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monomiales de módulos: %Poner otra ecuación de definición 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}.\todo{Creo que es mejor gastar energía en otras cosas.} + Véase \cite[Teorema 1.7.3 y 2.5.9]{SINGULARLIBRO}. \end{proof} Un ejemplo de cálculo de base de \Groebner{} puede encontrarse en apéndice \ref{Sección A mano}. @@ -1599,39 +1599,6 @@ 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 $0$. - - La implicación contraria es la que conlleva más trabajo. Lo primero que debemos hacer es introducir un orden monomial nuevo en función del conjunto generador. Como $\Kx^r$ es noetheriano, podemos asumir desde el principio que $G = \conjunto{g_1, g_2, \dots, g_k}$. Definimos, por tanto, el siguiente orden monomial de módulos en $\Kx^k$: - \begin{equation*} - x^\mathbf{\alpha}\vectorbasecanonica{i} >_\mathrm{Schreyer} x^\mathbf{\beta}\vectorbasecanonica{j} \iff \begin{cases} - \LM{x^\mathbf{\alpha}g_i} > \LM{x^\mathbf{\beta}} \quad \text{o} \\ - \LM{x^\mathbf{\alpha}g_i} = \LM{x^\mathbf{\beta}} \quad \text{e} \quad $i < j$. - \end{cases} - \end{equation*} - Ahora, para dos generadores $g_i$ y $g_j$ de forma que $\LM{g_i}$ y $\LM{g_j}$ estén en la misma componente, tomamos $c_jm_{ji}$ y $c_im_{ij}$ elementos de \Kx{} de forma que $c_i, c_j \in K$ y $m_{ji}, m_{ij}$ son monomios de \Kx y cumplen - \begin{equation*} - \spoly{g_i}{g_j} = c_jm_{ji}g_i - c_im_{ij}g_j. - \end{equation*} - Esto último lo podemos expresar en términos de los elementos de $G$ como - \begin{equation*} - c_jm_{ji}g_i - c_im_{ij}g_j = \sum_{\nu = 1}^{k} a_{\nu}^{(ij)} g_{\nu}. - \end{equation*} - Con todo esto, definimos la expresión - \begin{equation*} - s_{ij} := c_jm_{ji}\vectorbasecanonica{i} - c_im_{ij}\vectorbasecanonica{j} - \sum_{\nu = 1}^{k} a_{\nu}^{(ij)} \vectorbasecanonica{\nu} \in \Kx^k. - \end{equation*} - Que puede entenderse como un vector que registra una combinación lineal que anula los generadores, es decir, $s_{ij} \in \syz{g_1, g_2, \dots, g_k}$. Además, con respecto al orden de Schreyer, $\LM{s_{ij}} = m_{ji}\vectorbasecanonica{i}$. Esto se debe por un lado a que $\LM{m_{ij}g_j} = \LM{m_{ji}g_i}$ pero $i < j$; y por otro a que - \begin{equation*} - \LM{a_{\nu}^{(ij)} g_{\nu}} \leq \LM{c_jm_{ji}g_i - c_im_{ij}g_j} < \LM{m_{ji}g_i}. - \end{equation*} - - Con los preliminares terminados, pasamos a demostrar que si el resto de \spoly{g_i}{g_j} entre $G$ es $0$ para todo $i$ y $j$, entonces tenemos una base de \Groebner. Lo primero es tomar un $f \in \engendrado{G}$ y un $a = \left( a_1, a_2, \dots, a_k \right) \in \Kx^k$ de forma que $f = \sum_{i = 1}^{k} a_i g_i$. Ahora consideremos la expresión - \begin{equation*} - a = h + \sum - \end{equation*} - \todo{Completar la demostración.} \ifbibtex \bibliography{bibliografia} \bibliographystyle{plain}