4}\\right) \\cdot (a_{2} - \\nu_{2})^{-1} \\cdot (-a_{1} + \\nu_{1})^{-1} \\cdot (\\mathit{cd}^{2} + 2)$"
      ],
      "text/plain": [
       "(-3/4) * (a2 - nu2)^-1 * (-a1 + nu1)^-1 * (cd^2 + 2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cf23 = (prt1 / (a2 * t + nu2) / (a1 * t + nu1))(t=-1)\n",
    "show(cf23.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "33c604e3-cef3-4785-a4ba-5481905f7662",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-1} \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot a_{1} \\cdot (a_{1} + 2 \\nu_{1}) \\cdot (-a_{2} \\nu_{1} + a_{1} \\nu_{2})^{-1} \\cdot (\\nu_{1} \\mathit{cd}^{2} + 2 a_{1})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-1} \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot a_{1} \\cdot (a_{1} + 2 \\nu_{1}) \\cdot (-a_{2} \\nu_{1} + a_{1} \\nu_{2})^{-1} \\cdot (\\nu_{1} \\mathit{cd}^{2} + 2 a_{1})$"
      ],
      "text/plain": [
       "(1/4) * nu1^-1 * (a1 - nu1)^-1 * a1 * (a1 + 2*nu1) * (-a2*nu1 + a1*nu2)^-1 * (nu1*cd^2 + 2*a1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cf1 = (prt1 / (t + 1) / (a2 * t + nu2))(t=-nu1 / a1)\n",
    "show(cf1.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "eefdd19d-2b7e-4037-b448-70b360f99324",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\nu_{2}^{-1} \\cdot (a_{2} - \\nu_{2})^{-1} \\cdot a_{2} \\cdot (a_{2} + 2 \\nu_{2}) \\cdot (a_{2} \\nu_{1} - a_{1} \\nu_{2})^{-1} \\cdot (\\nu_{2} \\mathit{cd}^{2} + 2 a_{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\nu_{2}^{-1} \\cdot (a_{2} - \\nu_{2})^{-1} \\cdot a_{2} \\cdot (a_{2} + 2 \\nu_{2}) \\cdot (a_{2} \\nu_{1} - a_{1} \\nu_{2})^{-1} \\cdot (\\nu_{2} \\mathit{cd}^{2} + 2 a_{2})$"
      ],
      "text/plain": [
       "(1/4) * nu2^-1 * (a2 - nu2)^-1 * a2 * (a2 + 2*nu2) * (a2*nu1 - a1*nu2)^-1 * (nu2*cd^2 + 2*a2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cf2 = (prt1 / (t + 1) / (a1 * t + nu1))(t=-nu2 / a2)\n",
    "show(cf2.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "b9b27e01-26a3-4ce1-8c45-b1cef61e241e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cf23 / (t + 1) + cf1 / (a1 * t + nu1) + cf2 / (a2 * t + nu2) - prt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "32c69f2c-559d-444d-9ba2-28ebe87860b7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cf23(a2=0, nu2=1) / (t + 1) + cf1(a2=0, nu2=1) / (a1 * t + nu1) - prt(a2=0, nu2=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "3a8f4455-9759-4266-80fb-72bbc9eb5f9d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{3}{4}\\right) \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot (\\mathit{cd}^{2} + 2)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{3}{4}\\right) \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot (\\mathit{cd}^{2} + 2)$"
      ],
      "text/plain": [
       "(-3/4) * (a1 - nu1)^-1 * (cd^2 + 2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(cf23(a2=0, nu2=1).factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "5f833a09-ae0c-468e-9c48-f7de392135b5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-1} \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot (a_{1} + 2 \\nu_{1}) \\cdot (\\nu_{1} \\mathit{cd}^{2} + 2 a_{1})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-1} \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot (a_{1} + 2 \\nu_{1}) \\cdot (\\nu_{1} \\mathit{cd}^{2} + 2 a_{1})$"
      ],
      "text/plain": [
       "(1/4) * nu1^-1 * (a1 - nu1)^-1 * (a1 + 2*nu1) * (nu1*cd^2 + 2*a1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(cf1(a2=0, nu2=1).factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "id": "fa15ead6-04ac-47dc-be90-56c6ed7c24db",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-2} \\cdot \\mathit{rt}^{-1} \\cdot (t + 1)^{-1} \\cdot (a_{1} t + \\nu_{1})^{-2} \\cdot (2 \\nu_{1}^{2} t \\mathit{cd}^{2} - 2 a_{1}^{2} t^{2} - \\nu_{1}^{2} \\mathit{cd}^{2} - 2 a_{1}^{2} t - 4 a_{1} \\nu_{1} t - 4 a_{1} \\nu_{1} - 6 \\nu_{1}^{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-2} \\cdot \\mathit{rt}^{-1} \\cdot (t + 1)^{-1} \\cdot (a_{1} t + \\nu_{1})^{-2} \\cdot (2 \\nu_{1}^{2} t \\mathit{cd}^{2} - 2 a_{1}^{2} t^{2} - \\nu_{1}^{2} \\mathit{cd}^{2} - 2 a_{1}^{2} t - 4 a_{1} \\nu_{1} t - 4 a_{1} \\nu_{1} - 6 \\nu_{1}^{2})$"
      ],
      "text/plain": [
       "(-1/4) * nu1^-2 * rt^-1 * (t + 1)^-1 * (a1*t + nu1)^-2 * (2*nu1^2*t*cd^2 - 2*a1^2*t^2 - nu1^2*cd^2 - 2*a1^2*t - 4*a1*nu1*t - 4*a1*nu1 - 6*nu1^2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "prta = prt(a2=rt * a1, nu2=rt * nu1)\n",
    "show(prta.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "id": "def3aa24-7451-4a55-8f00-dfe19eb69ca8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{3}{4}\\right) \\cdot (a_{1} - \\nu_{1})^{-2} \\cdot \\mathit{rt}^{-1} \\cdot (\\mathit{cd}^{2} + 2)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{3}{4}\\right) \\cdot (a_{1} - \\nu_{1})^{-2} \\cdot \\mathit{rt}^{-1} \\cdot (\\mathit{cd}^{2} + 2)$"
      ],
      "text/plain": [
       "(3/4) * (a1 - nu1)^-2 * rt^-1 * (cd^2 + 2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cf23a = cf23(a2=rt * a1, nu2=rt * nu1)\n",
    "show(cf23a.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "id": "522409a6-3cd4-44d5-9647-b78bcbe86007",
   "metadata": {},
   "outputs": [],
   "source": [
    "prtb = prta - cf23a / (t + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "id": "e17948dd-26fb-48e0-9740-0bc20c584dbb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\mathit{rt}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot (a_{1} + 2 \\nu_{1}) \\cdot (\\nu_{1} \\mathit{cd}^{2} + 2 a_{1})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{4}\\right) \\cdot \\mathit{rt}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (a_{1} - \\nu_{1})^{-1} \\cdot (a_{1} + 2 \\nu_{1}) \\cdot (\\nu_{1} \\mathit{cd}^{2} + 2 a_{1})$"
      ],
      "text/plain": [
       "(1/4) * rt^-1 * nu1^-1 * (a1 - nu1)^-1 * (a1 + 2*nu1) * (nu1*cd^2 + 2*a1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-2} \\cdot (a_{1} - \\nu_{1})^{-2} \\cdot \\mathit{rt}^{-1} \\cdot a_{1} \\cdot (3 \\nu_{1}^{2} \\mathit{cd}^{2} - 2 a_{1}^{2} + 4 a_{1} \\nu_{1} + 4 \\nu_{1}^{2})\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-\\frac{1}{4}\\right) \\cdot \\nu_{1}^{-2} \\cdot (a_{1} - \\nu_{1})^{-2} \\cdot \\mathit{rt}^{-1} \\cdot a_{1} \\cdot (3 \\nu_{1}^{2} \\mathit{cd}^{2} - 2 a_{1}^{2} + 4 a_{1} \\nu_{1} + 4 \\nu_{1}^{2})$"
      ],
      "text/plain": [
       "(-1/4) * nu1^-2 * (a1 - nu1)^-2 * rt^-1 * a1 * (3*nu1^2*cd^2 - 2*a1^2 + 4*a1*nu1 + 4*nu1^2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cf1a = (prtb * (a1 * t + nu1)^2)(t=-nu1 / a1)\n",
    "show(cf1a.factor())\n",
    "prtc = prtb - cf1a / (a1 * t + nu1)^2\n",
    "cf2a = prtc * (a1 * t + nu1)\n",
    "show(cf2a.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "id": "5c6d7d36-ca2d-41d1-8272-80908564cbbc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cf23a / (t + 1) + cf1a / (a1 * t + nu1)^2 + cf2a / (a1 * t + nu1) - prta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "632d9fd5-b8a2-4b48-be93-54a23f3418a7",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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