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TNC=CDz|G8xKuIaCD1q=ldEU-L diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 3894b07..ef93d45 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -588,7 +588,7 @@ urlcolor=black]{hyperref} \end{equation*} Este orden monomial es el \nolbreaks{graduado-lexicográfico} y se denota por \mayorquedeglex. \end{ejemplo} - Para poder generalizar este concepto a un módulo como $\Kx^r$ primero tenemos que aclarar qué es un monomio en este anillo. La elección es bastante natural cuando consideramos que un elemento de, por ejemplo, $\anillodepolinomios{\reales}{x, y}^3$ como puede ser $\begin{psmallmatrix} + Para poder generalizar estos órdenes a un módulo como $\Kx^r$ primero tenemos que aclarar qué es un monomio en este anillo. La elección es bastante natural cuando consideramos que un elemento de, por ejemplo, $\anillodepolinomios{\reales}{x, y}^3$ como puede ser $\begin{psmallmatrix} 2xy + y^2 \\ x^3 \\ y \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos + y\etres$. \begin{definicion} @@ -616,7 +616,7 @@ urlcolor=black]{hyperref} Esta generalización que hemos hecho no está limitada a \mayorquelex, también se podría realizar con otros órdenes como \mayorquedeglex. Incluso se podrían permutar las posiciones para obtener todavía más órdenes. \end{ejemplo} - Con el concepto de orden monomial de módulos damos el primer paso en dirección a la respuesta de las preguntas con las que abríamos el capítulo. + Con el concepto de orden monomial de módulos damos el primer paso en dirección a la generalización del algoritmo de la división formalizando esa noción de <> que mencionábamos. \begin{definicion} Sea $p := c_1x^\mathbf{s_1}\vectorbasecanonica{i_1} + c_2x^\mathbf{s_2}\vectorbasecanonica{i_2} + \dots + c_mx^\mathbf{s_m}\vectorbasecanonica{i_m}$ un polinomio de $\Kx^r$ y $>$ un orden monomial de módulos. Sea $k \in \left\lbrace 1, \dots, m \right\rbrace$ de forma que $x^\mathbf{s_k}$ sea el máximo del conjunto de monomios de $p$ según el orden $>$, entonces \begin{itemize} @@ -626,7 +626,9 @@ urlcolor=black]{hyperref} \end{itemize} \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 construir el cociente, utilizamos los términos líderes. + Necesitamos también concretar la noción de divisibilidad para poder integrar las distintas componentes que puedan tener los elementos dentro de la división. 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 $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. + + Con todo esto, ya podemos definir un algoritmo de la división de un elemento por otro en $\Kx^r$. En esencia, imitaremos al algoritmo tradicional solo que, en este caso, nos guiaremos por los monomios líderes y no por el grado. \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. \begin{algorithmic}[1] @@ -661,17 +663,19 @@ urlcolor=black]{hyperref} \end{block} \end{blockarray} \end{center} - - La primera iteración ejecuta el bucle interno dos veces dejando $D = 2x^2y\edos - 6x^2y\etres + x^2\euno - 4xy\etres$; después, se encuentra con el término líder $2x^2y\edos$, que no es divisible por $\LT{g} = x^2\euno$, así que lo elimina del dividendo, que pasa a ser $D = - 6x^2y\etres + x^2\euno - 4xy\etres$ y lo añade al resto, que pasa a ser $r = 2x^2y\edos$. En la segunda iteración el término líder sigue sin ser divisible por \LT{g}; así que lo eliminamos del dividendo y lo añadimos al resto. En la tercera iteración, $\LT{D} = x^2\euno$ sí es divisible por \LT{g}, por lo que procedemos como en la primera iteración primero reduciendo y después pasando el término líder que queda al resto. En la última iteración, volvemos a retirar el término líder y ya nos encontramos en la situación de salida del algoritmo. - - El resultado puede leerse como + \begin{description} + \item[1\textsuperscript{a} iteración:] reducimos $D$ dos veces, lo cual se corresponde con dos ejecuciones del bucle interno o las dos primeras restas que aparecen arriba. Después, eliminamos el término líder de $D$, $2x^2y\edos$, ya que no es divisible por $\LM{x^2\euno + 2y\etres} = x^2\euno$ y se lo añadimos a $r$. Esto deja $D = - 6x^2y\etres + x^2\euno - 4xy\etres$, $c = 3x^2 + 2x$ y $r = 2x^2y\edos$. + \item[2\textsuperscript{a} iteración:] el término líder de $D$, $-6x^2y\etres$, sigue sin ser divisible por $x^2\euno$, por lo que no reducimos. Después, eliminamos este de $D$ y se lo añadimos a $r$. Esto deja $D = x^2\euno - 4xy\etres$, $c = 3x^2 + 2x$ y $r = 2x^2y\edos - 6x^2y\etres$. + \item[3\textsuperscript{a} iteración:] reducimos una vez. Después, eliminamos el término líder de $D$, $-4xy\etres$, ya que no es divisible por $x^2\euno$ y se lo añadimos a $r$. Esto deja $D = -2y\etres$, $c = 3x^2 + 2x + 1$ y $r = 2x^2y\edos - 6x^2y\etres -4xy\etres$. + \item[4\textsuperscript{a} iteración:] el término líder de $D$, $-2y\etres$, el último que queda, no es divisible por $x^2\euno$, por lo que no reducimos. Después, eliminamos el término líder de $D$ y se lo añadimos a $r$. Esto deja $D = 0$, $c = 3x^2 + 2x + 1$ y $r = 2x^2y\edos - 6x^2y\etres -4xy\etres - 2y\etres$. Al ser ya $D = 0$, el algoritmo para. + \end{description} + Queda por lo tanto \begin{equation*} - f = \left( 3x^2 + 2x + 1 \right)g + \underbrace{2x^2y\edos - 6x^2y\etres - 4xy\etres - 2y\etres} + f = \left( 3x^2 + 2x + 1 \right)g + \underbrace{2x^2y\edos - 6x^2y\etres - 4xy\etres - 2y\etres}_{\text{resto $r$}} \end{equation*} - donde ninguno de los términos en la llave es divisible por \LT{g}. + donde ninguno de los términos en $r$ es divisible por \LT{g}. \end{ejemplo} - 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 $G$ un subconjunto de $\Kx^r$ con un orden monomial de módulos $>$. El \submodulo{\Kx} de los monomios líderes de $G$ lo denotaremos por \begin{equation*}