1066 lines
20 KiB
Plaintext
1066 lines
20 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"# Diagramas de Rudolf de curvas planas\n",
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"\n",
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"<p style=\"text-align:center\">Seminario Rubio de Francia</p>\n",
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"\n",
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"<p style=\"text-align:center\">Miguel Angel Marco Buzunariz</p>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Geometría algebraica\n",
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"\n",
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"Considerar un sistema de ecuaciones polinómicas\n",
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"\n",
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"$$f_1(x_1,\\ldots,x_n)$$\n",
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"$$f_2(x_1,\\ldots,x_n)$$\n",
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"$$\\vdots$$\n",
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"$$f_m(x_1,\\ldots,x_m)$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"El conjunto $V$ de sus soluciones forman una *variedad algebraica*."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Problema general de la geometría algebraica\n",
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"\n",
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"Estudiar la geometría de $V$ usando la estructura algebraica de las ecuaciones que la definen."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"Nos vamos a restringir a los casos más fáciles:"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- $m=1, n=2$ (curvas planas)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- $\\mathbb{K}=\\mathbb{C}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Resultados clásicos conocidos"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- Si la curva es lisa: teoría de las superficies de Riemann\n",
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" - Superficies de género $g=\\frac{d(d-1)}{2}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Ejemplo \n",
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"\n",
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"$y^2= x^2-1$\n",
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"\n",
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""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"La proyección sobre el eje $x$ es $2:1$ excepto en $x=\\pm 1$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Ejemplo\n",
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"\n",
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""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Ejemplo\n",
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"\n",
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"<video data-autoplay src=\"supriemannelev.mp4\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Topología del encaje\n",
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"\n",
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"$C\\subseteq \\mathbb{C}^2$\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"Codimensión 2: situación análoga a los nudos"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"Uno de los invariantes principales es el grupo fundamental del complementario $\\pi_1(\\mathbb{C}^2\\setminus C)$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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""
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<video data-autoplay src=\"monodtrenzas4.webm\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<video data-autoplay src=\"monodtrenzas5.webm\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<video data-autoplay src=\"monodtrenzas6.webm\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<video data-autoplay src=\"monodtrenzas7.webm\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<video data-autoplay src=\"monodtrenzas8.webm\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<div class=\"thm\">\n",
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" <h3>\n",
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" <b>Lema</b>\n",
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" </h3>\n",
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" <p>\n",
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" Esta construcción determina un morfismo de grupos $\\pi_1(\\mathbb{C}\\setminus \\Delta)\\longrightarrow B_d$.\n",
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" </p>\n",
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"</div>\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"<div class=\"thm\">\n",
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" <h3>\n",
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" <b>Definición</b>\n",
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" </h3>\n",
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" <p>\n",
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" Este morfismo recibe el nombre de <i>monodromía de trenzas</i> de la curva.\n",
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" </p>\n",
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"</div>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"Se puede representar como una lista de trenzas correspondientes a una base geométrica de $\\pi_1(\\mathbb{C}\\setminus \\Delta)$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<video data-autoplay src=\"monodtrenzas9.webm\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<video data-autoplay src=\"monodtrenzas10.webm\"/>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Monodromía de trenzas\n",
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"\n",
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"<div class=\"thm\">\n",
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" <h3>\n",
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" <b>Teorema</b> (Zariski-Van Kampen)\n",
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" </h3>\n",
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" <p>\n",
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" El grupo fundamental del complementario $\\pi_1(\\mathbb{C}^2\\setminus C)$ es el cociente del grupo libre $F_d$ por la acción de la monodromía de trenzas.\n",
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" </p>\n",
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"</div>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"<div class=\"thm\">\n",
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" <h3>\n",
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" <b>Teorema</b>\n",
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" </h3>\n",
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" <p>\n",
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" La monodromía de trenzas determina completamente la topología del par $(\\mathbb{C}^2,C)$.\n",
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" </p>\n",
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"</div>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Pares de Zariski\n",
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"\n",
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"La monodromía de trenzas es un objeto algebraico (morfismo de grupos) definido a partir de otro objeto algebraico (ideal en el anillo de polinomios)."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
|
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"slide_type": "fragment"
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}
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},
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"source": [
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"<div class=\"thm\">\n",
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" <h3>\n",
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" <b>Definición</b>\n",
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" </h3>\n",
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" <p>\n",
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" Dos curvas algebraicas $C_1$ y $C_2$ forman un <i>par de Zariski aritmético</i> si\n",
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" <ul>\n",
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" <li class=\"fragment\">Los polinomios que las definen $f_1,f_2$ son conjugados de Galois.</li>\n",
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" <li class=\"fragment\">Los grupos fundamentales $\\pi_1(\\mathbb{C}^2\\setminus C_1)$ y $\\pi_1(\\mathbb{C}^2\\setminus C_1)$ son no isomorfos.</li>\n",
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" </ul>\n",
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" </p>\n",
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"</div>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
|
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Pares de Zariski\n",
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"\n",
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"<div class=\"thm\">\n",
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" <h3>\n",
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" <b>Teorema</b> (ACG_)\n",
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" </h3>\n",
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" <p>\n",
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" Existen pares de Zariski aritméticos\n",
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" </p>\n",
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"</div>"
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]
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},
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{
|
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"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
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"slide_type": "fragment"
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}
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},
|
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"source": [
|
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"Esto implica que la monodromía de trenzas (o el grupo fundamental) no se puede calcular de manera puramente algebraica."
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]
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},
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{
|
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"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
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"source": [
|
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"Se necesitan cálculos **numéricos**, pero **certificados**."
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]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
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"source": [
|
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"## Cálculo de la monodromía de trenzas\n",
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"\n",
|
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"Primer paso: elegir los caminos\n",
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"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo de la monodromía de trenzas\n",
|
|
"\n",
|
|
"Primer paso: elegir los caminos\n",
|
|
"\n",
|
|
"\n",
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"\n",
|
|
"El *Diagrama de Voronoi* nos da segmentos que rodean los puntos del discriminante, acercándose lo menos posible."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo de la trenza sobre un segmento\n",
|
|
"\n",
|
|
"Mediante cambio de variable, podemos normalizar a $[0,1]$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo de la trenza sobre un segmento\n",
|
|
"\n",
|
|
"Mediante cambio de variable, podemos normalizar a $[0,1]$\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo de la trenza sobre un segmento\n",
|
|
"\n",
|
|
"Mediante cambio de variable, podemos normalizar a $[0,1]$\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Aritmética intervalar\n",
|
|
"\n",
|
|
"<div class=\"thm\">\n",
|
|
" <h3>\n",
|
|
" <b>Definición</b>\n",
|
|
" </h3>\n",
|
|
" <p>\n",
|
|
" Un <i>intervalo complejo</i> es un conjunto $$[r_0,r_1]+[i_0,i_1]\\cdot i:=\\{a+b\\cdot i \\mid a \\in [r_0,r_1], b\\in [i_0,i_1]\\}\\subseteq \\mathbb{C}$$\n",
|
|
" </p>\n",
|
|
"</div>"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"<div class=\"thm\">\n",
|
|
" <h3>\n",
|
|
" <b>Definición</b>\n",
|
|
" </h3>\n",
|
|
" <p>\n",
|
|
" Dado $Y\\subseteq \\mathbb{C}$, su <i>clausura intervalar</i> $[Y]$ es el menor intervalo que lo contiene.\n",
|
|
" </p>\n",
|
|
"</div>"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"Se puede operar con intervalos tomando la clausura intervalar de las operaciones punto a punto."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"Los intervalos con estas operaciones **no** forman un anillo (pero para ciertos efectos, se puede operar como si lo fuera)."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Criterio del intervalo de Newton\n",
|
|
"\n",
|
|
"<div class=\"thm\">\n",
|
|
" <h3>\n",
|
|
" <b>Teorema</b>\n",
|
|
" </h3>\n",
|
|
" <p>\n",
|
|
" Sea $Y$ un intervalo complejo, $y0\\in Y$. Sea $f: Y\\longrightarrow \\mathbb{C}$ una función holomorfa, tal que $0\\in [f'(Y)]$. Si\n",
|
|
" $$\n",
|
|
" N(f,y_0,Y):=y_0-\\frac{f(y_0)}{[f'(Y)]}\\subseteq Y\n",
|
|
" $$\n",
|
|
" entonces $f$ tiene un único cero en $Y$. Además, este cero está en $N(f,y_0,Y)$.\n",
|
|
" </p>\n",
|
|
"</div>"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"Podemos usar este teorema para asegurar que tenemos un entorno alrededor de una raíz aproximada, que contiene a la raíz auténtica, pero sin tocar a las demás raíces."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"Notar que, ya que estamos operando con intervalos, podemos usar intervalos en los propios coeficientes."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Ejemplo\n",
|
|
"\n",
|
|
"$$f = [0.9,1.1]\\cdot y^2-1-x$$\n",
|
|
"$$y_0=1$$\n",
|
|
"$$Y=[0.5,1.5]$$\n",
|
|
"$$x_0=0$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"$$N(f,y_0,Y)=1-\\frac{[0.9,1.1]-1}{2\\cdot[0.9,1.1][0.5,1.5]}=$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"$$=1-\\frac{[-0.1,0.1]}{[0.9,3.3]}\\approx [0.888,1.111]$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Ejemplo: extender $x$ a un intervalo\n",
|
|
"\n",
|
|
"$$f = [0.9,1.1]\\cdot y^2-1-x$$\n",
|
|
"$$y_0=1$$\n",
|
|
"$$Y=[0.5,1.5]$$\n",
|
|
"$$x=[0,0.25]$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"$$N(f,y_0,Y)=1-\\frac{[0.9,1.1]-1-[0,0.25]}{2\\cdot[0.9,1.1][0.5,1.5]}=$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"$$=1-\\frac{[-0.35,0.1]}{[0.9,3.3]}\\approx [0.888,1.13888]$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo numérico certificado de la trenza\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo numérico certificado de la trenza\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo numérico certificado de la trenza\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo numérico certificado de la trenza\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo numérico certificado de la trenza\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Cálculo numérico certificado de la trenza\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## SIROCCO: demostración"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"A.<x,y> = AffineSpace(QQ,2)\n",
|
|
"C = A.curve(y^2-x^3-x^2)\n",
|
|
"C.plot((x,-3,3),(y,-3,3))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"show(C.fundamental_group())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## SIROCCO: demostración"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"C2 = A.curve((x^2+y^2)^2+18*(x^2+y^2) - 27 -8*(x^3-3*x*y^2))\n",
|
|
"C2.plot((x,-3,3),(y,-3,3))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"show(C2.fundamental_group())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Diagramas de Rudolf\n",
|
|
"\n",
|
|
"Volviendo a las superficies de Riemann:\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Diagramas de Rudolf\n",
|
|
"\n",
|
|
"Volviendo a las superficies de Riemann:\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Diagramas de Rudolf\n",
|
|
"\n",
|
|
"<div class=\"thm\">\n",
|
|
" <h3>\n",
|
|
" <b>Definición</b> \n",
|
|
" </h3>\n",
|
|
" <p>\n",
|
|
" Sea $f\\in\\mathbb{C}[x,y]$, su <i>diagrama de Rudolf</i> es el conjunto\n",
|
|
" $$\\{x\\in\\mathbb{C} \\mid \\exists y_0\\neq y_1, Re(y_0)=Re(y_1), f(x,y_0)=f(x,y_1)=0\\}$$\n",
|
|
" </p>\n",
|
|
"</div>"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"- Como conjunto de $\\mathbb{R}^2\\cong \\mathbb{C}$, es un conjunto semialgebraico"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"- En general, son difíciles de calcular"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Diagramas de Rudolf\n",
|
|
"\n",
|
|
"Diagrama de Rudolf de $(4z^3+12z+3-w^2)^2-8w-48z+52$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "fragment"
|
|
}
|
|
},
|
|
"source": [
|
|
"<img src=\"diagramarudolforevkov.svg\" style=\"max-height:450px !important; height:450px;\"/>"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Diagramas de Rudolf\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"## Diagramas de Rudolf\n",
|
|
"\n",
|
|
""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {
|
|
"slideshow": {
|
|
"slide_type": "slide"
|
|
}
|
|
},
|
|
"source": [
|
|
"# Muchas gracias"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"celltoolbar": "Slideshow",
|
|
"kernelspec": {
|
|
"display_name": "SageMath 9.1",
|
|
"language": "sage",
|
|
"name": "sagemath"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.7.9"
|
|
},
|
|
"rise": {
|
|
"transition": "none"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 2
|
|
}
|