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ef93d45..5c653d3 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -676,48 +676,44 @@ urlcolor=black]{hyperref} donde ninguno de los términos en $r$ es divisible por \LT{g}. \end{ejemplo} - \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*} - \LM{N} := \engendrado{\LM{g} \vert g \in N}. - \end{equation*} + A partir de este algoritmo, podemos generalizar más para afrontar un número finito de divisores. La clave para esta generalización será buscar entre estos divisores a alguno que pudiera reducir $D$ y, caso de encontrarlo, reducir por ese. Puede que haya casos donde haya más de un divisor que pueda reducir, en cuyo caso se puede escoger el que se prefiera. + \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. + \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} - - + Un ejemplo como el de \eqref{Ejemplo División Compuesta} muestra cómo funciona este algoritmo. Ahora, haría falta generalizar todavía más para ser capaces también de dividir por un número infinito de divisores, pero los siguientes resultado nos permitirán ver que esto no es necesario. \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 nos asegura que es suficiente con tener un algoritmo de la división para un número finito de divisores. - 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} \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. - \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. - - 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: + Con respecto a los problemas que se presentaban al principio, ya hemos resuelto varios: tenemos un orden para los monomios que nos permite tener un convenio de cuáles son los términos principales de los polinomios, hemos integrado las distintas componentes de un vector de $\Kx^r$ al concepto de monomio y sabemos cómo dividir por ellas, y también sabemos cómo afrontar la división con más de un divisor. Sin embargo, todavía podemos encontrarnos con casos como el siguiente: \begin{ejemplo} \label{Ejemplo Fallo División} 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. \end{ejemplo} + Para entender bien lo que ocurre en este ejemplo, necesitamos el siguiente objeto. + \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*} + \LM{N} := \engendrado{\LM{g} \vert g \in N}. + \end{equation*} + \end{definicion} + ¿Qué es lo que ocurre en el ejemplo \ref{Ejemplo Fallo División}? 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 $\engendrado{\LM{x\euno + y\edos}, \LM{y\euno + x\edos}}$. - ¿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. + 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 que sí sea compatible con el algoritmo de la división. Es decir, que cuando dividamos por los generadores y el resto sea distinto de $0$, podamos decir que el elemento no pertenece. \section{Bases de Gröbner} @@ -739,7 +735,7 @@ urlcolor=black]{hyperref} &= \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. + 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 algunos monomios del conjunto generador original. \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 $x^{\mathbf{\alpha}_j}\vectorbasecanonica{i_j} \vert x^\mathbf{\beta}\vectorbasecanonica{i}$. @@ -751,16 +747,16 @@ urlcolor=black]{hyperref} 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 $f \notin N$. + Como ya mencionamos antes, si el resto es $0$ sabemos que $f \in N$. Asumamos ahora que el resto no es $0$. 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 $G$ ---los cocientes por los divisores---. 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, $f-g = D \notin N$ 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. - Recordemos el problema que nos habíamos encontrado en \ref{Ejemplo Fallo División}. En este ejemplo, había monomios líderes ocultos en los términos de los generadores. Para desenmascararlos, lo que haremos será lo que vimos en ese ejemplo: anular los términos libres para forzar a que estos aparezcan. Para eso, dados $f = x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $g = x^\mathbf{\alpha}\vectorbasecanonica{j}$ elementos de $\Kx^r$ y $a$ y $b$ elementos de \Kx{} de forma que $a\LM{f} = b\LM{g} = \mathrm{mcm}\left(\LM{f}, \LM{g}\right)$\footnote{Aquí podría haber un pequeño malentendido con la noción de mínimo común múltiplo: por ejemplo, ¿cuál es el mínimo común múltiplo de $xy^2$ y $x^2y$ en \anillodepolinomios{\reales}{x, y}? Lo natural sería decir $x^2y^2$, que es nuestra convención, pero también sería válido decir $3x^2y^2$ ya que ambos tienen el mismo orden monomial. Ambos serían válidos, pero siempre nos referiremos a aquel que no está multiplicado por constante.}, definimos la siguiente expresión: + Recordemos el problema que nos habíamos encontrado en \ref{Ejemplo Fallo División}. En este ejemplo, había monomios líderes ocultos en los términos de los generadores. Para desenmascararlos, lo que haremos será lo que vimos en ese ejemplo: anular los términos libres para forzar a que estos aparezcan. Para eso, dados $f = x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $g = x^\mathbf{\alpha}\vectorbasecanonica{j}$ elementos de $\Kx^r$ y $a$ y $b$ elementos de \Kx{} de forma que $a\LM{f} = b\LM{g} = \mathrm{mcm}\left(\LM{f}, \LM{g}\right)$\footnote{Aquí podría haber un pequeño malentendido con la noción de mínimo común múltiplo: por ejemplo, ¿cuál es el mínimo común múltiplo de $xy^2$ y $x^2y$ en \anillodepolinomios{\reales}{x, y}? Lo natural sería decir $x^2y^2$, que es nuestra convención, pero también sería válido decir $3x^2y^2$ ya que ambos tienen el mismo orden monomial. Por simplicidad, siempre nos referiremos a aquel que no está multiplicado por constante.}, definimos la siguiente expresión: \begin{equation*} \spoly{f}{g} = \begin{cases} - \frac{a}{\LC{f}}f - \frac{b}{\LC{g}}g \ \text{si $i = j$;} \\ - 0 \ \text{si $i \neq j$.} + \frac{a}{\LC{f}}f - \frac{b}{\LC{g}}g &\text{si $i = j$;} \\ + 0 &\text{si $i \neq j$.} \end{cases} \end{equation*} \begin{ejemplo} @@ -785,15 +781,22 @@ urlcolor=black]{hyperref} \end{algorithmic} Al final del algoritmo, $\left\lbrace d_1, \dots, d_t \right\rbrace$ es una base de Gröbner de $N$. \end{definicion} - El funcionamiento de este algoritmo se sustenta en dos pilares. El primero es que, como cada vez que añadimos un nuevo elemento este es un resto, tenemos que los \submodulo{\Kx} \engendrado{\LM{d_1}, \dots, \LM{d_t}} forman una cadena ascendente la cual, por ser $\Kx^r$ noetheriano, debe finalizar en algún momento. Por esto mismo, el algoritmo termina. El segundo es el teorema \ref{Criterio de Buchberger}, que nos garantiza que cuando este algoritmo pare, el conjunto que tendremos será una base de \Groebner. + El funcionamiento de este algoritmo se sustenta en dos pilares. El primero es que, como cada vez que añadimos un nuevo elemento este es un resto cuyo término líder por definición no es múltiplo de ningún \LM{d_i}, tenemos que los \submodulo{\Kx} \engendrado{\LM{d_1}, \dots, \LM{d_t}} forman una cadena ascendente la cual, por ser $\Kx^r$ noetheriano, debe finalizar en algún momento. Por esto mismo, el algoritmo termina. El segundo es el teorema \ref{Criterio de Buchberger}, que nos garantiza que cuando este algoritmo pare, el conjunto que tendremos será una base de \Groebner. + \begin{teorema}[criterio de Buchberger] \label{Criterio de Buchberger} + Sea $G$ un subconjunto de $\Kx^r$, este último equipado con un orden monomial de módulos $>$. $G$ es una base de Gröbner de \engendrado{G} si y solo si para toda pareja de elementos $f$ y $g$ en $G$ se tiene que el resto de $\spoly{f}{g} = 0$ entre $G$ es $0$. + \end{teorema} + \begin{proof} + Véase el apéndice \ref{Apéndice Demostración del Criterio de Buchberger}. + \end{proof} + Un ejemplo de cálculo de base de \Groebner{} puede encontrarse en apéndice \ref{Sección A mano}. \section{Sicigias} En la sección anterior hemos solucionado el problema de pertenencia a un \submodulo{\Kx} de $\Kx^r$, pero necesitamos conocer los generadores de este submódulo. Por lo tanto, hasta el momento solo hemos resuelto la mitad del problema ya que para $\derivaciones{f}$ todavía no sabemos calcular todavía un conjunto de generadores. Para poder hacer esto, introducimos el concepto de sicigia, una noción que extiende el concepto de solución de un sistema lineal en el caso de un espacio vectorial. \begin{definicion} - Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Una sicigia de los elementos $f_1, f_2, \dots, f_k$ es una \nolbreaks{$k$-tupla} $\left( g_1, g_2, \dots, g_k \right)$ de elementos de $A$ de forma que $f_1g_1 + f_2g_2 + \dots + f_kg_k = 0$. + Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Una sicigia de los elementos $f_1, f_2, \dots, f_k$ es una \nolbreaks{$k$-tupla} $\left( g_1, g_2, \dots, g_k \right)$ de $A^k$ de forma que $f_1g_1 + f_2g_2 + \dots + f_kg_k = 0$. \end{definicion} Nótese que el conjunto de sicigias de $f_1, f_2, \dots, f_k$ forma un \submodulo{A} de $A^k$ puesto que es el kernel del homomorfismo de \modulo{A}s \begin{equation} \label{Definición Funcón Sicigias} @@ -810,22 +813,11 @@ urlcolor=black]{hyperref} Véase que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse en términos de sicigias: \begin{align*} \derivaciones{f} &= \definicionconjunto{a_1\partial_1 + a_2\partial_2 + \dots + a_n\partial_n}{a_1\partial_1f + a_2\partial_2f + \dots + a_n\partial_nf = pf, p \in \Kx}; \\ - \intertext{recordemos que podemos ver $\partial_i$ como \vectorbasecanonica{i} a través del isomorfismo $\kappa$ de \eqref{Secuencia Exacta Definición Módulo Derivaciones};} + \intertext{recordemos que podemos ver $\partial_i$ como \vectorbasecanonica{i};} &\simeq \definicionconjunto{\left( a_1, a_2, \dots, a_n \right) \in \Kx^n}{a_1\partial_1f + a_2\partial_2f + \dots + a_n\partial_nf - pf = 0, p \in \Kx} \\ &= \definicionconjunto{\left( a_1, a_2, \dots, a_n \right) \in \Kx^n}{\left( a_1, a_2, \dots, a_n, p \right) \in \syz{\partial_1f, \partial_2f, \dots, \partial_nf, f}, p \in \Kx}. \end{align*} - - Teniendo ya e - \begin{teorema}[criterio de Buchberger] \label{Criterio de Buchberger} - Sea $G$ un subconjunto de $\Kx^r$, este último equipado con un orden monomial de módulos $>$. $G$ es una base de Gröbner de \engendrado{G} si y solo si para toda pareja de elementos $f$ y $g$ en $G$ se tiene que el resto de $\spoly{f}{g} = 0$ entre $G$ es $0$. - \end{teorema} - \begin{proof} - Si $G$ es una base de \Groebner, entonces, como $\spoly{f}{g} \in \engendrado{G}$, por el teorema \ref{Teorema Buen Funcionamiento División Groebner} tenemos que el resto de su división entre los elementos de $G$ será $0$. - - Como $\Kx^n$ es noetheriano, podemos asumir desde el principio que $G = \conjunto{g_1, g_2, \dots, g_k}$ - \todo[inline]{Ver cómo lo metemos.} - \end{proof} - Para poder calcular las sicigias necesitaremos dos resultados. El primero nos permite restringir una base de Gröbner a solamente unas coordenadas. + Por lo tanto, para calcular \derivaciones{f} obtendremos primero \syz{\partial_1f, \partial_2f, \dots, \partial_nf, f} y nos olvidaremos de la última componente. Ahora nuestro objetivo será calcular sicigias, para lo cual necesitaremos dos resultados. El primero nos permite restringir una base de Gröbner a solamente unas coordenadas. \begin{lema} \label{Lema Restricción Groebner} Sea $>$ un orden monomial de módulos que satisface \begin{equation} \label{Condición Orden Eliminación} @@ -873,7 +865,7 @@ urlcolor=black]{hyperref} \end{equation*} con sus primeras $r$ componentes nulas y las últimas $k$ coinciden con las de $m_{r+1}$. Por lo tanto, $m_{r+1} \in M_{r+1}$. Concluimos, pues, que $M_{r+1} = \syz{f_1, f_2, \dots, f_k}$. \end{proof} - + Por lo tanto, para calcular \syz{f_1, f_2, \dots, f_k}, primero calcularemos una base de \Groebner{} $G$ de $$\engendrado{f_1^+ + \vectorbasecanonica{r + 1}, f_2^+ + \vectorbasecanonica{r + 2}, \dots, f_k^+ + \vectorbasecanonica{r + k}}.$$ Por este último teorema, sabemos que $G_{k+1}$ es un conjunto generador de \syz{f_1, f_2, \dots, f_k} y en particular una base de \Groebner{} con respecto al orden que hayamos escogido. Si deseamos una base de \Groebner{} con respecto a otro orden, como decíamos antes de este teorema, basta con aplicar el algoritmo de Buchberger sobre $G_{k+1}$. \chapter{El Criterio de Saito} En el capítulo anterior concluíamos con un método para calcular \derivaciones{f} utilizando las sicigias, una herramienta que hemos derivado de las bases de Gröbner. Con estas, podemos obtener un conjunto generador de \derivaciones{f} e incluso una base de Gröbner de este mismo que nos permita determinar si una derivación está o no en el conjunto. Pero otra pregunta de gran importancia aparece cuando consideramos una gran diferencia entre módulos y espacios vectoriales. Así como para estos últimos siempre tenemos que los subespacios admiten una base, en el caso de los módulos no tiene por qué ser así. @@ -1185,7 +1177,7 @@ urlcolor=black]{hyperref} El algoritmo para calcular las sicigias que nos permite obtener \derivaciones{f}, en esencia el algoritmo de Buchberger, tienen un alto coste computacional que hace impracticable ejecutarlos <>. En la práctica, se utilizan sistemas de Álgebra Computacional que ya vienen preparados con estos algoritmos en sus versiones más óptimas. Sin embargo, como ocurre con muchas otras operaciones a lo largo de nuestra experiencia matemática, conviene saber cómo hacerlas a mano para tener una mayor intuición. En muchos textos que tratan sobre estos asuntos se suelen omitir los ejemplos a este tipo de operaciones o se dan algunos en casos demasiado sencillos ---ideales de \Kx{} en vez de \modulo{\Kx} de $\Kx^n$ donde da la casualidad que todo se anula a la primera--- por lo que creemos conveniente desarrollar en este apéndice por completo un caso no trivial para que el lector se lleve una imagen realista de qué ocurre detrás de la pantalla. Además, ilustraremos también después cómo se resuelve utilizando un sistema dedicado a este tipo de cálculos. - \section{A mano} + \section{A mano} \label{Sección A mano} En concreto, intentaremos calcular \derivaciones{f} donde $f = x^2z + y^3$ en el anillo \anillodepolinomios{\complejos}{x, y, z} con el orden \mayorqueposdeglex. Recordemos que lo que buscamos es una descripción total del conjunto \begin{equation*} \derivaciones{f} = \definicionconjunto{\vectorhorizontaltres{a}{b}{c} \in \anillodepolinomios{\complejos}{x, y, z}^3}{a\diffp{f}{x} + b \diffp{f}{y} + c\diffp{f}{z} = kf, \ k \in \anillodepolinomios{\complejos}{x, y, z}}. @@ -1199,22 +1191,22 @@ urlcolor=black]{hyperref} g_1 := \vectorvertical{2xz \\ 1 \\ 0 \\ 0 \\ 0}\!\!, \ g_2 := \vectorvertical{3y^2 \\ 0 \\ 1 \\ 0 \\ 0}\!\!, \ g_3 := \vectorvertical{x^2 \\ 0 \\ 0 \\ 1 \\ 0}\ \text{y} \ \ g_4' := \vectorvertical{x^2z + y^3 \\ 0 \\ 0 \\ 0 \\ 1}. \end{equation*} Calculamos ahora una base de Gr\"obner a partir de estos elementos. Para empezar con el mejor pie posible, lo que haremos será convertir este conjunto en un conjunto reducido, es decir, que buscaremos garantizar que ninguno de los términos de $g_i$ sea divisible por algún \LM{g_j} donde $j \neq i$. En este caso, vemos que $g_1$, $g_2$ y $g_3$ no tienen ningún termino divisible, sin embargo $g_4'$ sí, así que lo reducimos: - \begin{center} + \begin{equation} \label{Ejemplo División Compuesta} \begin{blockarray}{rrrrrcl} \begin{block}{rrrrrc|l} - $x^2z\euno$ & +$y^3\euno$ & & & +$\ecinco$ & & \cocientecaja{2xz\euno + \edos} \\ - $-(x^2z\euno$ & & $+\frac{1}{2}x\edos)$ & & & & $\frac{1}{2}x$ \\ + x^2z\euno & +y^3\euno & & & +\ecinco & & \cocientecaja{2xz\euno + \edos} \\ + -(x^2z\euno & & +\frac{1}{2}x\edos) & & & & \frac{1}{2}x \\ \end{block} \begin{block}{rrrrrcl} \cmidrule{1-5} - & $y^3\euno$ & $-\frac{1}{2}x\edos\phantom{)}$ & & +$\ecinco$ & & \\ + & y^3\euno & -\frac{1}{2}x\edos\phantom{)} & & +\ecinco & & \\ \end{block} \begin{block}{rrrrrc|l} - & $-(y^3\euno $ & & $+\frac{1}{3}y\etres)$ & & & \cocientecaja{3y^2\euno + \etres} \\ \cmidrule{1-5} - & & $-\frac{1}{2}x\edos$ & $-\frac{1}{3}y\etres\phantom{)}$ & +$\ecinco$ & & $\frac{1}{3}y$ \\ + & -(y^3\euno & & +\frac{1}{3}y\etres) & & & \cocientecaja{3y^2\euno + \etres} \\ \cmidrule{1-5} + & & -\frac{1}{2}x\edos & -\frac{1}{3}y\etres\phantom{)} & +\ecinco & & \frac{1}{3}y \\ \end{block} \end{blockarray} - \end{center} + \end{equation} Añadimos <> conforme nos vayan haciendo falta ya que no solo dividimos por un elemento sino por varios a la vez, por lo que puede ser que para anular un término hagamos uso de uno, para anular otro usemos un elemento distinto e incluso puede que necesitemos volver a utilizar algún elemento de nuevo. Pero en esencia, el procedimiento de división es el mismo pero ahora podemos <> de caja en caja. El resto es irreducible por los demás elementos, así que lo usaremos como elemento $g_4$. Multiplicar por un escalar un elemento no afecta al resultado, por lo tanto, intentamos obtener la expresión más sencilla para este y así @@ -1385,6 +1377,13 @@ print(GS);\end{lstlisting} x2z+y3,3z,-3y2,-2z,0, -x2, -3,0, 0, 0 \end{lstlisting} + +\chapter{Demostración del Criterio de Buchberger} \label{Apéndice Demostración del Criterio de Buchberger} + Si $G$ es una base de \Groebner, entonces, como $\spoly{f}{g} \in \engendrado{G}$, por el teorema \ref{Teorema Buen Funcionamiento División Groebner} tenemos que el resto de su división entre los elementos de $G$ será $0$. + + Como $\Kx^n$ es noetheriano, podemos asumir desde el principio que $G = \conjunto{g_1, g_2, \dots, g_k}$ + \todo[inline]{Terminar} + \nocite{*} \printbibliography \end{document} \ No newline at end of file