From 3dc699a7f54baf62d8d316e08b8980977d181cae Mon Sep 17 00:00:00 2001 From: Sixto Date: Sun, 25 May 2025 20:11:43 +0200 Subject: [PATCH] =?UTF-8?q?Cambio=20de=20localizaci=C3=B3n=20del=20Lema=20?= =?UTF-8?q?de=20Dickson,=20del=20Teorema=20de=20la=20Base=20de=20Hilbert;?= =?UTF-8?q?=20inclusi=C3=B3n=20de=20la=20proposici=C3=B3n=20que=20demuestr?= =?UTF-8?q?a=20que=20K[x=5F1,=20...,=20x=5Fn]^r=20es=20noetheriano;=20y=20?= =?UTF-8?q?definici=C3=B3n=20del=20algoritmo=20de=20la=20divisi=C3=B3n=20p?= =?UTF-8?q?ara=20subm=C3=B3dulos=20en=20general.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 213690 -> 216235 bytes TFG_Sixto_José_Martínez_Gan.tex | 64 ++++++++++++++++++++++---------- 2 files changed, 44 insertions(+), 20 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index 74ce2da64310866cfac6fc8fde810198f786182d..6f80f8354348cb4a60e7acc7c248fb3574e64004 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zBeS*4bhIXDX10!*PTzk5^P$zPtVJJ0n_JIyRqbwhzp5fE;5LT>25vpV0?Pp&ypP04#?Uuo70mYFGnnVI2^fd`a5xv-bNH zNK?KEQkq7GbzviHgTXZWN3+emM+qOvaX3cHg_wk+EZNFz8r6RgY~x#Bq6@oVH|$}TlT@Ry7xuw^;F0;5+aGnh?I0Y2ahQN3pko|kD;;Cf?ZR2zaRRjc z?_AZv9Wp3@-i<>>?uF`mT0(ZC%zpQ+xLtOcQoKDXDcT1VD`a0THC!W9H5R=BKD1!b=+MxqFVIg!uw-;7+r$?^8Z9f1BDYhsIWo~41baG{3Z3<;>WN%_> b3UhQ}a&&ldWo8O8F)}tZG72RnMNdWwKyn3~ diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 055b714..fa9951c 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -398,12 +398,10 @@ urlcolor=black]{hyperref} Visto así, el ejemplo \eqref{Ejemplo Pertenencia Ideal Una Variable} supone una situación idílica: por ser un dominio de ideales principales, sabemos que vamos a poder representar al conjunto generador con un solo elemento, para el cual tenemos el algoritmo de la división usual definido. Además, sabemos que el resto de este algoritmo de la división arroja la conclusión que buscamos. El problema aparece cuando empezamos a investigar conjuntos más complicados. - \begin{ejemplo} \label{Ejemplo Ideal Principal Tres Variables} - Consideremos ahora el anillo de polinomios $\anillodepolinomios{\reales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Aquí, no hay problema en conseguir un sistema de generadores que nos permita <> puesto que solo hay un elemento. Sin embargo, a la hora de intentar dividir, nos encontramos dudando sobre cómo hacerlo. Tal y como está escrito, podríamos inclinarnos por tomar como el término principal a $4x^2y$ e intentar dividir como lo hacemos en \anillodepolinomios{\reales}{x}, pero hay términos que tienen mayor grado, así que puede que esta forma no nos termine llevando a un resto como buscamos. Pero si decidimos poner a un término de grado máximo, al principio, ¿cuál de los dos deberíamos elegir, $xz^3$ o $x^2yz$? ¿son los dos válidos y nos llevarán al mismo resultado? En resumen, ¿cómo debemos tomar el valor que tenemos que tomar como <> para dirigir la división? - \end{ejemplo} - \begin{ejemplo} \label{Ejemplo Ideal Tres Variables Varios Generadores} - En el mismo anillo que el ejemplo anterior, consideremos el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. Con este ideal la situación es todavía peor que con el anterior. ¿Nos sirven estos generadores? ¿Y cómo hacemos para dividir con ellos? - \end{ejemplo} + + Consideremos ahora el anillo de polinomios $\anillodepolinomios{\reales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Aquí, no hay problema en conseguir un sistema de generadores que nos permita <> puesto que solo hay un elemento. Sin embargo, a la hora de intentar dividir, nos encontramos dudando sobre cómo hacerlo. Tal y como está escrito, podríamos inclinarnos por tomar como el término principal a $4x^2y$ e intentar dividir como lo hacemos en \anillodepolinomios{\reales}{x}, pero hay términos que tienen mayor grado, así que puede que esta forma no nos termine llevando a un resto como buscamos. Pero si decidimos poner a un término de grado máximo, al principio, ¿cuál de los dos deberíamos elegir, $xz^3$ o $x^2yz$? ¿son los dos válidos y nos llevarán al mismo resultado? En resumen, ¿cómo debemos tomar el valor que tenemos que tomar como <> para dirigir la división? + + En el mismo anillo que el ejemplo anterior, consideremos el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. Con este ideal la situación es todavía peor que con el anterior. ¿Nos sirven estos generadores? ¿Y cómo hacemos para dividir con ellos? Vemos por lo tanto que en el momento en el que introducimos más variables, perdemos la capacidad de resolver el problema de una forma razonable. Para poder solucionar este problema, deberemos generalizar el algoritmo de la división siguiendo las claves del ejercicio \eqref{Ejemplo Pertenencia Ideal Dos Variables} para que este pueda aplicarse a estos nuevos objetos y, asimismo, deberemos trabajar en una representación del ideal que también nos permita que esto funcione como es debido, para evitar las ambigüedades vistas en este mismo ejercicio. @@ -495,7 +493,7 @@ urlcolor=black]{hyperref} \end{definicion} Con este aparataje ya tenemos una noción de cuáles son los términos <> dentro de un polinomio, lo cual nos hace empezar a vislumbrar cómo dividir uno por otro. Lo primero es entender cuándo pueden dividirse dos monomios de $\Kx^r$. Según la definición \ref{Definición Divisibilidad Módulos}, para $x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $x^\mathbf{\beta}\vectorbasecanonica{j}$ dos monomios de \Kx{} tenemos que $x^\mathbf{\alpha}\vectorbasecanonica{i}$ divide a $x^\mathbf{\beta}\vectorbasecanonica{j}$ si $i = j$ y $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. Ahora sí, podemos dar un algoritmo de la división en $\Kx^r$. Este consiste en imitar el algoritmo usual para dividir un polinomio por otro en una variable; pero en este caso, en vez de guiarnos por el grado para encontrar los términos <> del cociente y el dividendo, utilizamos el término líder. - \begin{definicion}[algoritmo de la división] + \begin{definicion} \label{Algoritmo de la División Simple} Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^n$ junto con un orden monomial sobre este. Para dividir $f$ entre $g$ aplicaremos un algoritmo que consiste en replicar el algoritmo que ya conocemos para los polinomios en una variable. Sin embargo, ahora los términos que guían los coeficientes que vamos poniendo son $\LT{f}$ y $\LT{g}$: \begin{algorithmic}[1] \State Definimos $D \gets f$, $d \gets g$, $c \gets 0$ y $r \gets 0$ como las variables que representan el dividendo, divisor, cociente y resto, respectivamente. @@ -539,24 +537,50 @@ urlcolor=black]{hyperref} donde ninguno de los términos en la llave es divisible por \LT{g}. \end{ejemplo} - Para completar estas herramientas, introducimos las bases de Gröbner. + Cuando obtenemos el resto de $f$ dividido por $g$ lo que esencialmente buscamos es un representante $r$ <> de la clase de equivalencia de $f$ en \cociente{\Kx^r}{\engendrado{g}}. En este caso, entendemos canónico como que no tenga ningún término divisible por $\LT{g}$, es decir, que sea <>. Entendida así, la noción de <> puede extenderse a submódulos de $\Kx^r$ en general. Es decir, dado cualquier \submodulo{\Kx} $N$ de $\Kx^r$, podemos <> $f$ por $N$ encontrando el representante <> de la clase de $f$ en \cociente{\Kx^r}{N}. En este caso, nuestra idea de canónico será que no haya ningún elemento $g$ en $N$ de forma que $\LT{g}$ sea uno de los términos de este representante. Para formalizar esta idea de división más general, necesitamos de algunos resultados previos. + \begin{definicion} + Sea $N$ un \submodulo{\Kx} de $\Kx^r$ con un orden monomial de módulos $>$. El \submodulo{\Kx} de los términos líderes de $N$ lo denotaremos por + \begin{equation*} + \LT{N} := \definicionconjunto{\LT{g}}{g \in N}. + \end{equation*} + \end{definicion} + + \begin{lema}[de Dickson] + Todo ideal monomial está finitamente generado. Más formalmente: + sea $N \subseteq \naturales^n$ e $I := \engendrado{x^\mathbf{\alpha} \ \vert \ \mathbf{\alpha} \in N}$ un ideal monomial. Entonces existen $\mathbf{\alpha}_1, \mathbf{\alpha}_2, \dots, \mathbf{\alpha}_k$ en $N$ de forma que $I = \engendrado{x^{\mathbf{\alpha}_1}, \dots, x^{\mathbf{\alpha}_k}}$. + \end{lema} + \begin{teorema}[de la base de Hilbert] + Sea $A$ un anillo noetheriano. Entonces el anillo de polinomios \anillodepolinomios{A}{x_1, \dots, x_n} es también noetheriano. + \end{teorema} + \begin{proposicion} \label{Proposición Kx^r noetheriano}%Proposición 2.1.29 de Singular. + Sea $A$ un anillo noetheriano y $M$ un \modulo{A} finitamente generado. Entonces, $M$ es un \modulo{A} noetheriano. + \end{proposicion} + + Como $K$ es un cuerpo, todos sus ideales son principales, así que es noetheriano. Por el teorema de la base de Hilbert, \Kx{} es noetheriano también. Y por último, por la proposición \ref{Proposición Kx^r noetheriano}, $\Kx^r$ es igualmente noetheriano. Esto es de gran importancia, ya que nos asegura que es suficiente con tener un algoritmo de la división para $N$ finitamente generado. + + + \begin{definicion} + Sean $f$ un elemento y $N = \engendrado{g_1, \dots, g_k}$ un \submodulo{\Kx} de $\Kx^r$. Definimos el algoritmo de la división de $f$ entre $N$ como sigue: + \begin{algorithmic}[1] + \State Definimos $D \gets f$, $d_i \gets g_i$, $c_i \gets 0$ y $r \gets 0$ como las variables que representan el dividendo, los divisores, los cocientes y el resto, respectivamente. + \Repeat + \While{haya algún $i$ de forma que $\LM{d_i} \vert \LM{D}$} + \State Sea $i$ de forma que $\LM{d_i} \vert \LM{D}$. + \State Sea $m \in \Kx$ de forma que $\LM{D} = m\LM{d_i}$. + \State $D \gets D - m d_i$, $c_i \gets c_i + m$ + \EndWhile + \State $D \gets D - \LM{D}$, $r \gets r + \LM{D}$ + \Until{$D = 0$} + \end{algorithmic} + \end{definicion} + En este caso, procedemos como en el algoritmo \ref{Algoritmo de la División Simple} pero, en vez de intentar reducir en cada paso por un solo divisor, buscamos si alguno de todos ellos nos sirve para anular \LM{D} y, en caso de que sí, lo utilizamos para reducir. Nótese que puede haber varios, en cuyo caso, puede elegirse el que se prefiera. + + \section{Bases de Gröbner} - El primer obstáculo a superar, como ya hemos mencionado antes, es el tener una buena representación <> de los generadores de un ideal $I$ del módulo $\Kx^r$. ¿Pero qué buscamos cuando decimos que queremos una representación favorable? - Antes de nada, veamos que siempre podremos encontrar un sistema generador finito, lo cual es de gran ayuda. - \begin{lema}[de Dickson] - Todo ideal monomial está finitamente generado. Más formalmente: - sea $N \subseteq \naturales^n$ e $I := \engendrado{x^\mathbf{\alpha} \ \vert \ \mathbf{\alpha} \in N}$ un ideal monomial. Entonces existen $\mathbf{\alpha}_1, \mathbf{\alpha}_2, \dots, \mathbf{\alpha}_k$ en $N$ de forma que $I = \engendrado{x^{\mathbf{\alpha}_1}, \dots, x^{\mathbf{\alpha}_k}}$. - \end{lema} - \begin{proof} - \todo[inline]{Para poner la demostración, además de que es larga por inducción, hacen falta algunos resultados previos. ¿La ponemos o es suficiente con un dibujo ilustrativo de la escalera?} - \end{proof} - \begin{teorema}[de la base de Hilbert] - contenidos... - \end{teorema} \section{Syzygias}