From f8427045bb69dd4266a51cbf36d4395eba7fa893 Mon Sep 17 00:00:00 2001 From: Jorge Martin Morales Date: Mon, 9 Jun 2025 13:08:41 +0200 Subject: [PATCH] cambios y comentarios de Kike y Jorge --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 470808 -> 442690 bytes TFG_Sixto_José_Martínez_Gan.tex | 49 +++++++++++++++++++++----------- 2 files changed, 33 insertions(+), 16 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index c71117e169aa88cec6ff792783c6343eab1c9ca6..dbb95ee8e696592a8238ccb6b63effa913a99bc8 100644 GIT binary patch delta 269026 zcmV(#K;*xe-W3dHT zMOs4GNSkD}T=pd)4|D7RjK*Vwj|nGVzg2yki+k*2Kr6Ib2^ddLb#-;!tGc1HH#0i> z>57Ku4==AC{#}Y0r$lS6W-njQ3?-%3Gi?+RR?c3oX1^Cts@*H0)a{|!&KF$Kg3teW z`42x+5>tN~hBN7Gp)4^vO#1be&|(Es_6e9tBBbU4SQ}!5^1z%hGG8!CY4K#+F6-Ug z7PoU*z+|`J%_||56vsf{ahO(YG-e!*Env0}Mx-yPc9IWKx6_=6Tr9C<%y zqBzw4<@N)>RyWnWEZ$Z4yWT{&eVln{D?jpP&QyO)9q#T1d8!;=xs#ca8`dOrjG{b}Zu_=AjBJ1Z 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zef2ZCon8odUm!V74>WnF-ADnNix&`%_izZ({`maZo4O@(qnh$fe7@iLYw-~AuSfp< zD&jBJ9hzY*MfV5G&4=)A$vA3H64g}LUxy`%sRwSlx?DH23l#5_MWw6SWcJQs(0`&{ zm}3pf{-j9*trv{}T@bk&Wc6JRK`1=}g&#XHXg`5E$zc-)xq`ZmYquzKJRr7gw~oQl&2 TAb{uO=4NGqrJ#^dl7#(VGL31{ diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index fe14700..0d54ac2 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -48,8 +48,8 @@ \usepackage{stix} % Bibliografía. -\usepackage[backend=biber]{biblatex} -\addbibresource{bibliografia.bib} +%\usepackage[backend=biber]{biblatex} +%\addbibresource{bibliografia.bib} \usepackage{algpseudocodex} \algrenewcommand\algorithmicwhile{\textbf{mientras}} @@ -101,7 +101,7 @@ urlcolor=black]{hyperref} \newpagestyle{miestilo}{ %-- Cabecera par y cabecera impar \sethead[\thepage][][\textsl{\chaptername\ \thechapter. \chaptertitle}] -{\textsl{Divisores Libres y Confirguraciones de Hiperplanos \ - \ Sixto José Martínez Gan}}{}{\thepage} +{\textsl{Divisores Libres y Configuraciones de Hiperplanos \ - \ Sixto José Martínez Gan}}{}{\thepage} %-- Pie de página par y pie de página impar %\setfoot[][\textsl{\thesection. \sectiontitle}][] %{}{\textsl{\thesection. \sectiontitle}}{} @@ -126,9 +126,10 @@ urlcolor=black]{hyperref} \newtheorem{teorema}{Teorema}[chapter] \newtheorem{proposicion}[teorema]{Proposición} \newtheorem{lema}[teorema]{Lema} % La misma numeración que 'teorema' -\newtheorem{definicion}[teorema]{Definición} + \theoremstyle{definition} % Entornos tipo definición +\newtheorem{definicion}[teorema]{Definición} \newtheorem*{defin}{Definición} % El asterisco hace que no se numere %\newtheorem{ejemplo}{Ejemplo} \newtheorem{ejemplo}[teorema]{Ejemplo} @@ -267,6 +268,7 @@ urlcolor=black]{hyperref} \chapter{Módulos} \section{Conceptos básicos} Los conjuntos con los que trabajaremos son una generalización de los espacios vectoriales donde los coeficientes pertenecen a un anillo en vez de a un cuerpo. A esta generalización la llamamos módulo. + \todo{Añadir referencia para todo el capítulo de módulos, etc.} \begin{definicion} Sea $A$ un anillo y $M$ un conjunto sobre el que hay definidas dos operaciones: una suma, $+: M \times M \rightarrow M$; y una multiplicación escalar, $\cdot: A \times M \rightarrow M$. El conjunto $M$ es un \modulo{A} si \begin{itemize} @@ -282,9 +284,11 @@ urlcolor=black]{hyperref} \begin{ejemplo} $\enteros^n$ es un \modulo{\enteros} con las operaciones naturales. En general, dado cualquier anillo $A$, $A^n$ es también un \modulo{A} con las operaciones componente a componente. Trivialmente, el propio $A$ es un \modulo{A}. + Un módulo sobre un cuerpo es un espacio vectorial pero no todas las propiedades de espacios vectoriales + se trasladan a módulos. La definición de espacio vectorial es más restrictiva que la de módulo, por lo que todo \nolbreaks{$C$-espacio vectorial} donde $C$ es un cuerpo es también un \modulo{C}. Por ejemplo, $\reales^n$ es un \modulo{\reales}. - A lo largo de este trabajo será de gran importancia el anillo de polinomios sobre un cuerpo, $\anillodepolinomios{K}{x_1, x_2, \dots, x_n}$, al que denotaremos \anillodepolinomios{K}{\mathbf{x}}. Sobre todo porque a partir de este construimos el conjunto de vectores de polinomios $\anillodepolinomios{K}{\mathbf{x}}^n$, que es un \modulo{\anillodepolinomios{K}{\mathbf{x}}}. + A lo largo de este trabajo será de gran importancia el anillo de polinomios $\anillodepolinomios{K}{x_1, x_2, \dots, x_n}$ sobre un cuerpo $K$, al que denotaremos \anillodepolinomios{K}{\mathbf{x}}. Sobre todo porque a partir de este construimos el conjunto de vectores de polinomios $\anillodepolinomios{K}{\mathbf{x}}^n$, que es un \modulo{\anillodepolinomios{K}{\mathbf{x}}}. \end{ejemplo} De aquí en adelante, para todos los resultados, daremos por sentado que $A$ sea un anillo y $M$ un \modulo{A}. @@ -337,7 +341,7 @@ urlcolor=black]{hyperref} Por otro lado, la aplicación $\phi: \anillodepolinomios{\complejos}{x, y} \longrightarrow \complejos$ que envía a cada polinomio $p\!\left( x, y \right)$ a $p\!\left(0, 1\right)$ es un homomorfismo de \modulo{\complejos}s. \end{ejemplo} \begin{lema} - Sean dos \modulo{A}s $M$ y $N$ y un homomorfismo $\phi$ de \modulo{A}s. $\ker \phi$ es un submódulo de $M$ e $\imagen \phi$ es un submódulo de $N$. + Sean dos \modulo{A}s $M$ y $N$ y un homomorfismo $\phi$ de \modulo{A}s. El núcleo $\ker \phi$ es un submódulo de $M$ y la imagen $\imagen \phi$ es un submódulo de $N$. \end{lema} @@ -346,7 +350,9 @@ urlcolor=black]{hyperref} \partial_i : \anillodepolinomios{K}{\mathbf{x}} &\rightarrow \anillodepolinomios{K}{\mathbf{x}} \\ p &\mapsto \diffp{p}{x_i}. \end{align*} - Con estas aplicaciones, pasamos a definir el módulo de derivaciones, el conjunto donde residen todas las posibles combinaciones lineales de derivadas parciales: + Observamos que $\partial_i$ no es un homomorfismo de \modulo{\anillodepolinomios{K}{\mathbf{x}}}. + + Con estas aplicaciones, pasamos a definir el módulo de derivaciones, el conjunto de todas las posibles combinaciones lineales de derivadas parciales: \begin{equation*} \derivaciones{\anillodepolinomios{K}{\mathbf{x}}} := \definicionconjunto{a_1\partial_1 + a_2\partial_2 + \dots + a_n\partial_n}{a_i \in \anillodepolinomios{K}{\mathbf{x}} \, \forall i}. \end{equation*} @@ -355,6 +361,7 @@ urlcolor=black]{hyperref} \derivaciones{f} := \definicionconjunto{\delta \in \derivaciones{\Kx}}{\delta\!\left( f \right) \in \engendrado{f}}. \end{equation} Puede verse que si identificamos $\partial_i$ con \vectorbasecanonica{i}, tenemos que \derivaciones{\Kx} es isomorfo a $\Kx^n$ y, por esta misma razón, \derivaciones{f} puede verse como un \submodulo{\Kx} de $\Kx^n$. + \todo{Caracterizar las derivaciones con la regla de Leibniz} \begin{definicion} Diremos que un anillo $A$ es noetheriano si todos sus ideales son finitamente generados. Diremos también que un \modulo{A} $M$ es noetheriano si todo submódulo $N \subseteq M$ está finitamente generado. Es decir, si existen $n_1, n_2, \dots, n_k$ en $N$ de forma que $N = \engendrado{n_1, n_2, \dots, n_k}$. @@ -374,6 +381,7 @@ urlcolor=black]{hyperref} \begin{equation*} \Hom{A}{M}{N} := \definicionconjunto{\phi : M \longrightarrow N}{\text{$\phi$ es un homomorfismo de \modulo{A}s}}. \end{equation*} + \todo{Decir que es un $A$-módulo} \end{definicion} \begin{definicion} Se define el módulo dual \dual{M} como el conjunto de homomorfismos de \modulo{A}s de $M$ en $A$: @@ -477,6 +485,8 @@ urlcolor=black]{hyperref} \end{equation} La función $\Phi$ es una inclusión, $\xi$ envía una derivación $\delta$ a su evaluación en $f$ ---$\delta\!\left( f \right)$--- y $\pi$ es simplemente la aplicación canónica de $J$ a \cociente{J}{J \cap \engendrado{f}}. + \todo{Decir que es isomorfo a $J+(f)/(f)$} + \begin{wrapfigure}[3]{r}{0.3\textwidth} \vspace{-1.3cm} \begin{equation*} @@ -528,7 +538,7 @@ urlcolor=black]{hyperref} \end{ejemplo} Este ejemplo también ilustra un cambio de perspectiva que será muy importante a lo largo de toda esta sección: la división deja de ser importante por los valores exactos que arroja y su relevancia es más bien cualitativa. Con esto nos referimos a que, como tal, los valores concretos del cociente y el resto no tienen ninguna relevancia a la hora de determinar si $p$ pertenece o no a \engendrado{f, g}. Conocidas las propiedades de este algoritmo de la división en \anillodepolinomios{\reales}{x}, solo nos interesa si el resto es $0$ o no, puesto que es lo que determina si $p$ pertenece o no a \engendrado{f, g}. En base a esto, nuestra motivación para la división debe cambiar de ser <> a un más abstracto <>. - Visto así, el ejemplo \ref{Ejemplo Pertenencia Ideal Una Variable} supone una situación idílica: por ser un dominio de ideales principales, sabemos que vamos a poder representar a cualquier conjunto generador con un solo elemento; por lo tanto, sobre cualquier conjunto generador siempre habrá definida una división, que consistirá en dividir como ya conocemos por el elemento generador. Además, por el Teorema de la División, conocemos que el representante $r$ que satisfaga $\grado r < \grado d$ es único, lo cual nos permite discernir si $p$ está en \engendrado{f, g} viendo si $r = 0$ o no. + Visto así, el Ejemplo \ref{Ejemplo Pertenencia Ideal Una Variable} supone una situación idílica: por ser un dominio de ideales principales, sabemos que vamos a poder representar a cualquier conjunto generador con un solo elemento; por lo tanto, sobre cualquier conjunto generador siempre habrá definida una división, que consistirá en dividir como ya conocemos por el elemento generador. Además, por el Teorema de la División, conocemos que el representante $r$ que satisfaga $\grado r < \grado d$ es único, lo cual nos permite discernir si $p$ está en \engendrado{f, g} viendo si $r = 0$ o no. En espacios más complicados, muchas de estas buenas propiedades desaparecen e impiden extender el proceso anterior. \begin{description} @@ -551,6 +561,7 @@ urlcolor=black]{hyperref} x^\mathbf{\alpha} > x^\mathbf{\beta} \implies x^{\mathbf{\gamma}} x^\mathbf{\alpha} > x^{\mathbf{\gamma}}x^\mathbf{\beta} \end{equation*} para todo $\mathbf{\alpha}, \mathbf{\beta}$ y $\mathbf{\gamma}$ en $\naturales^{n}$. + \todo{Decir que el orden es bueno o hablar de órdenes locales y globales y mixtos.} \end{definicion} \begin{ejemplo}[Orden lexicográfico] Una posibilidad de ordenar los monomios es como si estos fueran palabras en el diccionario. Por ejemplo, podemos ver $x^3y^2z^4$ como $xxxyyzzzz$ y en $x^2y^4z^4$ como $xxyyyyzzzz$. Así, $x^3y^2z^4 > x^2y^4z^4$. De manera más formal, @@ -611,6 +622,7 @@ urlcolor=black]{hyperref} 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} + \todo{diferenciar vectores y polinomios. Por ejemplo $f$ es un vector pero cociente $c$ es un polinomio.} Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^n$ junto con un orden monomial sobre este. \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. @@ -733,7 +745,7 @@ urlcolor=black]{hyperref} 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. Por simplicidad, siempre nos referiremos a aquel que no está multiplicado por constante.}, definimos la siguiente expresión: + Recordemos el problema que nos habíamos encontrado en el Ejemplo~\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 $i = j$;} \\ @@ -864,7 +876,7 @@ urlcolor=black]{hyperref} \begin{align*} xp_1'a_1 + xp_2'a_2 + \dots + xp_k'a_k &= x \\ x\left( p_1'a_1 + p_2a_2' + \dots + p_k'a_k \right) &= x, \\ - \intertext{$\Kx$ es un dominio de integridad por lo que podemos simplificar,} + \intertext{El anillo $\Kx$ es un dominio de integridad por lo que podemos simplificar,} p_1'a_1 + p_2a_2' + \dots + p_k'a_k &= 1. \end{align*} Esto último es una contradicción, ya que $1 \notin \engendrado{x, y}$. @@ -876,7 +888,7 @@ urlcolor=black]{hyperref} Un polinomio $p \in \Kx$ se dice libre de cuadrados si, en caso de poder expresarlo de la forma $p = a^2b$, se tiene que $a$ es una unidad de \Kx. \end{definicion} \begin{lema} \label{Lema MCD Parciales} - Un polinomio $p \in \Kx$ es libre de cuadrados si y solo si $\mcd{\diffp{p}{x_1}, \diffp{p}{x_2}, \dots, \diffp{p}{x_n}}$ es una unidad de \Kx. + Un polinomio $p \in \Kx$ es libre de cuadrados si y solo si $\mcd{p, \diffp{p}{x_1}, \diffp{p}{x_2}, \dots, \diffp{p}{x_n}}$ es una unidad de \Kx. \end{lema} El criterio de Saito nos da una forma bastante rápida de determinar si \derivaciones{f} es o no libre. Nos permitimos la licencia de denotar por $\delta_i$ tanto a la derivación como al vector de coeficientes asociado en $\Kx^n$. @@ -945,6 +957,8 @@ urlcolor=black]{hyperref} \end{ejemplo} Estas mismas ideas pueden extenderse a aplicaciones \nolbreaks{$n$-lineales} para construir productos como $\bigotimes^n M = M \otimes M \otimes \dots \otimes M$ y, consecuentemente a aplicaciones \nolbreaks{$n$-lineales} alternadas para construir $\bigwedge^n M := M \wedge M \wedge \dots \wedge M$. + \todo{Añadir la relación con el determinante y decir que si $n$ es mayor que el número de generadores del módulo entonces el producto exterior es cero} + Aunque pueda parecer que los productos exteriores hacen un espacio <>, esto no tiene por qué ser así. \begin{ejemplo} @@ -986,7 +1000,7 @@ urlcolor=black]{hyperref} \subsection{El papel del rango} En un módulo libre, la noción de rango es simplemente la cantidad de elementos que tiene la base. Pero como ya hemos visto a lo largo de este trabajo, hay módulos que no tienen por qué ser libres. Por ende, necesitamos generalizar la noción de rango para poder tratar con estos. \begin{definicion} - Sea $A$ un anillo y $\sim$ la relación de equivalencia sobre $A\times \left( A \setminus \left\lbrace 0 \right\rbrace \right)$ dada por $\left( a, b \right) \sim \left( \alpha, \beta \right) \iff a\beta = \alpha b$. A la clase de equivalencia de $\left( a, b \right)$ la denotaremos $\frac{a}{b}$. El conjunto + Sea $A$ un dominio de integridad y $\sim$ la relación de equivalencia sobre $A\times \left( A \setminus \left\lbrace 0 \right\rbrace \right)$ dada por $\left( a, b \right) \sim \left( \alpha, \beta \right) \iff a\beta = \alpha b$. A la clase de equivalencia de $\left( a, b \right)$ la denotaremos $\frac{a}{b}$. El conjunto \begin{equation*} \cuerpofracciones{A} := \cociente{A\times \left( A \setminus \left\lbrace 0 \right\rbrace \right)}{\sim} \end{equation*} @@ -994,8 +1008,9 @@ urlcolor=black]{hyperref} \begin{equation*} \frac{a}{b} + \frac{\alpha}{\beta} := \frac{a\beta + \alpha b}{b\beta} \quad \text{y} \quad \frac{a}{b} \cdot \frac{\alpha}{\beta} := \frac{a\alpha}{b\beta} \end{equation*} - forma un cuerpo al cual llamaremos cuerpo de fracciones de $A$. + forma un cuerpo al cual llamaremos cuerpo de fracciones de $A$. El homomorfismo natural $A \hookrightarrow \cuerpofracciones{A}$ es inyectivo. \end{definicion} + \todo{Definir la extensión de escalares como $M \otimes \cuerpofracciones{A}$ y decir cuándo es inyectiva.} \begin{definicion} Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene rango $r$ si $M \otimes \cuerpofracciones{A}$ es un \modulo{\cuerpofracciones{A}} libre de rango $r$. \end{definicion} @@ -1043,7 +1058,7 @@ urlcolor=black]{hyperref} Recordemos lo que ocurría en \ref{Aparición Determinante Función Exterior}. De forma similar, aquí tendremos que $\bigwedge^n \Phi$ se corresponde con multiplicar por el escalar $p$ ---o $up$ donde $u$ es una unidad de \Kx---. Sin embargo, el hecho de que no tengamos base de \derivaciones{f} impide que se pueda ver $\Phi$ como una matriz. Aun así, nos permitimos el abuso de notación de llamar $\det\Phi := p$, pero no es más que una convención, no se trata de un determinante de una matriz como ocurre en el caso de las aplicaciones entre espacios vectoriales. Tenemos pues que $\bigwedge^n\derivaciones{f} = \engendrado{\det\Phi \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$. \subsection{\derivaciones{f} libre implica $\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$} - Ahora, por un lado, definamos $N := \engendrado{\delta_1, \delta_2, \dots, \delta_n}$ como un \submodulo{\Kx} de \derivaciones{f} donde los $\delta_i$ forman una base de \derivaciones{f} y consideremos la inclusión $N \xrightarrow{\iota} \derivaciones{f}$. Para aclararnos algo más, escribimos el siguiente esquema de los homomorfismos ---¡ojo!, esto no es una secuencia exacta---: + Ahora, por un lado, definamos $N := \engendrado{\delta_1, \delta_2, \dots, \delta_n}$ como un \submodulo{\Kx} de \derivaciones{f} donde los $\delta_i$ forman una base de \derivaciones{f} y consideremos la inclusión $N \xrightarrow{\iota} \derivaciones{f}$. Para aclararnos algo más, escribimos el siguiente esquema de los homomorfismos ---¡ojo!, esto no es una sucesión exacta---: \begin{equation}\label{Secuencia Demostración Saito} N = \engendrado{\delta_1, \delta_2, \dots, \delta_n} \xrightarrow{\phantom{s}\iota\phantom{s}} \derivaciones{f} \xrightarrow{\Phi\phantom{s}} \derivaciones{\Kx}. \end{equation} @@ -1073,7 +1088,7 @@ urlcolor=black]{hyperref} \end{equation*} Esto quiere decir que $\det\Phi \vert f\left( \diffp{f}{x_i} \right)^{n-1}$ para todo $i$. Asimismo, el máximo común divisor de todos ellos también será divisible por $\det\Phi$: \begin{equation*} - \det\Phi \left\lvert f\mcd{\left( \diffp{f}{x_1} \right)^{n-1}, \left(\diffp{f}{x_2} \right)^{n-1}, \dots, \left( \diffp{f}{x_n} \right)^{n-1}}\right.. + \det\Phi \left\lvert f\mcd{f^{n-1}, \left( \diffp{f}{x_1} \right)^{n-1}, \left(\diffp{f}{x_2} \right)^{n-1}, \dots, \left( \diffp{f}{x_n} \right)^{n-1}}\right.. \end{equation*} No nos olvidemos de que gracias al lema \ref{Lema MCD Parciales} sabemos que ese máximo común divisor es $1$. Por esta razón, $\det\Phi$ debe dividir a $f$. @@ -1386,6 +1401,8 @@ x2z+y3,3z,-3y2,-2z,0, Como $\Kx^n$ es noetheriano, podemos asumir desde el principio que $G = \conjunto{g_1, g_2, \dots, g_k}$ \todo[inline]{Terminar} + \bibliography{bibliografia} + \bibliographystyle{plain} \nocite{*} - \printbibliography + %\printbibliography \end{document} \ No newline at end of file