{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "60263160-66d3-45f7-a6cd-0042e5c5d37e",
   "metadata": {},
   "outputs": [],
   "source": [
    "R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')\n",
    "R2.<u, v> = PolynomialRing(QQ, order='negdeglex')\n",
    "R1.<s> = QQ[]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "3e206069-21d2-4ac7-80af-0f997622ce0e",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = u^5 + v^6\n",
    "F = z^10 + f(u=x, v=y)\n",
    "Z = ZetaFunctions(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "38a0780c-be08-4a0a-a310-55b279c5528f",
   "metadata": {},
   "outputs": [],
   "source": [
    "zt = Z.topological_zeta(local=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "e1e6e6cb-1720-4de5-8a9f-751891aa189e",
   "metadata": {},
   "outputs": [],
   "source": [
    "g0 = v^10 - u^30\n",
    "Z0 = ZetaFunctions(g0)\n",
    "zt0 = Z0.topological_zeta(local=True, weights=[11, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "9d2955da-b2cb-4787-b71b-1a435ca6ae5d",
   "metadata": {},
   "outputs": [],
   "source": [
    "g1 = v^10 - u^6\n",
    "Z1 = ZetaFunctions(g1)\n",
    "zt1 = Z1.topological_zeta(local=True, weights=[11 / 5, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "9dce165f-4968-45b0-82bd-9da8322439a9",
   "metadata": {},
   "outputs": [],
   "source": [
    "g2 = v^10 - u^5\n",
    "Z2 = ZetaFunctions(g2)\n",
    "zt2 = Z2.topological_zeta(local=True, weights=[11 / 6, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "69595c54-4c36-47d8-8b69-f3cc40b345b6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/110) * (s + 7/15)^-1 * (s + 1)^-1 * (s + 14/3)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g3 = z^10 - x^30 * y\n",
    "Z3 = ZetaFunctions(g3)\n",
    "zt3 = Z3.topological_zeta(local=True, weights=[11, 1, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "c7e0fa8e-f750-4945-aa62-6d7a82fed340",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt - (-zt0 + zt1 + zt2 + zt3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "042d2c61-0ddf-4b13-b1ab-ade1de1c430e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1/6) * (s + 7/15)^-1"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt10 = Z.topological_zeta(local=True, d=10)\n",
    "zt10.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "1bcb4435-1f60-49a0-b26a-14bd6750bb9b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/5/(s + 7/15)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt5 = Z.topological_zeta(local=True, d=5) - zt10\n",
    "zt5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "52bb0b78-f7a7-4717-9c11-25a4cec9de02",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/6/(s + 7/15)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zt2 = Z.topological_zeta(local=True, d=2) - zt10\n",
    "zt2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "e613a530-bfa1-4fa1-9c7c-c52e12ccc1cc",
   "metadata": {},
   "outputs": [],
   "source": [
    "Zf = ZetaFunctions(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "67800a77-a153-4e9d-9320-75d6db28cbbd",
   "metadata": {},
   "outputs": [],
   "source": [
    "ztf = Zf.topological_zeta(local=True)\n",
    "ztf10 = Zf.topological_zeta(local=True, d=10)\n",
    "ztf5 = Zf.topological_zeta(local=True, d=5) - ztf10\n",
    "ztf2 = Zf.topological_zeta(local=True, d=2) - ztf10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "c9ba3ae1-9f6f-440d-961e-a16b3815287e",
   "metadata": {},
   "outputs": [],
   "source": [
    "p = s + 1 / 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "faed1d1e-b23e-4da7-9fed-dd470f6fd1d9",
   "metadata": {},
   "outputs": [],
   "source": [
    "h = {}\n",
    "h[1, 0] = 1 / 10 / p\n",
    "h[1, 1] = 9 / 10 * s / (s + 1) * (p + 1) / p\n",
    "h[1, 2] = 3 / 10 * s / (s + 1)\n",
    "h[1, 5] = 24 / 10 * s / (s + 1)\n",
    "h[1, 10] = 99 / 10 * s / (s + 1) \n",
    "aux =  h[1, 0] +  h[1, 1] * ztf(s=p) -  h[1, 2] * ztf2(s=p) -  h[1, 5] * ztf5(s=p) -  h[1, 10] * ztf10(s=p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "2c8cd3e8-bc4a-417d-99c6-6903442387ba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "aux - zt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "39ffdc08-85a3-48db-88f6-edb2b6dcd5c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "h[2, 0] = 1 / 10 / p\n",
    "h[2, 1] = 1 / 10 * (p + 1) / p\n",
    "h[2, 2] = 7 / 10\n",
    "h[2, 5] = -14 / 10\n",
    "h[2, 10] = -89 / 10\n",
    "aux =  h[2, 0] +  h[2, 1] * ztf(s=p) +  h[2, 2] * ztf2(s=p) +  h[2, 5] * ztf5(s=p) +  h[2, 10] * ztf10(s=p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "52a2a318-3955-4fad-b5ea-83244a0282d0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/10) * (s + 1/10)^-1 * (s + 7/15)^-1 * (s + 5/6)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(aux - zt2).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3a775a53-ee34-4241-acca-7e768f3cf51d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "SageMath 10.4.beta3",
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   "name": "sagemath-310"
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