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--- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -50,8 +50,13 @@ \usepackage{enumitem} % Bibliografía. +\newif\ifbibtex +\bibtextrue +\ifbibtex +\else \usepackage[backend=biber]{biblatex} \addbibresource{bibliografia.bib} +\fi \usepackage{algpseudocodex} \algrenewcommand\algorithmicwhile{\textbf{mientras}} @@ -215,8 +220,8 @@ urlcolor=black]{hyperref} \noindent \begin{figure}[h] \centering -\includegraphics[width=100mm]{Identidad_Corporativa_Universidad_de_Zaragoza/ciencias.png}\\% -\includegraphics[width=100mm]{Identidad_Corporativa_Universidad_de_Zaragoza/logoUZ.png} +\includegraphics[width=100mm]{Imagenes_y_Figuras/Identidad_Corporativa_Universidad_de_Zaragoza/ciencias.png}\\% +\includegraphics[width=100mm]{Imagenes_y_Figuras/Identidad_Corporativa_Universidad_de_Zaragoza/logoUZ.png} \end{figure} \vfill @@ -267,13 +272,13 @@ urlcolor=black]{hyperref} \end{equation*} Es decir, con estas derivaciones de \derivaciones{f} estamos obteniendo campos de vectores tangentes a la hipersuperficie $f=0$. Este conjunto será el principal objeto de estudio de este documento. - A primera vista, es apreciable la relación con los vectores, pero no se tratará de espacios vectoriales, sino de una generalización de estos a coeficientes en un anillo: los módulos. Desarrollaremos una descripción de este tipo de objeto en el capítulo \ref{Capítulo de Módulos}. A continuación, notemos que la condición $\delta\!\left( f \right) = hf$ puede reescribirse como + A primera vista, es apreciable la relación con los vectores, pero no se tratará de espacios vectoriales, sino de una generalización de estos a coeficientes en un anillo: los módulos. Desarrollaremos una descripción de este tipo de objeto en el Capítulo \ref{Capítulo de Módulos}. A continuación, notemos que la condición $\delta\!\left( f \right) = hf$ puede reescribirse como \begin{equation*} \sum_{i = 1}^{n}a_i \diffp{f}{x_i} = hf \iff \sum_{i = 1}^{n}a_i \diffp{f}{x_i} - hf = 0 \iff \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} \vectorvertical{a_1 \\ a_2 \\ \vdots \\ a_n \\ -h} = 0. \end{equation*} - El vector de coeficientes expresa una <> entre los elementos de \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} al igual que una solución de un sistema lineal homogéneo expresa una relación entre las columnas de una matriz. El conjunto de todas estas relaciones o soluciones del sistema se conoce como el módulo de sicigias. Para poder calcularlo, necesitaremos las Bases de \Groebner: una herramienta que nos ayuda a la hora de determinar si un elemento pertenece o no a un ideal de \Kx, pero en este caso generalizada también a módulos. Todo esto lo desarrollaremos en el capítulo \ref{Capítulo de Bases de Groebner y Sicigias}. + El vector de coeficientes expresa una <> entre los elementos de \vectorhorizontal{\diffp{f}{x_1}, \diffp{f}{x_2}, \dots, \diffp{f}{x_n}, f} al igual que una solución de un sistema lineal homogéneo expresa una relación entre las columnas de una matriz. El conjunto de todas estas relaciones o soluciones del sistema se conoce como el módulo de sicigias. Para poder calcularlo, necesitaremos las Bases de \Groebner: una herramienta que nos ayuda a la hora de determinar si un elemento pertenece o no a un ideal de \Kx, pero en este caso generalizada también a módulos. Todo esto lo desarrollaremos en el Capítulo \ref{Capítulo de Bases de Groebner y Sicigias}. - Por último, una de las principales dificultades que presentan los módulos son las bases. Así como en cualquier espacio vectorial sabemos que siempre hay una base y que todo subespacio también la tiene, no tiene por qué ser así en el caso de módulos. Para poder discernir en particular si \derivaciones{f} admite una base, la herramienta a utilizar es el Criterio de Saito. Este resultado afirma que \derivaciones{f} admite una base si y solo si se pueden encontrar $\delta_1, \delta_2, \dots, \delta_n \in \derivaciones{f}$ de forma que ---viéndolos como vectores de $\Kx^n$--- se cumpla + Por último, una de las principales dificultades que presentan los módulos son las bases\todo{diría saber si son libres y encontrar una base}. Así como en cualquier espacio vectorial sabemos que siempre hay una base y que todo subespacio también la tiene, no tiene por qué ser así en el caso de módulos. Para poder discernir en particular si \derivaciones{f} admite una base, la herramienta a utilizar es el Criterio de Saito. Este resultado afirma que \derivaciones{f} admite una base si y solo si se pueden encontrar $\delta_1, \delta_2, \dots, \delta_n \in \derivaciones{f}$ de forma que ---viéndolos como vectores de $\Kx^n$--- se cumpla \begin{equation*} \det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf\quad \text{donde $u$ es una unidad de \Kx}. \end{equation*} @@ -282,16 +287,16 @@ urlcolor=black]{hyperref} De hecho, este concepto de que \derivaciones{f} admita una base no se comprende del todo bien actualmente. Por ejemplo, si tomamos $f := x^2z + y^3 \in \anillodepolinomios{K}{x,y,z}$ nos encontramos con la superficie \begin{figure}[h] \centering - \includegraphics[scale=0.3]{primerasuperficie.png}\\ + \includegraphics[scale=0.3]{Imagenes_y_Figuras/primerasuperficie.png}\\ $x^2z + y^3 = 0$ \end{figure} En este caso, \derivaciones{x^2z + y^3} no es libre. Puede encontrarse el cálculo de todo lo comentado hasta ahora para este caso en el apéndice \ref{Apéndice Cálculo}. Sin embargo, añadiendo un plano que cree una especie de <> a esas esquinas que aparecen, que se consigue multiplicando por $x$, nos queda la siguiente superficie: \begin{figure}[h] \centering - \includegraphics[scale=0.3]{segundasuperficie.png}\\ + \includegraphics[scale=0.3]{Imagenes_y_Figuras/segundasuperficie.png}\\ $x\left( x^2z + y^3 \right) = 0$ \end{figure} - + \todo[inline]{Si puedes, el plano de otro color y algo traslúcido.} Y en este caso, \derivaciones{f} sí que admite una base. Para garantizarlo, hacemos uso del Criterio de Saito con \begin{align*} \delta_1 &:= \left( x^2z + y^3 \right)\diffp{}{z},\\ @@ -307,6 +312,7 @@ urlcolor=black]{hyperref} \end{pmatrix} = -y\left[ x\left( x^2z + y^3 \right) \right] = -yf. \end{equation*} El criterio de Saito se cumple, y por lo tanto, $\delta_1$, $\delta_2$ y $\delta_3$ forman una base de \derivaciones{f}. + \todo[inline]{Haz referencia al apartado donde está el criterio} \chapter{Módulos} \label{Capítulo de Módulos} \section{Conceptos básicos} @@ -477,7 +483,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \end{align*} Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $\Psi$ es un isomorfismo. \begin{ejemplo} -\todo{Cambiaría la estructura de este ejemplo usando bases duales y lo simplificaría.} +\todo[inline]{Cambiaría la estructura de este ejemplo usando bases duales y lo simplificaría.} Veamos que todo espacio vectorial de dimensión finita es reflexivo. Sea $V$ un $K$-espacio vectorial de dimensión $n$ y con base $\conjunto{v_1, v_2, \dots, v_n}$. El conjunto \Hom{K}{V}{K} se compone de las aplicaciones lineales de $V$ en $K$. Cada elemento $\phi$ de \Hom{K}{V}{K} puede representarse como la multiplicación por una matriz, puesto que \begin{equation*} \phi\!\left( v \right) = \productoescalar{\vectorhorizontalcuatro{\phi\!\left( v_1 \right)}{\phi\!\left( v_2 \right)}{\hdots}{\phi\!\left( v_n \right)}}{v}; @@ -597,7 +603,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du Una introducción a los módulos proyectivos puede encontrarse en \cite[sección 1.2]{LIBROPROYECTIVOS}. \section{Producto Exterior} - \todo{Si haces lo de la bilineal lo puedes usar aquí} + \todo[inline]{Si haces lo de la bilineal lo puedes usar aquí} De cara a la demostración del Criterio de Saito, utilizaremos una operación que nos será de gran ayuda: el producto exterior. Para entender esta, necesitaremos entender primero el producto tensorial. En esencia, el producto tensorial aparece en el siguiente contexto: supongamos que $M$, $N$ y $O$ son \modulo{A}s y tenemos una aplicación $b : M \times N \rightarrow O$ que es bilineal, o sea, que $b\left( \bullet, n \right)$ y $b\left( m, \bullet \right)$ son homorfismos de \modulo{A}s para cualquier $m \in M$ y $n \in N$; nos gustaría encontrar un \modulo{A} $E$ a partir de $M \times N$ en el cual exista un homomorfismo de \modulo{A}s $b' : E \rightarrow O$ que se comporte como $b$. En resumen, queremos que para cada $\left( m, n \right)$ en $M \times N$ exista un elemento $m * n$ en $E$ de forma que $b'\left( m*n \right) = b\left( m, n \right)$. Esta es la motivación de la construcción del producto tensorial. \begin{definicion} Sea $A$ un anillo y $M$ y $N$ dos \modulo{A}s. Tomemos como $E$ al \modulo{A} definido como $\bigoplus_{\substack{m \in M \\ n \in N}} A \cdot \vectorbasecanonica{m, n}$ donde \vectorbasecanonica{m, n} debe entenderse como la tupla $\left( m, n \right)$ sin ningún tipo de operación asociada, es decir, no debemos pensar que $\vectorbasecanonica{m, n} + \vectorbasecanonica{m', n'} = \vectorbasecanonica{m+m', n+n'}$ al igual que no pensamos que $\euno + \edos = \etres$. Ahora, consideremos dentro de $E$ el \submodulo{A} $S$ definido por @@ -610,7 +616,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \end{gathered} \right\rvert a \in A, m \in M, n \in N}. \end{equation*} Definimos el p\emph{roducto tensorial} de $M$ por $N$ como $M \otimes N := \cociente{E}{S}$ y denotaremos a la clase de equivalencia de cada $\vectorbasecanonica{m, n}$ como $m \otimes n$ - \todo{Propiedad universal} + \todo[inline]{Propiedad universal} \end{definicion} \begin{ejemplo} Quizás el ejemplo más natural sea el producto escalar. Así pues, tomemos la aplicación $\productoescalarantilambda{\bullet}{\bullet} : \reales^3 \times \reales^3 \rightarrow \reales$, que es bilineal. En este caso, todo elemento de $\reales^3 \otimes \reales^3$ es de la siguiente forma ---entendiendo $S$ como en la definición de antes---: @@ -620,7 +626,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du &= ad \euno\otimes\euno + ae\euno\otimes\edos + af\euno\otimes\etres + bd\edos\otimes\euno \\ &\phantom{text} + be\edos\otimes\edos + bf\edos\otimes\etres + cd\etres\otimes\euno + ce\etres\otimes\edos + cf\etres\otimes\etres + S \end{aligned} \end{equation} - (De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente.) + De aquí en adelante omitiremos los <<$+S$>> o similares sumas por el \submodulo{A} que estemos haciendo el cociente. Vemos pues que \definicionconjunto{\vectorbasecanonica{i} \otimes \vectorbasecanonica{j}}{ i, j \in \conjunto{1, 2, 3}} es una base de $\reales^3 \otimes \reales^3$. Por lo tanto, para encontrar una aplicación lineal $\phi : \reales^3 \otimes \reales^3 \rightarrow \reales$ que coincida con el producto escalar debemos dar las imágenes de la base. En este caso, \begin{equation*} @@ -686,13 +692,14 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \begin{proposicion} \label{Aparición Determinante Función Exterior} Sea $\phi : V \rightarrow W$ como en el párrafo anterior. Entonces, la aplicación lineal $\bigwedge^n \phi : \bigwedge^n V \rightarrow \bigwedge^n W$ envía \begin{equation*} - v_1\wedge v_2\wedge \dots \wedge v_n \mapsto \det\Phi w_1\wedge w_2\wedge \dots \wedge w_n. + v_1\wedge v_2\wedge \dots \wedge v_n \mapsto \det\Phi\, w_1\wedge w_2\wedge \dots \wedge w_n. \end{equation*} \end{proposicion} \begin{proof} Es suficiente con el razonamiento del párrafo anterior y un desarrollo similar al de \eqref{Producto Exterior Como Determinante}, que como puede verse, es el determinante de una matriz $3\times 3$: \begin{equation} \funcionexterior{n}{\phi}{v_1\wedge v_2\wedge \dots \wedge v_n} = P_1 \wedge P_2 \wedge \dots \wedge P_n = \left( \det P \right) w_1\wedge w_2\wedge \dots \wedge w_n. + \qedhere \end{equation} \end{proof} @@ -712,7 +719,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du El cuerpo de fracciones de un anillo nos permite llevar un \modulo{A} a un \espaciovectorial{\cuerpofracciones{A}}. \begin{definicion} - Llamaremos extensión de escalares del \modulo{A} $M$ al conjunto $M \otimes \cuerpofracciones{A}$. + Llamaremos \emph{extensión de escalares} del \modulo{A} $M$ al \espaciovectorial{\cuerpofracciones{A}} $M \otimes \cuerpofracciones{A}$. \end{definicion} \begin{lema} La extensión de escalares de un \modulo{A} $M$ es un \espaciovectorial{\cuerpofracciones{A}}. Además, la aplicación $\varphi: M \rightarrow M \otimes \cuerpofracciones{A}$ dada por $m \mapsto m \otimes 1$ es inyectiva si y solo si $M$ es libre de torsión. @@ -728,13 +735,13 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du &\implies& am \neq 0 & \end{array} \end{equation*} - \todo{La otra implicación.} + \todo{La otra implicación. Creo que quedará más corto definiendo la extensión de escalares como fracciones} \end{proof} \begin{definicion} - Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene rango $r$ si $M \otimes \cuerpofracciones{A}$ es un \espaciovectorial{\cuerpofracciones{A}} de dimensión $r$. + Sea $M$ un \modulo{A} finitamente generado. Diremos que $M$ tiene \emph{rango} $r$ si $M \otimes \cuerpofracciones{A}$ es un \espaciovectorial{\cuerpofracciones{A}} de dimensión $r$. \end{definicion} - Recordemos que ya en \ref{Lema Rango Producto Exterior} había un resultado sobre el rango del producto exterior de los espacios vectoriales. Generalizamos esto a los módulos. + Recordemos que ya en el Lema~\ref{Lema Rango Producto Exterior} había un resultado sobre el rango del producto exterior de los espacios vectoriales. Generalizamos esto a los módulos. \begin{lema} \label{Lema Rango Producto Exterior Módulos} Sea $M$ un \modulo{A} de rango $n$ y $s \in \naturales$. Entonces, $\bigwedge^s M$ es un \modulo{A} de rango $\binom{n}{s}$. \end{lema} @@ -764,6 +771,7 @@ Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\du \text{$M$ es reflexivo} \end{aligned}\right\rbrace \implies \text{$M$ es un módulo libre} \end{gather} + \todo{Esto igual sí que tiene que estar en el último capítulo y aquí alguna cosa de reflexivos.} \chapter{Bases de Gröbner y Sicigias} \label{Capítulo de Bases de Groebner y Sicigias} @@ -1035,7 +1043,7 @@ distintos órdenes 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 el apéndice \ref{Apéndice Demostración del Criterio de Buchberger}.\todo[inline]{Creo que es mejor gastar energía en otras cosas.} \end{proof} Un ejemplo de cálculo de base de \Groebner{} puede encontrarse en apéndice \ref{Sección A mano}. @@ -1144,13 +1152,14 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{definicion} Un polinomio $p \in \Kx$ se dice \emph{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} + \todo{Al principio de 2} \begin{lema} \label{Lema MCD Parciales} 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 $\Kx^n$. \begin{teorema}[Criterio de Saito] \label{Criterio de Saito} - Sea $f$ un polinomio libre de cuadrados de \Kx{} y $\delta_1, \delta_2, \dots, \delta_n$ derivaciones de \derivaciones{f}. Entonces, \derivaciones{f} es libre si y solo si $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ donde $u$ es una unidad de $K$, es decir, un elemento de $K \setminus \left\lbrace 0 \right\rbrace$. + Sea $f$ un polinomio libre de cuadrados de \Kx{} y $\delta_1, \delta_2, \dots, \delta_n$ derivaciones de \derivaciones{f}. Entonces, \derivaciones{f} es libre si y solo si $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ donde $u$ es una unidad de $\Kx$, es decir, un elemento de $K \setminus \left\lbrace 0 \right\rbrace$. \end{teorema} En este capítulo explicaremos la demostración de este resultado que aparecen en \cite{VALLES}. Pero primero, deberemos ilustrar algunas construcciones que se utilizan en dicha demostración. @@ -1170,7 +1179,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 El Lema~\ref{Lema Rango Producto Exterior Módulos} nos garantiza que $\bigwedge^n \derivaciones{f}$ es un \modulo{\Kx} reflexivo de rango $1$. Siguiendo el esquema en \ref{Esquema Demostración}, tenemos que $\bigwedge^n \derivaciones{f}$ es también libre de torsión y, por lo tanto, $\bigwedge^n \derivaciones{f}$ es isomorfo a un ideal de \Kx. Por último, aplicando la última de las implicaciones, tenemos que $\bigwedge^n \derivaciones{f}$ es un \modulo{\Kx} libre y además de rango $1$. Sumado a esto tenemos que $\bigwedge^n \derivaciones{f}$ es un \submodulo{\Kx} de $\bigwedge^n \derivaciones{\Kx} = \engendrado{\partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$, por lo tanto, $\derivaciones{f} = \engendrado{p \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$ para algún $p$. Nótese que este $p$ está bien definido salvo por alguna unidad de \Kx, es decir, tanto nos daría poner $p$ que $kp$ siempre y cuando exista $k^{-1} \in \Kx$. - Ahora bien, recordemos la sucesión exacta de \ref{Secuencia Exacta Definición Módulo Derivaciones}. Podemos aplicar productos exteriores a los elementos, pero esto hace que la sucesión pueda dejar de ser exacta. Sin embargo, habrá un homomorfismo + Ahora bien, recordemos la sucesión exacta \eqref{Secuencia Exacta Definición Módulo Derivaciones}. Podemos aplicar productos exteriores a los elementos, pero esto hace que la sucesión pueda dejar de ser exacta. Sin embargo, habrá un homomorfismo \begin{equation*} \bigwedge^n \derivaciones{f} \xrightarrow{\bigwedge^n \Phi} \bigwedge^n \derivaciones{\Kx}. \end{equation*} @@ -1188,7 +1197,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 Por lo tanto, podemos pensar en que la aplicación $\Phi \circ \iota$ envía a cada $\delta_i$ a $\vectorhorizontalcuatro{\partial_1}{\partial_2}{\dots}{\partial_n} A_i$ donde $A_i$ es la \nolbreaks{$i$-ésima} columna de $A$. Es decir, que las columnas de $A$ se entienden como las coordenadas de los $\delta_i$ en función de la base \conjunto{\partial_1, \partial_2, \dots, \partial_n}\footnote{Es decir, que $\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = \det A$.}. Pensemos ahora en la aplicación \begin{equation*} - \begin{tikzcd}[sep=small] + \begin{tikzcd}[row sep=0pt] N \arrow[r, "\bigwedge^n \left( \Phi \circ \iota \right)"] & \derivaciones{\Kx} \\ \delta_1 \wedge \delta_2 \wedge \dots \wedge \delta_n \arrow[r, mapsto] & \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n \det A \end{tikzcd} @@ -1324,7 +1333,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} Lo primero que tendremos que hacer será calcular el submódulo de sicigias \syz{2xz, 3y^2, x^2, x^2z + y^3} siguiendo el método indicado en \ref{Teorema Calculo Sicigias}. Comenzamos ampliando los distintos elementos a vectores para que sus componentes extra <> de las anulaciones que hacemos. Así, tomamos \begin{equation*} - 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}. + 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{equation} \label{Ejemplo División Compuesta} @@ -1345,14 +1354,14 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{blockarray} \end{equation} Por ser esta la primera división con más de un divisor que hacemos, la desglosaremos algo más siguiendo la notación del algoritmo en \ref{Algoritmo de la División Compuesta}. - \begin{description} + \begin{description}[wide, leftmargin=0pt,itemindent=6pt] \item[1\textsuperscript{a} iteración: ] $\LT{D} = x^2z\euno$ es divisible por $\LM{g_1}$ y $\LM{g_3}$. Escogemos, por pura preferencia personal, $g_1$ y reducimos ---la primera resta---. Después de reducir, $\LT{D} = y^3\euno$ es divisible por $\LM{g_2}$ así que reducimos por este. Tras esta reducción, $\LT{D} = -\frac{1}{2}x\edos$, que no es divisible por ninguno de los monomios líderes de $g_1$, $g_2$ y $g_3$. Se lo sustraemos a $D$ y lo añadimos al resto. Esto deja $D = -\frac{1}{3}y\etres + \ecinco$, $c_1 = \frac{1}{2}x$, $c_2 = \frac{1}{3}y$, $c_3 = 0$ y $r = -\frac{1}{2}x\edos$. \item[2\textsuperscript{a} iteración: ] $\LT{D} = -\frac{1}{3}y\etres$ no es divisible por ninguno de los términos líderes de los divisores. Así pues, no reducimos y sustraemos este término de $D$ para pasarlo a $r$. Esto deja $D = \ecinco$, $r = -\frac{1}{2}x\edos - \frac{1}{3}y\etres$ y los cocientes como antes. \item[3\textsuperscript{a} iteración: ] $\LT{D} = \ecinco$ y nos encontramos con la misma situación que antes. Al sustraer este de $D$, $D$ pasa a ser $0$, por lo que el algoritmo termina. Nos queda $r = -\frac{1}{2}x\edos - \frac{1}{3}y\etres + \ecinco$. \end{description} De aquí en adelante, no escribiremos todas las <> desde el principio, sino que las iremos añadiendo conforme las necesitemos. Tampoco seguiremos tachando ni indicando hasta dónde alcanza cada iteración para descargar las divisiones. - El resto de la anterior división 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 + El resto de la anterior división 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 \begin{equation*} g_4 := -6\left( -\frac{1}{2}x\edos - \frac{1}{3}y\etres +\ecinco\right) = \vectorvertical{0 \\ 3x \\ 2y \\ 0 \\ -6}. \end{equation*} @@ -1553,8 +1562,11 @@ x2z+y3,3z,-3y2,-2z,0, a = h + \sum \end{equation*} \todo{Completar la demostración.} - %\bibliography{bibliografia} - %\bibliographystyle{plain} +\ifbibtex + \bibliography{bibliografia} + \bibliographystyle{plain} +\else \nocite{*} \printbibliography +\fi \end{document}

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