571 lines
12 KiB
Plaintext
571 lines
12 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "e881bcba-75d3-415b-bfe0-5e0f50a49d1d",
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"metadata": {},
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"outputs": [],
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"source": [
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"S.<m3, nu1, g3, s> = PolynomialRing(QQ, order='neglex')"
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]
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},
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{
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"cell_type": "markdown",
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"id": "b3884a71-e7bb-4a29-95c1-141e364eec85",
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"metadata": {},
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"source": [
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"Primer caso: $f=z^3 - x^{m_3}$, $\\omega=x^{\\nu_1-1} d x d z$ y $g_3=\\gcd(3,m_3)$."
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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": 17,
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"id": "f12e8928-4d29-4295-add0-c61c40785580",
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"metadata": {},
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"outputs": [],
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"source": [
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"tau1 = vector([m3, 0])\n",
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"tau0 = vector(S, [0, 3])\n",
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"V1 = vector(S, [0, 1])\n",
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"V0 = vector(S, [1, 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": 39,
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"id": "bbfe6b4f-5d07-43de-8fea-0d8f62ef60f3",
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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*nu1 + m3 + 3*m3*s)/g3, 1, nu1)"
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]
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},
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"execution_count": 39,
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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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"Vgamma = vector([3, m3]) / g3\n",
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"omega1 = vector([nu1, 1])\n",
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"mu1 = 3 / g3\n",
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"Pgamma = (tau0 * s + omega1) * Vgamma\n",
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"volgamma = g3\n",
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"P1 = (tau1 * s + omega1) * V1\n",
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"P0 = (tau0 * s + omega1) * V0\n",
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"Pgamma, P1, P0"
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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": 32,
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"id": "2701f743-4d04-4b66-b8f1-ae978bdf5ac7",
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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/g3, m3/g3)"
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]
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},
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"execution_count": 32,
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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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"mu1 = Matrix([Vgamma, V1]).det()\n",
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"mu0 = Matrix([V0, Vgamma]).det()\n",
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"mu1, mu0"
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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": "40a5decf-c4ec-40ad-be1c-a452c8ecf178",
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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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"g3/(3*nu1 + m3 + 3*m3*s)"
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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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"Jtau1 = mu1 / P1 / Pgamma\n",
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"Jtau0 = mu0 / P0 / Pgamma\n",
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"Jgamma = 1 / Pgamma"
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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": "a4754c02-1b72-4725-85b1-ab158979cd32",
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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) * nu1^-1 * (1 + s)^-1 * (3*nu1 + m3 + 3*m3*s)^-1 * (-3*nu1 - 3*nu1*s + nu1*g3^2*s - m3 - m3*s)"
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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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"Zeta = Jtau0 + Jtau1 - s / (s + 1) * Jgamma * volgamma\n",
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"Zeta.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": 49,
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"id": "34d5f080-c0d6-46a9-a665-4d89fc659b59",
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"metadata": {},
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"outputs": [],
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"source": [
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"R1 = (Zeta * (s + 1))(s=-1)"
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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": 57,
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"id": "d4083af7-c4f6-4b70-b303-91598f08029a",
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"metadata": {},
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"outputs": [],
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"source": [
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"Rp = (Zeta * Pgamma)(s=-Pgamma(s=0) / Pgamma.derivative(s))"
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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": 58,
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"id": "680d1454-1098-4481-90ef-a849cf3ee595",
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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"
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]
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},
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"execution_count": 58,
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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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"Zeta - R1 / (s + 1) - Rp / Pgamma"
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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": 64,
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"id": "6b03816c-0216-43b0-b7d0-e25bffde0f27",
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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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"(-1/6) * (1 + s)^-2 * m3^-1 * (-9 - 9*s + 2*g3^2*s)"
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]
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},
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"execution_count": 64,
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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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"Zeta2 = Zeta(nu1=2 * m3 / 3)\n",
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"Zeta2.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": 76,
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"id": "deee339d-836e-4572-94aa-a16b3f95bd0f",
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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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"(-1/6) * m3^-1 * (-9 + 2*g3^2)"
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]
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},
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"execution_count": 76,
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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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"R2a = (Zeta2 * (s + 1)^2)(s=-1)\n",
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"R1a = ((s + 1)^2 * Zeta2 - R2a) / (s + 1)\n",
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"R1a.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": 77,
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"id": "4c078f46-552b-40d9-a9bf-a133ba42e8be",
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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"
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]
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},
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"execution_count": 77,
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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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"Zeta2 - R1a / (s + 1) - R2a / (s + 1)^2"
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]
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},
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{
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"cell_type": "markdown",
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"id": "883017aa-31fa-45ea-9f47-90ced9eaacf1",
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"metadata": {},
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"source": [
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"Segundo caso: $f=z^3 + x^{m_2}z + x^{m_3}$"
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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": 79,
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"id": "c9ebb2ca-6ea3-4492-84b4-7057e22184cb",
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"metadata": {},
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"outputs": [],
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"source": [
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"S.<m2, nu1, g2, s> = PolynomialRing(QQ, order='neglex')"
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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": 80,
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"id": "4fd45644-e41a-4b74-9f38-75ae1120e59b",
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"metadata": {},
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"outputs": [],
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"source": [
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"tau1 = vector([m2, 1])\n",
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"tau0 = vector(S, [0, 3])\n",
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"V1 = vector(S, [0, 1])\n",
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"V0 = vector(S, [1, 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": 86,
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"id": "13a43344-f27a-425b-8ce6-eb9d452651d0",
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"metadata": {},
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"outputs": [],
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"source": [
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"Vgamma = vector([2, m2]) / g2\n",
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"omega1 = vector([nu1, 1])\n",
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"mu1 = Matrix([Vgamma, V1]).det()\n",
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"mu0 = Matrix([V0, Vgamma]).det()\n",
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"Pgamma = (tau0 * s + omega1) * Vgamma\n",
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"volgamma = g2\n",
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"P1 = (tau1 * s + omega1) * V1\n",
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"P0 = (tau0 * s + omega1) * V0"
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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": 87,
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"id": "04b82282-4fd4-4e7f-9982-c59c380c0a61",
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"metadata": {},
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"outputs": [],
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"source": [
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"Jtau1 = mu1 / P1 / Pgamma\n",
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"Jtau0 = mu0 / P0 / Pgamma\n",
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"Jgamma = 1 / Pgamma"
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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": 89,
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"id": "cb8f8ee3-6f40-423d-80bc-f3985c0dc70e",
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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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"g2 * (2*nu1 + m2 + 3*m2*s)^-1"
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]
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},
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"execution_count": 89,
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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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"Jgamma.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": 91,
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"id": "0ef91b6f-62c6-4b39-9651-1e5f60729893",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<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>"
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],
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"text/latex": [
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"$\\displaystyle \\left(-1\\right) \\cdot \\nu_{1}^{-1} \\cdot (1 + s)^{-1} \\cdot (2 \\nu_{1} + m_{2} + 3 m_{2} s)^{-1} \\cdot (-2 \\nu_{1} + \\nu_{1} g_{2}^{2} s - m_{2} - m_{2} s)$"
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],
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"text/plain": [
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"(-1) * nu1^-1 * (1 + s)^-1 * (2*nu1 + m2 + 3*m2*s)^-1 * (-2*nu1 + nu1*g2^2*s - m2 - m2*s)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"Zeta = Jtau0 + Jtau1 - s / (s + 1) * Jgamma * volgamma\n",
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"show(Zeta.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": 93,
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"id": "0719a53b-fa90-42b1-ac78-418acb0fd0c8",
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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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"(1/2) * (-nu1 + m2)^-1 * nu1^-1 * g2^-1 * (2*nu1 + m2) * (nu1*g2^2 + 2*m2)"
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]
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},
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"execution_count": 93,
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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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"R1 = (Zeta * (s + 1))(s=-1)\n",
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"Rp = (Zeta * Pgamma)(s=-Pgamma(s=0) / Pgamma.derivative(s))\n",
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"Rp.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": 94,
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"id": "37c88cb0-c2e1-4c18-ada0-a593435bd250",
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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"
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]
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},
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"execution_count": 94,
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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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"Zeta - R1 / (s + 1) - Rp / Pgamma"
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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": 95,
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"id": "54e9c4b8-a3f7-45d8-b48e-de8234f4d829",
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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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"(2*nu1 - 2*m2)/g2"
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]
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},
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"execution_count": 95,
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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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"Pgamma(s=-1)"
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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": 96,
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"id": "66efc352-66e3-4c39-8dd0-17c85c37ccf1",
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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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"(-1/3) * (1 + s)^-2 * m2^-1 * (-3 - s + g2^2*s)"
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]
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},
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"execution_count": 96,
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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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"Zeta2 = Zeta(nu1=m2)\n",
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"Zeta2.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": 99,
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"id": "8ac89af7-786c-4504-923d-938ab424400b",
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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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"((-1/3) * m2^-1 * (-1 + g2) * (1 + g2), (1/3) * m2^-1 * (2 + g2^2))"
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]
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},
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"execution_count": 99,
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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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"R2a = (Zeta2 * (s + 1)^2)(s=-1)\n",
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"R1a = ((s + 1)^2 * Zeta2 - R2a) / (s + 1)\n",
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"R1a.factor(), R2a.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": 98,
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"id": "afd51c00-c529-416a-98c3-13c6cfaf3760",
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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"
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]
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},
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"execution_count": 98,
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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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"Zeta2 - R1a / (s + 1) - R2a / (s + 1)^2"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8301561c-7e6a-4fd2-8bfb-d8690c1f9700",
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"metadata": {},
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"source": [
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"Tercer caso: $f=z^3 + x^{m_1}z^2 + x^{m_3}$, $m_3 > 3 m_1$."
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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": 100,
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"id": "1a73caca-514d-4133-9345-ccb80b276d00",
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"metadata": {},
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"outputs": [],
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"source": [
|
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"S.<m1, m3, nu1, g2, s> = PolynomialRing(QQ, order='neglex')"
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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": 11,
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"id": "9d30e804-9dec-4e73-8f8b-4927873a8445",
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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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"(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)"
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]
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},
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"execution_count": 11,
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"metadata": {},
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|
"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": []
|
|
}
|
|
],
|
|
"metadata": {
|
|
"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",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.12.5"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|