610 lines
15 KiB
Plaintext
610 lines
15 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "a6a83522-f953-4884-a2b4-da2b1ed5f484",
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"metadata": {},
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"source": []
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},
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{
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"cell_type": "code",
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"execution_count": 59,
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"id": "731aa3f2-2822-491a-b5c8-150ad760c626",
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"metadata": {},
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"outputs": [],
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"source": [
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"# load(\"ZetaFunctionsNewtonND.py\")\n",
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"load(\"zetas_triples_general.py\")\n",
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"R.<x, y, z> = QQ[]\n",
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"f=(y^2-x^3)^2+x^5*y\n",
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"g=(y^2-x^3)^3+x^10\n",
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"d=f^3-g^2"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "a49fdf82-1388-403c-8cd5-4bbd7e3bafd3",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"-x^20 + 2*x^19 + 3*x^17*y - 3*x^16*y^2 + x^15*y^3 - 12*x^14*y^3 + 18*x^11*y^5 + x^10*y^6 - 12*x^8*y^7 + 3*x^5*y^9"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"d"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 34,
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"id": "3be58442-a908-4fb2-ba36-1894bd963e12",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Newton's polyhedron of -x^20 + 2*x^19 + 3*x^17*y - 3*x^16*y^2 + x^15*y^3 - 12*x^14*y^3 + 18*x^11*y^5 + x^10*y^6 - 12*x^8*y^7 + 3*x^5*y^9:\n",
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" support points = [(20, 0, 0), (19, 0, 0), (17, 1, 0), (16, 2, 0), (15, 3, 0), (14, 3, 0), (11, 5, 0), (10, 6, 0), (8, 7, 0), (5, 9, 0)]\n",
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" vertices = [(5, 9, 0), (19, 0, 0), (17, 1, 0)]\n",
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" number of proper faces = 15\n",
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" Facet 0: x + 2*y - 19 >= 0\n",
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" Facet 1: 2*x + 3*y - 37 >= 0\n",
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" Facet 2: x - 5 >= 0\n",
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" Facet 3: y >= 0\n",
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" Facet 4: z >= 0\n"
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]
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}
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],
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"source": [
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"Zd=ZetaFunctions(d)\n",
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"Zd.give_info_newton(faces=False, cones=False, compact = False)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 35,
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"id": "bcc41b1a-ea09-4f75-9e53-71b87a47d0e2",
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"metadata": {},
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"outputs": [],
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"source": [
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"f1 = R(f(x=x^2, y=x^3 * (y + 1)) / x^12)\n",
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"g1 = R(g(x=x^2, y=x^3 * (y - 1)) / x^18)\n",
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"d1 = R(d(x=x^2, y=x^3 * (y - 1)) / x^37)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 36,
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"id": "328f85f7-6cd6-40ed-9d60-c1be613e32ea",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"y^4 + 4*y^3 + x*y + 4*y^2 + x"
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]
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},
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"execution_count": 36,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"f1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 37,
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"id": "2eb04dbc-e5eb-4631-9531-92b01fe8f37d",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"y^6 - 6*y^5 + 12*y^4 - 8*y^3 + x^2"
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]
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},
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"execution_count": 37,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"g1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 38,
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"id": "b502dd16-e5fc-41dd-894d-c4744b1fc1b4",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"3*y^9 - 27*y^8 + x*y^6 + 96*y^7 - 6*x*y^5 - 168*y^6 + x^2*y^3 + 15*x*y^4 + 144*y^5 - 3*x^2*y^2 - 20*x*y^3 - 48*y^4 - x^3 + 3*x^2*y + 12*x*y^2 - x^2"
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]
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},
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"execution_count": 38,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"d1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 39,
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"id": "113114bc-af4a-4c9e-b3bf-93541e3ad24f",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Newton's polyhedron of 3*y^9 - 27*y^8 + x*y^6 + 96*y^7 - 6*x*y^5 - 168*y^6 + x^2*y^3 + 15*x*y^4 + 144*y^5 - 3*x^2*y^2 - 20*x*y^3 - 48*y^4 - x^3 + 3*x^2*y + 12*x*y^2 - x^2:\n",
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" support points = [(0, 9, 0), (0, 8, 0), (1, 6, 0), (0, 7, 0), (1, 5, 0), (0, 6, 0), (2, 3, 0), (1, 4, 0), (0, 5, 0), (2, 2, 0), (1, 3, 0), (0, 4, 0), (3, 0, 0), (2, 1, 0), (1, 2, 0), (2, 0, 0)]\n",
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" vertices = [(0, 4, 0), (2, 0, 0)]\n",
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" number of proper faces = 11\n",
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" Facet 0: 2*x + y - 4 >= 0\n",
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" Facet 1: x >= 0\n",
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" Facet 2: y >= 0\n",
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" Facet 3: z >= 0\n"
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]
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}
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],
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"source": [
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"Zd1=ZetaFunctions(d1)\n",
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"Zd1.give_info_newton(faces=False, cones=False, compact = False)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 40,
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"id": "5ff51403-dcee-4f63-936a-e4311f2e6ac3",
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"metadata": {},
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"outputs": [],
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"source": [
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"A = Matrix(QQ, 6, [ [-3, 0, 1, 0, 0, 0], [0, -2, 1, 0, 0, 0], [1, 1, -3, 0, 1, 0], [0, 0, 0, -3, 0, 1], [0, 0, 1, 0, -2, 1], [0, 0, 0, 1, 1, -1] ])\n",
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"Vf = vector(4 * [0] + [1] + [0] )\n",
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"Vg = vector( 5 * [0] + [1] )\n",
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"Vd = vector([5] + [1] + 2 * [0] + [2] + [0] )"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 41,
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"id": "13ba1ce2-83c5-4a88-9785-a769a2e32f73",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1"
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]
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},
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"execution_count": 41,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"det(-A)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 45,
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"id": "733c243c-b7c1-454b-bd6e-e88aba1e74d5",
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"metadata": {},
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"outputs": [],
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"source": [
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"n, Nf, Ng, Nd, Nhs, euler, nus, pols = trinomiales_general(A, Vf, Vg, [], Vd, total=False)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 46,
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"id": "69fa70da-afa4-4cca-91e8-c723aae47e68",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"((4, 6, 12, 13, 26, 39), (6, 9, 18, 20, 39, 60), (14, 19, 37, 39, 78, 117))"
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]
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},
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"execution_count": 46,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"Nf, Ng, Nd"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 47,
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"id": "0fb421e2-8ad0-40f8-ab3f-a1048ec3a12a",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{0: (z^3 + x^2*z^2 + x^8, (2, 1, 1), -4),\n",
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" 1: (z^3 + x^3*z^2 + x^10, (3, 1, 1), 0),\n",
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" 2: (z^3 + x^6*z^2 + x^19, (5, 1, 1), -1),\n",
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" 3: (z^3 + x^13*z + x^20, (6, 1, 1), 1),\n",
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" 4: (z^3 + x^26*z + x^39, (11, 1, 1), -2),\n",
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" 5: (z^3 + x^39*z + x^60, (17, 1, 1), 0),\n",
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" (0, 2): (z^3 + x^2*y^6*z^2 + x^8*y^19, (2, 5, 1), 1),\n",
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" (1, 2): (z^3 + x^3*y^6*z^2 + x^10*y^19, (3, 5, 1), 1),\n",
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" (2, 4): (z^3 + x^6*y^13*z^2 + x^19*y^39, (5, 11, 1), 1),\n",
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" (3, 5): (z^3 + x^13*y^39*z + x^20*y^60, (6, 17, 1), 1),\n",
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" (4, 5): (z^3 + x^26*y^39*z + x^39*y^60, (11, 17, 1), 1),\n",
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" (0, 'd'): (z^3 + x^2*z^2 + x^8*y, (2, 1, 1), 5),\n",
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" (1, 'd'): (z^3 + x^3*z^2 + x^10*y, (3, 1, 1), 1),\n",
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" (4, 'd'): (z^3 + x^13*z^2 + x^39*y, (11, 1, 1), 2)}"
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]
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},
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"execution_count": 47,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"pols"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 48,
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"id": "116a3b7e-17a1-45dd-a501-2d7430385922",
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"metadata": {},
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"outputs": [],
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"source": [
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"zt, zm, dic = zetas(n, pols, info=True)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 49,
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"id": "8ca2025c-c4c4-40f5-a690-74ed194507f4",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(1357/2223) * (s + 23/38)^-1 * (s + 11/18)^-1 * (s + 8/13)^-1 * (s + 5/8)^-1 * (s + 1)^-1 * (s^4 + 182245/65136*s^3 + 126929/43424*s^2 + 175993/130272*s + 55/236)"
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]
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},
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"execution_count": 49,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"zt.factor()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 50,
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"id": "d5321785-1e6d-48ab-b02c-d81f3b514034",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{1: 5, 2: 5, 3: 1, 4: 4, 8: 4, 18: 1, 13: 1, 38: 1, 39: 1}"
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]
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},
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"execution_count": 50,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"dic"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 51,
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"id": "d82d56e7-1f23-448b-b8af-16751f7d5db1",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"El factor ciclotómico de 1 tiene multiplicidad 5\n",
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"El factor ciclotómico de 2 tiene multiplicidad 5\n",
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"El factor ciclotómico de 3 tiene multiplicidad 1\n",
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"El factor ciclotómico de 4 tiene multiplicidad 4\n",
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"El factor ciclotómico de 8 tiene multiplicidad 4\n",
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"El factor ciclotómico de 18 tiene multiplicidad 1\n",
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"El factor ciclotómico de 13 tiene multiplicidad 1\n",
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"El factor ciclotómico de 38 tiene multiplicidad 1\n",
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"El factor ciclotómico de 39 tiene multiplicidad 1\n"
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]
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}
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],
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"source": [
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"for j in dic:\n",
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" print(\"El factor ciclotómico de \", j, \"tiene multiplicidad \", dic[j])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 60,
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"id": "18c07685-494b-4b67-8358-c322028d700e",
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"metadata": {},
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"outputs": [],
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"source": [
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"componentes = zetas_components(n, pols, A, Vf, Vg, Vd, Nf, Ng, Nd, info=True)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 61,
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"id": "bd6ac3c0-019a-4153-81f6-0e145158997c",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"('polos', (3, 5, (3, 5), (4, 5)))\n",
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"(5/143) * (s + 8/13)^-1 * (s + 1)^-1 * (s + 16/5)\n",
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"{13: -1, 39: -1}\n",
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"=================\n",
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"('unos', (0, 2, 1, (0, 2), (1, 2), (2, 4), (0, 'd'), (1, 'd')))\n",
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"(986/1881) * (s + 23/38)^-1 * (s + 11/18)^-1 * (s + 8/13)^-1 * (s + 5/8)^-1 * (s + 1)^-1 * (s^4 + 99233/36192*s^3 + 119869/42432*s^2 + 265381/205088*s + 11385/51272)\n",
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"{1: 4, 2: 5, 4: 4, 8: 4, 18: 1, 38: 1}\n",
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"=================\n",
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"('unos', ((4, 'd'),))\n",
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"(4/429) * (s + 8/13)^-1 * (s + 1)^-1 * (s + 12)\n",
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"{13: -2, 39: -2}\n",
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"=================\n",
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"('dic', (4,))\n",
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"(6/143) * (s + 8/13)^-1 * (s + 1)^-1 * (s - 8/3)\n",
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"{1: 1, 3: 1, 13: 4, 39: 4}\n",
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"=================\n"
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]
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}
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],
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"source": [
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"for a in componentes:\n",
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" zetatop, zetamon, dic = componentes[a]\n",
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" print(a)\n",
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" print(zetatop.factor())\n",
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" print(dic)\n",
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" print(\"=================\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 22,
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"id": "1ba9bd3d-e05b-4f00-80c9-9f543a622ee2",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(5, 1, 0, 0, 2, 0)"
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]
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},
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"execution_count": 22,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"Vd"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"id": "ac643c88-e81a-47a9-aa65-71094947049e",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(32/855*s^2 + 491/3420*s + 253/3420)/(s^3 + 379/171*s^2 + 1085/684*s + 253/684)"
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]
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},
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"execution_count": 18,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"f2=z^3 + x^6*z^2 + x^19\n",
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"zf2=ZetaFunctions(f2)\n",
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"zf2.topological_zeta(local=true, weights=[5,1,1])"
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]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 19,
|
|
"id": "fe441461-5766-438d-9b0f-151db04e4b9b",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(s + 23/38) * (s + 11/18) * (s + 1)"
|
|
]
|
|
},
|
|
"execution_count": 19,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"(s^3 + 379/171*s^2 + 1085/684*s + 253/684).factor()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 20,
|
|
"id": "8d480cde-88b7-43ce-bc01-a90c66eab0c7",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(-3/143*s + 8/143)/(s^2 + 21/13*s + 8/13)"
|
|
]
|
|
},
|
|
"execution_count": 20,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"f4=z^3 + x^26*z + x^39\n",
|
|
"zf4=ZetaFunctions(f4)\n",
|
|
"zf4.topological_zeta(local=true, weights=[11,1,1])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 21,
|
|
"id": "07b6c70f-84c8-48a8-95ef-21fa53ebce2f",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(22/1989*s + 73/1989)/(s^2 + 190/117*s + 73/117)"
|
|
]
|
|
},
|
|
"execution_count": 21,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"f5=z^3 + x^39*z + x^60\n",
|
|
"zf5=ZetaFunctions(f5)\n",
|
|
"zf5.topological_zeta(local=true, weights=[17,1,1])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 22,
|
|
"id": "aafa5255-b9a6-493b-b8e0-30bade89c987",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(7/234*s + 25/234)/(s^2 + 64/39*s + 25/39)"
|
|
]
|
|
},
|
|
"execution_count": 22,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"f3=z^3 + x^13*z + x^20\n",
|
|
"zf3=ZetaFunctions(f3)\n",
|
|
"zf3.topological_zeta(local=true, weights=[6,1,1])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 23,
|
|
"id": "e71214a3-4cf7-4ff7-b691-9bc0f077bc3b",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(-1/8*s + 5/16)/(s^2 + 13/8*s + 5/8)"
|
|
]
|
|
},
|
|
"execution_count": 23,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"f0=z^3 + x^2*z^2 + x^8\n",
|
|
"zf0=ZetaFunctions(f0)\n",
|
|
"zf0.topological_zeta(local=true, weights=[2,1,1])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 24,
|
|
"id": "3b080c43-5e06-49c8-b649-5659c9a1acd7",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(1/20*s + 13/60)/(s^2 + 33/20*s + 13/20)"
|
|
]
|
|
},
|
|
"execution_count": 24,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"f1=z^3 + x^3*z^2 + x^10\n",
|
|
"zf1=ZetaFunctions(f1)\n",
|
|
"zf1.topological_zeta(local=true, weights=[3,1,1])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"id": "14950aeb-aa98-494b-b0f6-d6f02cbaec13",
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "SageMath 10.5.beta8",
|
|
"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": "3.12.7"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|