{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e881bcba-75d3-415b-bfe0-5e0f50a49d1d",
   "metadata": {},
   "outputs": [],
   "source": [
    "S.<m3, nu1, g3, s> = PolynomialRing(QQ, order='neglex')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3884a71-e7bb-4a29-95c1-141e364eec85",
   "metadata": {},
   "source": [
    "Primer caso: $f=z^3 - x^{m_3}$, $\\omega=x^{\\nu_1-1} d x d z$ y $g_3=\\gcd(3,m_3)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "f12e8928-4d29-4295-add0-c61c40785580",
   "metadata": {},
   "outputs": [],
   "source": [
    "tau1 = vector([m3, 0])\n",
    "tau0 = vector(S, [0, 3])\n",
    "V1 = vector(S, [0, 1])\n",
    "V0 = vector(S, [1, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "bbfe6b4f-5d07-43de-8fea-0d8f62ef60f3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((3*nu1 + m3 + 3*m3*s)/g3, 1, nu1)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vgamma = vector([3, m3]) / g3\n",
    "omega1 = vector([nu1, 1])\n",
    "mu1 = 3 / g3\n",
    "Pgamma = (tau0 * s + omega1) * Vgamma\n",
    "volgamma = g3\n",
    "P1 = (tau1 * s + omega1) * V1\n",
    "P0 = (tau0 * s + omega1) * V0\n",
    "Pgamma, P1, P0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "2701f743-4d04-4b66-b8f1-ae978bdf5ac7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3/g3, m3/g3)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu1 = Matrix([Vgamma, V1]).det()\n",
    "mu0 = Matrix([V0, Vgamma]).det()\n",
    "mu1, mu0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "40a5decf-c4ec-40ad-be1c-a452c8ecf178",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "g3/(3*nu1 + m3 + 3*m3*s)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Jtau1 = mu1 / P1 / Pgamma\n",
    "Jtau0 = mu0 / P0 / Pgamma\n",
    "Jgamma = 1 / Pgamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "a4754c02-1b72-4725-85b1-ab158979cd32",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1) * nu1^-1 * (1 + s)^-1 * (3*nu1 + m3 + 3*m3*s)^-1 * (-3*nu1 - 3*nu1*s + nu1*g3^2*s - m3 - m3*s)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zeta = Jtau0 + Jtau1 - s / (s + 1) * Jgamma * volgamma\n",
    "Zeta.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "34d5f080-c0d6-46a9-a665-4d89fc659b59",
   "metadata": {},
   "outputs": [],
   "source": [
    "R1 = (Zeta * (s + 1))(s=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "d4083af7-c4f6-4b70-b303-91598f08029a",
   "metadata": {},
   "outputs": [],
   "source": [
    "Rp = (Zeta * Pgamma)(s=-Pgamma(s=0) / Pgamma.derivative(s))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "680d1454-1098-4481-90ef-a849cf3ee595",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zeta - R1 / (s + 1) - Rp / Pgamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "6b03816c-0216-43b0-b7d0-e25bffde0f27",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1/6) * (1 + s)^-2 * m3^-1 * (-9 - 9*s + 2*g3^2*s)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zeta2 = Zeta(nu1=2 * m3 / 3)\n",
    "Zeta2.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "deee339d-836e-4572-94aa-a16b3f95bd0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1/6) * m3^-1 * (-9 + 2*g3^2)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R2a = (Zeta2 * (s + 1)^2)(s=-1)\n",
    "R1a = ((s + 1)^2 * Zeta2 - R2a) / (s + 1)\n",
    "R1a.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "4c078f46-552b-40d9-a9bf-a133ba42e8be",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zeta2 - R1a / (s + 1) - R2a / (s + 1)^2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "883017aa-31fa-45ea-9f47-90ced9eaacf1",
   "metadata": {},
   "source": [
    "Segundo caso: $f=z^3 + x^{m_2}z + x^{m_3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "c9ebb2ca-6ea3-4492-84b4-7057e22184cb",
   "metadata": {},
   "outputs": [],
   "source": [
    "S.<m2, nu1, g2, s> = PolynomialRing(QQ, order='neglex')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "id": "4fd45644-e41a-4b74-9f38-75ae1120e59b",
   "metadata": {},
   "outputs": [],
   "source": [
    "tau1 = vector([m2, 1])\n",
    "tau0 = vector(S, [0, 3])\n",
    "V1 = vector(S, [0, 1])\n",
    "V0 = vector(S, [1, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "13a43344-f27a-425b-8ce6-eb9d452651d0",
   "metadata": {},
   "outputs": [],
   "source": [
    "Vgamma = vector([2, m2]) / g2\n",
    "omega1 = vector([nu1, 1])\n",
    "mu1 = Matrix([Vgamma, V1]).det()\n",
    "mu0 = Matrix([V0, Vgamma]).det()\n",
    "Pgamma = (tau0 * s + omega1) * Vgamma\n",
    "volgamma = g2\n",
    "P1 = (tau1 * s + omega1) * V1\n",
    "P0 = (tau0 * s + omega1) * V0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "04b82282-4fd4-4e7f-9982-c59c380c0a61",
   "metadata": {},
   "outputs": [],
   "source": [
    "Jtau1 = mu1 / P1 / Pgamma\n",
    "Jtau0 = mu0 / P0 / Pgamma\n",
    "Jgamma = 1 / Pgamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "id": "cb8f8ee3-6f40-423d-80bc-f3985c0dc70e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "g2 * (2*nu1 + m2 + 3*m2*s)^-1"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Jgamma.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "id": "0ef91b6f-62c6-4b39-9651-1e5f60729893",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(-1\\right) \\cdot \\nu_{1}^{-1} \\cdot (1 + s)^{-1} \\cdot (2 \\nu_{1} + m_{2} + 3 m_{2} s)^{-1} \\cdot (-2 \\nu_{1} + \\nu_{1} g_{2}^{2} s - m_{2} - m_{2} s)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(-1\\right) \\cdot \\nu_{1}^{-1} \\cdot (1 + s)^{-1} \\cdot (2 \\nu_{1} + m_{2} + 3 m_{2} s)^{-1} \\cdot (-2 \\nu_{1} + \\nu_{1} g_{2}^{2} s - m_{2} - m_{2} s)$"
      ],
      "text/plain": [
       "(-1) * nu1^-1 * (1 + s)^-1 * (2*nu1 + m2 + 3*m2*s)^-1 * (-2*nu1 + nu1*g2^2*s - m2 - m2*s)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Zeta = Jtau0 + Jtau1 - s / (s + 1) * Jgamma * volgamma\n",
    "show(Zeta.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "id": "0719a53b-fa90-42b1-ac78-418acb0fd0c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/2) * (-nu1 + m2)^-1 * nu1^-1 * g2^-1 * (2*nu1 + m2) * (nu1*g2^2 + 2*m2)"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R1 = (Zeta * (s + 1))(s=-1)\n",
    "Rp = (Zeta * Pgamma)(s=-Pgamma(s=0) / Pgamma.derivative(s))\n",
    "Rp.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "id": "37c88cb0-c2e1-4c18-ada0-a593435bd250",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zeta - R1 / (s + 1) - Rp / Pgamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "id": "54e9c4b8-a3f7-45d8-b48e-de8234f4d829",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2*nu1 - 2*m2)/g2"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pgamma(s=-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "id": "66efc352-66e3-4c39-8dd0-17c85c37ccf1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1/3) * (1 + s)^-2 * m2^-1 * (-3 - s + g2^2*s)"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zeta2 = Zeta(nu1=m2)\n",
    "Zeta2.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "8ac89af7-786c-4504-923d-938ab424400b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((-1/3) * m2^-1 * (-1 + g2) * (1 + g2), (1/3) * m2^-1 * (2 + g2^2))"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R2a = (Zeta2 * (s + 1)^2)(s=-1)\n",
    "R1a = ((s + 1)^2 * Zeta2 - R2a) / (s + 1)\n",
    "R1a.factor(), R2a.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "id": "afd51c00-c529-416a-98c3-13c6cfaf3760",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zeta2 - R1a / (s + 1) - R2a / (s + 1)^2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8301561c-7e6a-4fd2-8bfb-d8690c1f9700",
   "metadata": {},
   "source": [
    "Tercer caso: $f=z^3 + x^{m_1}z^2 + x^{m_3}$, $m_3 > 3 m_1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "id": "1a73caca-514d-4133-9345-ccb80b276d00",
   "metadata": {},
   "outputs": [],
   "source": [
    "S.<m1, m3, nu1, g2, s> = PolynomialRing(QQ, order='neglex')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "9d30e804-9dec-4e73-8f8b-4927873a8445",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2*c - c*d^2*s + a + a*s)/(2*c^2 + 2*c^2*s + a*c + 4*a*c*s + 3*a*c*s^2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tau2 - Gamma2*s / (s+1) + Tau1 - d*Gamma1*s / (s+1) + Tau0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "id": "ff7811d0-081a-4cd9-9a18-83d39c6d0e4b",
   "metadata": {},
   "outputs": [],
   "source": [
    "tau2 = vector([m3, 0])\n",
    "tau1 = vector([m1, 2])\n",
    "tau0 = vector(S, [0, 3])\n",
    "V2 = vector(S, [0, 1])\n",
    "V0 = vector(S, [1, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e8199a7d-0205-4e39-8c56-c378dc664164",
   "metadata": {},
   "outputs": [],
   "source": [
    "Vgamma1 = vector([1, m1]) \n",
    "Vgamma2 = vector([2, m3 - m1]) / g2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4fc1a581-aa5e-425a-99f4-f43dd992aaee",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
   "display_name": "SageMath 10.5.rc0",
   "language": "sage",
   "name": "sagemath-devel"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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}