{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "050758e0-be1e-497b-b19b-2daf03d253f2",
   "metadata": {},
   "outputs": [],
   "source": [
    "load('zetas_triples_general.py')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "7d68f9ce-8bfb-4a97-9cff-c79fa4295bcd",
   "metadata": {},
   "outputs": [],
   "source": [
    "R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')\n",
    "R2.<u, v> = PolynomialRing(QQ, order='negdeglex')\n",
    "def surface(f, g):\n",
    "    sb = {u: x, v: y}\n",
    "    return z^3 - 3 * f.subs(sb) * z + 2 * g.subs(sb)\n",
    "\n",
    "def discrim3(f, g):\n",
    "    return f^3 - g^2\n",
    "\n",
    "def milnor(F):\n",
    "    return F.jacobian_ideal().vector_space_dimension()\n",
    "\n",
    "def factoresNewton(f):\n",
    "    return [p.factor() for p in ZetaFunctions(f).get_polyfaces_dictionary(compact=True)\n",
    "            if p.number_of_terms() > 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "a13a69d1-427f-4cba-984f-04cfd3e3f733",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# f = ((v^2 + u^3)^2 + u^5*v) * ((u^2 + v^3)^2 + v^5*u)\n",
    "f = (v^6 + u^7) * (u^6 + v^7)\n",
    "# g = (v^4 + u^6) * (u^4 + v^6)\n",
    "g = ((v^2 + u^3)^2 + u^5*v) * ((u^2 + v^3)^2 + v^5*u)\n",
    "hs = []\n",
    "ks = [((u^2 + v^3) * (v^2 + u^3), 4)]\n",
    "ff = f * prod(p^j for p, j in hs) * prod(p for p, j in ks)\n",
    "gg = g * prod(p^j for p, j in ks) * prod(p for p, j in hs)\n",
    "d = R2(discrim3(ff, gg) / prod(p^2 for p, j in hs) /  prod(p^3 for p, j in ks))\n",
    "[(f, j) for f, j in d.factor() if j > 1]#, milnor(surface(f, g))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "6bb64d4a-4731-4902-8408-61e08a19bc5f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[u^25 * (3*v^12 + 3*u^7*v^6 + u^14),\n",
       " v^25 * (3*u^12 + 3*u^6*v^7 + v^14),\n",
       " (3) * (u + v) * v^12 * u^12 * (u^12 - u^11*v + u^10*v^2 - u^9*v^3 - 2*u^8*v^4 + 2*u^7*v^5 - 2*u^6*v^6 + 2*u^5*v^7 - 2*u^4*v^8 - u^3*v^9 + u^2*v^10 - u*v^11 + v^12)]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factoresNewton(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "c25f0826-7206-4767-883c-8e9c164a18c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# f = ((v^2 + u^3)^2 + u^5 * v) * ((u^2 + v^3)^2 + v^5 * u)\n",
    "f = (v^4 + u^5) * (v^5 + u^4)\n",
    "g = (v^6 + u^8) * (u^6 + v^8)\n",
    "# g = ((v^3 + u^5)^2 + u^2 * v^5) * ((u^3 + v^5)^2 + v^2 * u^5)\n",
    "hs = []\n",
    "ks = []\n",
    "ff = f * prod(p^j for p, j in hs) * prod(p for p, j in ks)\n",
    "gg = g * prod(p^j for p, j in ks) * prod(p for p, j in hs)\n",
    "d = R2(discrim3(ff, gg) / prod(p^2 for p, j in hs) /  prod(p^3 for p, j in ks))\n",
    "[(f, j) for f, j in d.factor() if j > 1]#, milnor(surface(f, g))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "949563f9-dd7e-4db3-9101-a4f3a2003481",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1) * v^19 * (2*u^6 + v^7),\n",
       " (-1) * u^19 * (2*v^6 + u^7),\n",
       " (-1) * (u + v) * v^6 * u^6 * (2*u^12 - 2*u^11*v - u^10*v^2 + u^9*v^3 - u^8*v^4 + u^7*v^5 - u^6*v^6 + u^5*v^7 - u^4*v^8 + u^3*v^9 - u^2*v^10 - 2*u*v^11 + 2*v^12)]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factoresNewton(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "66f80ac9-aef5-4ea7-8770-335fbdce37d0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1) * (49*v^2 + u^13)]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1 = R2(d(u=u^7, v=u^6 * (v - 1))/ u^156)\n",
    "factoresNewton(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "id": "1acb3dcb-8044-4672-8c44-2ec5c4173fa0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((-1) * (-u^4*v^3 + 3*u^6*v^2 + 3*u^3*v^5 - 3*u^8*v - 6*u^5*v^4 + u^4*v^5 - 3*u^2*v^7 + u^10 + u^7*v^3 + 3*u^4*v^6 - 2*u^2*v^8 + u*v^9 + u^10*v + 2*u^5*v^6 + v^11),)"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = u * v - u^3 - v^3\n",
    "g = u^2 * v^2 - u^5 - v^5\n",
    "hs = [(u, 1)]\n",
    "ks = [(v, 2)]\n",
    "ff = f * prod(p^j for p, j in hs) * prod(p for p, j in ks)\n",
    "gg = g * prod(p^j for p, j in ks) * prod(p for p, j in hs)\n",
    "d = R2(discrim3(ff, gg) / prod(p^2 for p, j in hs) /  prod(p^3 for p, j in ks))\n",
    "d.factor(), # milnor(surface(ff, gg))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "id": "f9fb1d7b-9e33-46ce-bf7b-776c7d04f6fa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1) * v^3 * (-u^4 + 3*u^3*v^2 - 3*u^2*v^4 + u*v^6 + v^8), u^4 * (v - u^2)^3]"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factoresNewton(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "id": "b32d46f1-1e18-403e-b813-e2221188c9ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1) * (-1 + 4*u) * (-1 + u)^2"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-u^4 + 3*u^3*v^2 - 3*u^2*v^4 + u*v^6 + v^8)(v=1).derivative(u).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "id": "51b8b296-2523-47c5-a5bb-b1f559d897f6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 143,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-u^4 + 3*u^3*v^2 - 3*u^2*v^4 + u*v^6 + v^8)(v=1).derivative(u)(u=1 / 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "id": "51298dd8-ab45-4244-a5d4-30cbaaabd783",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1 * 7^6 * 31^-4"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(ff(u=1 / 4, v=1)^3/gg(u=1 / 4, v=1)^2).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "id": "7fc67ac3-d76e-4c24-98b5-6c0d6057e37a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1/6912) * v^3 * (-4*u + v^2)^2 * (16*u^2 - 40*u*v^2 + 27*v^4),\n",
       " (-1/27) * u^4 * (v - u^2)^3]"
      ]
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dd = R2(discrim3(-ff / 3, gg / 16) / prod(p^2 for p, j in hs) /  prod(p^3 for p, j in ks))\n",
    "factoresNewton(dd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "id": "788a5bda-775a-4001-a390-4adfecd1450f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1/6144) * v^7 * (-256*u^2 + 3*v^5),\n",
       " (-1/27) * u^4 * (v - u^2)^3,\n",
       " (-1/216) * u^2 * v^3 * (8*u^2 - 16*u*v^2 + 9*v^4)]"
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factoresNewton(dd(u=u + v^2 / 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "id": "1acc71df-f964-4acd-ba7c-11f98c3c4f02",
   "metadata": {},
   "outputs": [],
   "source": [
    "d1 = R2(d(v=v^2, u=v^3 * (u + 1)) / v^22)\n",
    "g1 = R2(g(v=v^2, u=v^3 * (u + 1)) / v^10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "id": "7be2ed6b-2bdd-418a-8b14-6f6baf147c27",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-4*u^2 + v^5]"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factoresNewton(d1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "id": "2546d6ca-f3d1-4425-b6b8-54205f2af7c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1) * (-2*u + v^5)]"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factoresNewton(g1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "id": "ef51632f-a8a0-4655-b471-76a4b975e9d7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrr}\n",
       "-5 & 0 & 1 & 0 & 1 \\\\\n",
       "0 & -2 & 1 & 0 & 0 \\\\\n",
       "1 & 1 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & -2 & 1 \\\\\n",
       "1 & 0 & 0 & 1 & -1\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rrrrr}\n",
       "-5 & 0 & 1 & 0 & 1 \\\\\n",
       "0 & -2 & 1 & 0 & 0 \\\\\n",
       "1 & 1 & -1 & 0 & 0 \\\\\n",
       "0 & 0 & 0 & -2 & 1 \\\\\n",
       "1 & 0 & 0 & 1 & -1\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[-5  0  1  0  1]\n",
       "[ 0 -2  1  0  0]\n",
       "[ 1  1 -1  0  0]\n",
       "[ 0  0  0 -2  1]\n",
       "[ 1  0  0  1 -1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = Matrix(QQ, 5)\n",
    "for i, j in ((0, 2), (0, 4), (1, 2), (3, 4)):\n",
    "    A[i, j] = A[j, i] = 1\n",
    "A[0, 0] = -5\n",
    "A[1, 1] = A[3, 3] = -2\n",
    "A[2, 2] = A[4, 4] = -1\n",
    "show(A)\n",
    "Vf = vector(QQ, [0, 1, 0, 1, 0])\n",
    "Vg = vector(QQ, [0, 0, 1, 0, 1])\n",
    "Vd = vector(QQ, [5, 0, 1, 0, 1])\n",
    "Vhs = ((vector(QQ, [0, 1, 0, 1, 0]), 1),)\n",
    "Vks = ()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "id": "30f4affd-a57c-4c4a-b6a4-dc34bb42245e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5,\n",
       " (2, 3, 5, 3, 5),\n",
       " (4, 5, 10, 5, 10),\n",
       " (9, 10, 20, 10, 20),\n",
       " [((1, 2, 3, 1, 2), 1), ((1, 1, 2, 2, 3), 1)],\n",
       " [],\n",
       " (-5, -1, -2, -1, -2),\n",
       " (2, 3, 5, 3, 5),\n",
       " False)"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n, Nf, Ng, Nd, Nhs, Nks, euler, nus, pols == trinomiales_general(A, Vf, Vg, Vhs, Vks, Vd, total=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "id": "9947139a-5e98-478e-b649-971f96b3a5d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4, 6, 10, 6, 10)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nf + sum(j * V0 for V0, j in Nhs) + sum(V0 for V0, j in Nks)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "id": "5e717f0e-d183-4791-b92d-eb709b7b7452",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(6, 8, 15, 8, 15)"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ng + sum(j * V0 for V0, j in Nks) + sum(V0 for V0, j in Nhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "b54c5c8b-04e8-4c45-b82b-caeb3d73252e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(13, 16, 30, 16, 30)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nd + 3 * sum(V0 for V0, j in Nks) + 2 * sum(V0 for V0, j in Nhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "a1544358-9d8c-411a-b30c-18630767a9c1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "(z^3 + x^2*z^2 + x^7, (2, 1, 1), -5)\n",
      "===\n",
      "1\n",
      "(z^3 + x^6*z + x^8, (3, 1, 1), -1)\n",
      "===\n",
      "2\n",
      "(z^3 + x^10*z + x^15, (5, 1, 1), -2)\n",
      "===\n",
      "3\n",
      "(z^3 + x^6*z + x^8, (3, 1, 1), -1)\n",
      "===\n",
      "4\n",
      "(z^3 + x^10*z + x^15, (5, 1, 1), -2)\n",
      "===\n",
      "(0, 2)\n",
      "(z^3 + x^2*y^5*z^2 + x^7*y^15, (2, 5, 1), 1)\n",
      "===\n",
      "(0, 4)\n",
      "(z^3 + x^2*y^5*z^2 + x^7*y^15, (2, 5, 1), 1)\n",
      "===\n",
      "(1, 2)\n",
      "(z^3 + x^6*y^10*z + x^8*y^15, (3, 5, 1), 1)\n",
      "===\n",
      "(3, 4)\n",
      "(z^3 + x^6*y^10*z + x^8*y^15, (3, 5, 1), 1)\n",
      "===\n",
      "(0, 'd')\n",
      "(z^3 + x^2*z^2 + x^7*y, (2, 1, 1), 5)\n",
      "===\n",
      "(1, 'f')\n",
      "(z^3 + x^8 + x^6*y*z, (3, 1, 1), 1)\n",
      "===\n",
      "(2, 'g')\n",
      "(z^3 + x^10*z + x^15*y, (5, 1, 1), 1)\n",
      "===\n",
      "(2, 'd')\n",
      "(z^3 + x^5*z^2 + x^15*y, (5, 1, 1), 1)\n",
      "===\n",
      "(3, 'f')\n",
      "(z^3 + x^8 + x^6*y*z, (3, 1, 1), 1)\n",
      "===\n",
      "(4, 'g')\n",
      "(z^3 + x^10*z + x^15*y, (5, 1, 1), 1)\n",
      "===\n",
      "(4, 'd')\n",
      "(z^3 + x^5*z^2 + x^15*y, (5, 1, 1), 1)\n",
      "===\n",
      "(1, ('h', 0))\n",
      "(z^3 + x^6*y*z + x^8*y, (3, 1, 1), 1)\n",
      "===\n",
      "(3, ('h', 1))\n",
      "(z^3 + x^6*y*z + x^8*y, (3, 1, 1), 1)\n",
      "===\n"
     ]
    }
   ],
   "source": [
    "for k in pols:\n",
    "    print(k)\n",
    "    print(pols[k])\n",
    "    print(\"===\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "id": "e1d42c5c-80cf-4298-8026-31f8bfc2d5bb",
   "metadata": {},
   "outputs": [],
   "source": [
    "zt, zm, dic = zetas(n, pols, info=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "id": "3f0385d7-0397-485e-a870-09e9d7f65e27",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "(1737/350) * (s + 9/14)^-1\n",
      "(-56/15) * (s + 2/3)^-1\n",
      "(-28/25) * (s + 1)^-1\n"
     ]
    }
   ],
   "source": [
    "quo, L = zt.partial_fraction_decomposition()\n",
    "print(quo)\n",
    "for p in L:\n",
    "    print(p.factor())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "id": "196d3a0c-81cb-4fc3-b2ab-aa36016ae5b1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "El ciclotómico de orden  1  tiene multiplicidad  3\n",
      "El ciclotómico de orden  2  tiene multiplicidad  5\n",
      "El ciclotómico de orden  3  tiene multiplicidad  3\n",
      "El ciclotómico de orden  5  tiene multiplicidad  2\n",
      "El ciclotómico de orden  14  tiene multiplicidad  5\n",
      "El ciclotómico de orden  15  tiene multiplicidad  2\n"
     ]
    }
   ],
   "source": [
    "for j in dic:\n",
    "    print(\"El ciclotómico de orden \", j, \" tiene multiplicidad \", dic[j])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "id": "e9c684cd-c9fa-473a-bb3c-397bfc664735",
   "metadata": {},
   "outputs": [],
   "source": [
    "n1a, Nf1a, Ng1a, Nd1a, Nhs1a, Nks1a, euler1a, nus1a, pols1a = trinomiales_general(A, Vf, Vg, Vhs, Vks, Vd, total=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "id": "6bde71ac-74ea-414e-9ac2-7a2e62b07b16",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n1 == n1a and Nf1 == Nf1a and Ng1 == Ng1a and nus1 == nus1a and euler1 == euler1a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "id": "659bc2d8-84bd-4098-ad64-c7efbc6c32f2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[((1, 2, 3, 1, 2), 1), ((1, 1, 2, 2, 3), 1)]"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nhs1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "id": "a3337a8f-82e7-48a2-ae1d-2b6c08f02b52",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[((2, 3, 5, 3, 5), 1)]"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Nhs1a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "id": "7af9041a-217d-4350-8f0f-4f99be179258",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dict_keys([0, 1, 2, 3, 4, (0, 2), (0, 4), (1, 2), (3, 4), (0, 'd'), (2, 'd'), (4, 'd'), (1, ('h', 0)), (3, ('h', 0))])"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pols1a.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4bf1a1e5-e264-44fa-bd70-50345b91a7cc",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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