From 6d36dd168c502d0fc3509c1d95fe997840df4d7a Mon Sep 17 00:00:00 2001 From: Sixto Date: Mon, 26 May 2025 16:44:39 +0200 Subject: [PATCH] =?UTF-8?q?A=C3=B1adida=20la=20definici=C3=B3n=20de=20base?= =?UTF-8?q?=20de=20Gr=C3=B6bner=20y=20el=20teorema=20que=20garantiza=20que?= =?UTF-8?q?=20sirven=20para=20determinar=20la=20pertenencia=20a=20un=20sub?= =?UTF-8?q?m=C3=B3dulo.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 216235 -> 224246 bytes TFG_Sixto_José_Martínez_Gan.tex | 62 ++++++++++++++++++++++++-------- 2 files changed, 48 insertions(+), 14 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index 6f80f8354348cb4a60e7acc7c248fb3574e64004..19d4938b85cc4db5f1f155125d29e5e21668c6fa 100644 GIT binary patch delta 58243 zcmZUaQ+J*X)UBV`wvDE-?WD17+x8tN4I0~RY&N!?#8$Vj3SW#8krr-h5 zN=d#frTZ|Z`B1Kd_zM)$kKF!1z1-AIJF-qan^xFA;zLg9Y&yJT1S6BjdvGVB}psJeo)}fbYS5d2540erTF7~Ks 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-518,12 +518,12 @@ urlcolor=black]{hyperref} \end{block} \begin{block}{rrrrrrrcl} \cmidrule{1-5} - & +$2x^3\euno$ & +$2x^2y\edos$ & $-6x^2y\etres\phantom{)}$ & +$x^2\euno$ & & & & \\ - & $-(2x^3\euno$ & & & & $+4xy\etres)$ & & & \\ \cmidrule{2-6} - & \tiny 1\textsuperscript{a} iteración: & \sout{$2x^2y\edos$} & \sout{$-6x^2y\etres\phantom{)}$} & +$x^2\euno$ & $-4xy\etres\phantom{)}$ & & & \\ - & & \tiny $\uparrow$ 1\textsuperscript{a} iteración & \tiny $\uparrow$ 2\textsuperscript{a} iteración & $-(x^2\euno$ & & $+2y\etres)$ & & \\ \cmidrule{5-7} - & & & & \tiny 3\textsuperscript{a} iteración: & \sout{$-4xy\etres\phantom{)}$} & \sout{$-2y\etres$} & & \\ - & & & & & \tiny $\uparrow$ 3\textsuperscript{a} iteración & \tiny $\uparrow$ 4\textsuperscript{a} iteración & & \\ + & +$2x^3\euno$ & +$2x^2y\edos$ & $-6x^2y\etres\phantom{)}$ & +$x^2\euno$ & & & & \\ + & $-(2x^3\euno$ & & & & $+4xy\etres)$ & & & \\ \cmidrule{2-6} + & \tiny 1\textsuperscript{a} iteración: & \sout{$2x^2y\edos$} & \sout{$-6x^2y\etres\phantom{)}$} & +$x^2\euno$ & $-4xy\etres\phantom{)}$ & & & \\ + & & \tiny $\uparrow$ 1\textsuperscript{a} iteración & \tiny $\uparrow$ 2\textsuperscript{a} iteración & $-(x^2\euno$ & & $+2y\etres)$ & & \\ \cmidrule{5-7} + & & & & \tiny 3\textsuperscript{a} iteración: & \sout{$-4xy\etres\phantom{)}$} & \sout{$-2y\etres$} & & \\ + & & & & & \tiny $\uparrow$ 3\textsuperscript{a} iteración & \tiny $\uparrow$ 4\textsuperscript{a} iteración & & \\ \end{block} \end{blockarray} \end{center} @@ -539,16 +539,13 @@ urlcolor=black]{hyperref} 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 + Sea $N$ un \submodulo{\Kx} de $\Kx^r$ con un orden monomial de módulos $>$. El \submodulo{\Kx} de los monomios líderes de $N$ lo denotaremos por \begin{equation*} - \LT{N} := \definicionconjunto{\LT{g}}{g \in N}. + \LM{N} := \definicionconjunto{\LM{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} @@ -559,7 +556,7 @@ urlcolor=black]{hyperref} 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} + \begin{definicion} \label{Algoritmo de la División Compuesta} 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. @@ -575,13 +572,50 @@ urlcolor=black]{hyperref} \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. + Pudiera parecer que ya hemos solucionado el problema que planteábamos al principio del capítulo puesto que ya tenemos un algoritmo de la división y además sabemos que todos los submódulos de $\Kx^n$ son finitamente generados. Sin embargo, todavía podemos encontrarnos con casos como el siguiente: supongamos que $N := \engendrado{x\euno + y\edos, y\euno + x\edos}$ es un \submodulo{\anillodepolinomios{\reales}{x,y}} de $\anillodepolinomios{\reales}{x, y}^2$ y queremos saber si $f := x^2\edos - y^2\edos$ pertenece a $N$. Al aplicar el algoritmo de la división siguiendo el orden \mayorqueposlex{} de \ref{Algoritmo de la División Compuesta} nos encontramos que ninguno de los dos generadores de $N$ puede reducir a $f$, por lo tanto el resto es el propio $f$. Como el resto es distinto de $0$, estaríamos tentados a decir que $f$ no pertenece a $N$, pero démonos cuenta de que $f = -y\left( x\euno + y\edos \right) + x\left( y\euno + x\edos \right)$ y, por ende, sí que pertenece. + ¿Qué es lo que ocurre aquí? Si nos fijamos, en la operación $f = -y\left( x\euno + y\edos \right) + x\left( y\euno + x\edos \right)$, los términos líderes de sendos generadores ---$x\euno$ e $y\euno$--- se anulan, por lo que el término líder de esta combinación es menor que los términos líderes de los generadores. En general, esto lo que nos muestra es que $\LM{N}$ no tiene que ser igual a $\engendrado{\LM{x\euno + y\edos}, \LM{y\euno + x\edos}}$. Esto supone un duro golpe para nuestros algoritmos de la división, ya que cuando para un \submodulo{\Kx} $N = \engendrado{g_1, \dots, g_k}$ tengamos que $\LM{N} \neq \engendrado{\LM{g_1}, \dots, \LM{g_k}}$, aunque el resto sea distinto de $0$, no podremos garantizar que el elemento esté fuera de $N$. Por supuesto, si el resto es $0$ sí que podemos garantizar que sí está. + + Para solucionar este problema, necesitamos encontrar una representación de los submódulos algo mejor. Esta mejor representación se conoce como base de Gröbner. \section{Bases de Gröbner} + Para conseguir esta mejor representación, simplemente exigimos la condición que discutíamos en la sección anterior. + + \begin{definicion} + Sea $N$ un \submodulo{\Kx} de $\Kx^r$ con un orden monomial de módulos $>$ y $G$ un subconjunto de $N$. Este subconjunto $G$ es una base de Gröbner de $N$ si $N = \engendrado{G}$ y $\engendrado{\LM{G}} = \LM{\engendrado{G}}$. + \end{definicion} + + Para verificar si esta clase de conjuntos generadores soluciona de una vez por todas estos problemas, necesitamos unos resultados intermedios. + \begin{lema}[de Dickson] + Sea \nolbreaks{$N \subseteq \naturales^n$} 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 \nolbreaks{$I = \engendrado{x^{\mathbf{\alpha}_1}, \dots, x^{\mathbf{\alpha}_k}}$}. + \end{lema} + Aunque el lema de Dickson habla de ideales monomiales dentro del anillo de polinomios \Kx, extenderlo a submódulos es inmediato. Si pensamos en un \submodulo{\Kx} $N$ de $\Kx^r$ generado por monomios, es decir $N := \engendrado{x^\mathbf{\alpha}\vectorbasecanonica{i_\mathbf{\alpha}} \vert \mathbf{\alpha} \in J}$ donde $J \subset \naturales^n$, podemos expresarlo como + \begin{align*} + N &= \engendrado{x^\mathbf{\alpha} \vert \mathbf{\alpha} \in J \ \text{e} \ i_\mathbf{\alpha} = 1}\euno + \engendrado{x^\mathbf{\alpha} \vert \mathbf{\alpha} \in J \ \text{e} \ i_\mathbf{\alpha} = 2}\edos + \dots + \engendrado{x^\mathbf{\alpha} \vert \mathbf{\alpha} \in J \ \text{e} \ i_\mathbf{\alpha} = r}\vectorbasecanonica{r}; \\ + \intertext{por el lema anterior, cada uno de estos <> que acompañan a los vectores \vectorbasecanonica{i} es un ideal monomial finitamente generado;} + &= \engendrado{x^{\mathbf{\alpha}_1}, x^{\mathbf{\alpha}_2}, \dots, x^{\mathbf{\alpha}_{k_1}}}\euno + \engendrado{x^{\mathbf{\beta}_1}, x^{\mathbf{\beta}_2}, \dots, x^{\mathbf{\beta}_{k_2}}}\edos + \dots + \engendrado{x^{\mathbf{\gamma}_1}, x^{\mathbf{\gamma}_2}, \dots, x^{\mathbf{\gamma}_{k_r}}}\vectorbasecanonica{r} \\ + &= \engendrado{x^{\mathbf{\alpha}_1}\euno, x^{\mathbf{\alpha}_2}\euno, \dots, x^{\mathbf{\alpha}_{k_1}}\euno, x^{\mathbf{\beta}_1}\edos, x^{\mathbf{\beta}_2}\edos, \dots, x^{\mathbf{\beta}_{k_2}}\edos, \dots, x^{\mathbf{\gamma}_1}\vectorbasecanonica{r}, x^{\mathbf{\gamma}_2}\vectorbasecanonica{r}, \dots, x^{\mathbf{\gamma}_{k_r}}\vectorbasecanonica{r}}. + \end{align*} + + Este resultado nos permite ver una condición más fuerte sobre los \submodulo{\Kx}s monomiales de $\Kx^r$: no solo son finitamente generados, como ya sabíamos gracias a la proposición \ref{Proposición Kx^r noetheriano}; sino que además admiten conjuntos generadores finitos formados por monomios. + + \begin{lema} + Sea $N$ un \submodulo{\Kx} monomial de $\Kx^r$ de forma que $N = \engendrado{x^{\mathbf{\alpha}_1}\vectorbasecanonica{i_1}, x^{\mathbf{\alpha}_2}\vectorbasecanonica{i_2}, \dots, x^{\mathbf{\alpha}_k}\vectorbasecanonica{i_k}}$. Un monomio $x^\mathbf{\beta}\vectorbasecanonica{i}$ pertenece a $N$ si y solo si es un múltiplo de alguno de los monomios generadores. En otras palabras, existe $j \in \left\lbrace 1, \dots, k \right\rbrace$ de forma que $x^{\mathbf{\alpha}_j}\vectorbasecanonica{i_j} \vert x^\mathbf{\beta}\vectorbasecanonica{i}$. + \end{lema} + + Este pequeño resultado nos da una forma bastante rápida de determinar si un monomio pertenece o no a un \submodulo{\Kx} monomial de $\Kx^r$. Démonos cuenta de que esto es justo lo que hacemos cuando intentamos reducir el término líder de $f$ en el algoritmo de la división de \ref{Algoritmo de la División Compuesta}. Con todo esto, ya damos la respuesta final a la pregunta que abre el capítulo. + + \begin{teorema} + Sea $N$ un \submodulo{\Kx} de $\Kx^r$ y sea $G = \left\lbrace g_1, g_2, \dots, g_k \right\rbrace$ una base de Gröbner de $N$. Un elemento $f$ de $\Kx^r$ pertenece a $N$ si y solo si el resto de $f$ dividido por $G$ según el algoritmo \ref{Algoritmo de la División Compuesta} es $0$. + \end{teorema} + \begin{proof} + Como ya mencionamos antes, si el resto es $0$ ya sabemos que $f \in N$. Asumamos ahora que el resto no es cero. Eso quiere decir que en alguna iteración nos hemos encontrado con que $\LM{D}$ no es divisible por ninguno de los $\LM{g_i}$. En todo momento, $D$ es de la forma $f - g$ donde $g$ es una combinación lineal de los elementos de $G$. Por lo tanto, si $\LM{D}$ no pertenece a \engendrado{\LM{g_1}, \dots, \LM{g_k}}, como $G$ es una base de Gröbner, tampoco pertenecerá a $\LM{N}$ y, por lo tanto, $f-g \notin N$ lo cual, como $g$ sí que pertenece a $N$, es equivalente a decir que $f \notin N$. + \end{proof} + + Con todo esto, ya tenemos una solución al problema de la pertenencia de un elemento a un submódulo: obtener la base de Gröbner del submódulo y dividir por ella para ver si el resto es nulo o no. Sin embargo, no tenemos garantías de que existan bases de Gröbner para todos los submódulos y, en caso de que sí las haya, tampoco sabemos cómo calcularlas. - \section{Syzygias} \chapter{El Criterio de Saito}