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a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index c13e1fb..9df446b 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -204,6 +204,9 @@ urlcolor=black]{hyperref} \newcommand{\dertheta}[2]{\ensuremath{\theta_{#1, #2}}} \newcommand{\Groebner}{Gr\"obner} +\newlength{\ahorroecuacion} +\setlength{\ahorroecuacion}{-0.2cm} + \makeatletter \long\def\@makecaption#1#2{ \vskip\abovecaptionskip @@ -273,7 +276,6 @@ urlcolor=black]{hyperref} \selectlanguage{english} \chapter{Abstract} - \section*{English} This work is framed within the context of Terao’s Conjecture for hyperplane arrangements. In this conjecture, Terao asserts that the freeness of a hyperplane arrangement depends solely on its intersections and their degrees, i.e. that the dependency is purely combinatorial. To reach this conclusion, it is necessary to construct free divisors. In our case, in a less general space, these divisors materialize as the module of polynomial derivations tangent to a function. Thus, part of the path toward Terao’s Conjecture involves understanding these divisors and, in particular, determining when they are free. Given the vastness of this topic and that the dimensions of this work are not enough to make all of it fit, we will only focus on said two objectives. The first problem will be solved in the second section with the introduction of syzygies and the second problem in the third section using Saito’s Criterion. The structure of our document is divided into three. In the first part, we will familiarize ourselves with the type of object we are working with: the module. We will explore its basic properties, its relationships with similar spaces such as the vector space, and develop with moderate deepness some specific aspects more closely related to the context in which we find ourselves. @@ -370,15 +372,13 @@ urlcolor=black]{hyperref} \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 \emph{\modulo{A}} si - \begin{itemize} - \item $(M, +)$ es un grupo abeliano y - \item se cumple que $\left\lbrace \begin{aligned} - &\left( a + b \right) \cdot m = a\cdot m + b \cdot m, \\ - &a \cdot \left( m+n \right) = a\cdot m + a\cdot n, \\ - &\left( ab \right) \cdot m = a\cdot \left( b\cdot m \right) \;\; \text{y} \\ - &1 \cdot m = m - \end{aligned} \right. \quad \text{para todo $a$ y $b$ en $A$ y $m$ y $n$ en $M$.}$ - \end{itemize} + \vspace{0cm}\begin{equation*} + (M, +) \ \ \text{es un grupo abeliano y} \ \ \begin{cases} + \left( a + b \right) \cdot \left( m + m' \right) = a\cdot m + a \cdot m' + b \cdot m + b \cdot m', \\ + \left( ab \right) \cdot m = a\cdot \left( b\cdot m \right) \;\; \text{y} \\ + 1 \cdot m = m + \end{cases} \!\!\!\!\!\!\forall a,b \in A, \ \forall m,m'\in M. + \vspace{0cm}\end{equation*} \end{definicion} \begin{ejemplo} El producto cartesiano $\enteros^n$ 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}. @@ -937,7 +937,7 @@ Ahora sí, tenemos la siguiente propiedad universal que buscábamos al principio \end{equation*} Ahora responder la pregunta es mucho más sencillo, ya que $p \in \engendrado{f, g}$ si y solo si $p$ es un múltiplo de $x^2 -5x + 6$. Y eso podemos saberlo utilizando el algoritmo de la división para polinomios de una variable que conocemos: \begin{equation*} - p = \underbrace{\left( 3x^5 + 15x^4 + 61x^3 + 215x^2 + 709x + 2252 \right)}_{\text{cociente $q$}} \underbrace{\left( x^2 - 5x + 6 \right)}_{\text{divisor $d$}} + \underbrace{7007x - 13511}_{\text{resto $r$}}. + p = \left( 3x^5 + 15x^4 + 61x^3 + 215x^2 + 709x + 2252 \right)\left( x^2 - 5x + 6 \right) + 7007x - 13511. \end{equation*} Por el Teorema de la División, sabemos que este resto es el único polinomio de \anillodepolinomios{\racionales}{x} que satisface $p = q d + r$ con grado menor que $d$. Por lo tanto, al no ser $0$, podemos afirmar que $p$ no está en \engendrado{f,g}. \end{ejemplo} @@ -955,17 +955,9 @@ Ahora sí, tenemos la siguiente propiedad universal que buscábamos al principio En resumen, para poder construir un algoritmo de la división más general necesitaremos tener un orden total dentro de los monomios para saber cómo ir construyendo el cociente, ver cómo dividir por más de un generador e integrar a las distintas componentes. \section{Órdenes monomiales} - El concepto de orden monomial nos permite construir un orden total sobre los monomios de \Kx. Para poder definirlo bien, debemos identificar un monomio en $n$ variables con un vector en $\naturales^n$ que represente sus exponentes, es decir, - \begin{equation*} - x_1^{\alpha_1}x_2^{\alpha_2}\dots x_n^{\alpha_n} \longleftrightarrow \left( \alpha_1, \alpha_2, \dots, \alpha_n \right). - \end{equation*} - Tomando $\mathbf{\alpha} \in \naturales^n$, notaremos $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}$ como $x^\mathbf{\alpha}$. Así pues, un orden monomial se corresponde con un orden total en $\naturales^n$ de forma que sea compatible con la estructura de monoide que posee el conjunto de monomios. + El concepto de orden monomial nos permite construir un orden total sobre los monomios de \Kx. Para poder definirlo bien, debemos identificar un monomio en $n$ variables con un vector en $\naturales^n$ que represente sus exponentes, es decir, $x_1^{\alpha_1}x_2^{\alpha_2}\dots x_n^{\alpha_n} \longleftrightarrow \left( \alpha_1, \alpha_2, \dots, \alpha_n \right)$. Tomando $\mathbf{\alpha} \in \naturales^n$, notaremos $x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n}$ como $x^\mathbf{\alpha}$. Así pues, un orden monomial se corresponde con un orden total en $\naturales^n$ de forma que sea compatible con la estructura de monoide que posee el conjunto de monomios. \begin{definicion} - Un \emph{orden monomial} $>$ es un orden total sobre los monomios en $n$ variables que respeta la multiplicación, es decir, que satisface - \begin{equation*} - 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 $\naturales^{n}$. + Un \emph{orden monomial} $>$ es un orden total sobre los monomios en $n$ variables que respeta la multiplicación, es decir, que satisface $x^\mathbf{\alpha} > x^\mathbf{\beta} \implies x^{\mathbf{\gamma}} x^\mathbf{\alpha} > x^{\mathbf{\gamma}}x^\mathbf{\beta}$ para todo $\mathbf{\alpha}, \mathbf{\beta}$ y $\mathbf{\gamma}$ en $\naturales^{n}$. \end{definicion} Sin embargo esta propiedad deja en el aire el papel del $1$. ¿Cómo debemos ordenarlo? \begin{definicion} @@ -978,34 +970,30 @@ Ahora sí, tenemos la siguiente propiedad universal que buscábamos al principio \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, - $x^\mathbf{\alpha} > x^\mathbf{\beta}$ si la primera componente no nula por la izquierda de $\alpha - \beta$ es positiva. Con los valores de antes, $x^3y^2z^4 > x^2y^4z^4$ ya que la primera componente de $\left( 3, 2, 4 \right) - \left( 2, 4, 4 \right) = \left( 1, -2, 0 \right)$ es positiva. - - Es fácil ver que este orden es, en efecto, un orden monomial ya que si $x^\mathbf{\alpha} > x^\mathbf{\beta}$, tendremos que la primera componente de $\mathbf{\alpha} - \mathbf{\beta}$ será positiva, y lo mismo ocurrirá con la de $\mathbf{\alpha} + \mathbf{\gamma} - \left( \mathbf{\beta} + \mathbf{\gamma} \right) = \mathbf{\alpha} - \mathbf{\beta}$ por lo que $x^{\mathbf{\gamma}} x^\mathbf{\alpha} > x^{\mathbf{\gamma}}x^\mathbf{\beta}$. - - Para referirnos a este orden en concreto utilizaremos \mayorquelex. + $x^\mathbf{\alpha} > x^\mathbf{\beta}$ si la primera componente no nula por la izquierda de $\alpha - \beta$ es positiva. Con los valores de antes, $x^3y^2z^4 > x^2y^4z^4$ ya que la primera componente de $\left( 3, 2, 4 \right) - \left( 2, 4, 4 \right) = \left( 1, -2, 0 \right)$ es positiva. Para referirnos a este orden en concreto utilizaremos \mayorquelex. Es fácil ver que es, en efecto, un orden monomial ya que si $x^\mathbf{\alpha} > x^\mathbf{\beta}$, tendremos que la primera componente de $\mathbf{\alpha} - \mathbf{\beta}$ será positiva, y lo mismo ocurrirá con la de $\mathbf{\alpha} + \mathbf{\gamma} - \left( \mathbf{\beta} + \mathbf{\gamma} \right) = \mathbf{\alpha} - \mathbf{\beta}$ por lo que $x^{\mathbf{\gamma}} x^\mathbf{\alpha} > x^{\mathbf{\gamma}}x^\mathbf{\beta}$. \end{ejemplo} \begin{ejemplo}[Orden graduado-lexicográfico] - En ocasiones no nos interesará ordenar los monomios con el orden lexicográfico puesto que no tiene en cuenta el grado: $x^2 > xy^{129}z^{34}$. Por lo tanto, podemos definir un orden que tenga en cuenta el grado; en particular, que primero ordene por el grado y, en caso de empate, utilice el orden lexicográfico: + En ocasiones no nos interesará ordenar los monomios con el orden lexicográfico puesto que no tiene en cuenta el grado: $x^2 > xy^{129}z^{34}$. Por lo tanto, podemos definir un orden que tenga en cuenta el grado; en particular, que primero ordene por el grado y, en caso de empate, utilice el orden lexicográfico. Lo llamaremos <>: \begin{equation*} - x^\mathbf{\alpha} > x^\mathbf{\beta} \Longleftrightarrow \begin{cases} + x^\mathbf{\alpha} \mayorquedeglex x^\mathbf{\beta} \Longleftrightarrow \begin{cases} \grado{x^\mathbf{\alpha}} > \grado{x^\mathbf{\beta}} \quad \text{o} \\ \grado{x^\mathbf{\alpha}} = \grado{x^\mathbf{\beta}} \; \text{y} \; x^\mathbf{\alpha} \mayorquelex x^\mathbf{\beta}. \end{cases} \end{equation*} - Este orden monomial es el \nolbreaks{graduado-lexicográfico} y se denota por \mayorquedeglex. \end{ejemplo} 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, $\anillodepolinomios{\racionales}{x, y}^3$ como puede ser $\begin{psmallmatrix} - 2xy + y^2 \\ x^3 \\ y - \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos + y\etres$. + 2xy + y^2 \\ x^3 + \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos$. \begin{definicion} A las expresiones de la forma $x^\mathbf{\alpha}\vectorbasecanonica{i}$ con $i \in \left\lbrace 1, \dots, n \right\rbrace$ las llamaremos \emph{monomios} de \Kx. \end{definicion} \begin{definicion} Sea $>$ un orden monomial en \Kx. Un \emph{orden monomial de módulos} \mayorquemodulos{} es un orden total sobre los monomios de $\Kx^r$ que es compatible con la estructura de \modulo{\Kx} de $\Kx^r$ y con el orden $>$, es decir, que satisface - \begin{itemize} - \item $x^{\mathbf{\alpha}}\vectorbasecanonica{i} \mayorquemodulos x^{\mathbf{\beta}}\vectorbasecanonica{j} \implies x^{\mathbf{\alpha}}x^\mathbf{\gamma}\vectorbasecanonica{i} \mayorquemodulos x^{\mathbf{\beta}}x^\mathbf{\gamma}\vectorbasecanonica{j}\quad$ y - \item $x^\mathbf{\alpha} > x^\mathbf{\beta} \implies x^\mathbf{\alpha}\vectorbasecanonica{i} \mayorquemodulos x^\mathbf{\beta}\vectorbasecanonica{i}$ - \end{itemize} - para todo $\mathbf{\alpha}, \mathbf{\beta}$ y $\mathbf{\gamma}$ en $\naturales^n$ e $i$ y $j$ en $\left\lbrace 1, \dots, r \right\rbrace$. Por lo general, omitiremos el subíndice <<$\mathrm{m\acute{o}d}$>> ya que se sobreentenderá que estamos tratando con módulos. + \begin{equation*} + \begin{cases} + x^{\mathbf{\alpha}}\vectorbasecanonica{i} \mayorquemodulos x^{\mathbf{\beta}}\vectorbasecanonica{j} \implies x^{\mathbf{\alpha}}x^\mathbf{\gamma}\vectorbasecanonica{i} \mayorquemodulos x^{\mathbf{\beta}}x^\mathbf{\gamma}\vectorbasecanonica{j}\quad \text{y} \\ + x^\mathbf{\alpha} > x^\mathbf{\beta} \implies x^\mathbf{\alpha}\vectorbasecanonica{i} \mayorquemodulos x^\mathbf{\beta}\vectorbasecanonica{i} + \end{cases} \forall \mathbf{\alpha}, \mathbf{\beta},\mathbf{\gamma} \in \naturales \ \forall i, j \in \left\lbrace 1, \dots, r \right\rbrace. + \end{equation*}Por lo general, omitiremos el subíndice <<$\mathrm{m\acute{o}d}$>> por claridad. \end{definicion} \begin{ejemplo} Una manera de extender el orden monomial \mayorquelex{} a los monomios de $\Kx^n$ es ordenar primero en función de la posición, haciendo que las primeras posiciones vayan antes que las últimas ---por ejemplo $\vectorbasecanonica{2} > \vectorbasecanonica{3}$---, y si hubiese un empate, entonces se utilizase el orden \mayorquelex. Así definimos el siguiente orden monomial de módulos: @@ -1068,18 +1056,18 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición & & & & & \tiny $\uparrow$ 3\textsuperscript{a} iteración & \tiny $\uparrow$ 4\textsuperscript{a} iteración & & \\ \end{block} \end{blockarray} + \vspace{-0.4cm} \end{center} \begin{description}[wide, leftmargin=0pt,itemindent=6pt] \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. + \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 y nos queda + \begin{equation*} + f = \left( 3x^2 + 2x + 1 \right)g + 2x^2y\edos - 6x^2y\etres - 4xy\etres - 2y\etres + \end{equation*} + donde ninguno de los términos en $r$ es divisible por \LT{g}. \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}_{\text{resto $r$}} - \end{equation*} - donde ninguno de los términos en $r$ es divisible por \LT{g}. \end{ejemplo} 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. @@ -1112,10 +1100,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \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\emph{ \submodulo{\Kx} de los monomios líderes} de $G$ lo denotaremos por - \begin{equation*} - \LM{N} := \engendrado{\LM{g} \vert g \in N}. - \end{equation*} + Sea $G$ un subconjunto de $\Kx^r$ con un orden monomial de módulos $>$. El\emph{ \submodulo{\Kx} de los monomios líderes} de $G$ lo denotaremos por $\LM{N} := \engendrado{\LM{g} \vert g \in N}$. \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 $\engendrado{\LM{x\euno + y\edos}, \LM{y\euno + x\edos}}$. @@ -1152,6 +1137,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \begin{teorema} \label{Teorema Buen Funcionamiento División Groebner} 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} + \vspace{-0.3cm} \begin{proof} 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} @@ -1164,6 +1150,7 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \frac{a}{\LC{f}}f - \frac{b}{\LC{g}}g &\text{si $i = j$;} \\ 0 &\text{si $i \neq j$.} \end{cases} + \vspace{-0.3cm} \end{equation*} \begin{ejemplo} Sean $f = 3x^2y\euno + 2x\edos$ y $g = 13x\euno + y^5\edos$ elementos de $\anillodepolinomios{\racionales}{x, y}^2$ con el orden \mayorquelexpos. Como $\LM{f} = x^2y\euno$ y $\LM{g} = x\euno$ están en la misma componente ---\euno---, y $\mathrm{mcm}\left(\LM{f}, \LM{g}\right) = x^2y = \LM{f} = xy\LM{g}$, tenemos que @@ -1187,29 +1174,22 @@ distintos órdenes monomiales de módulos: %Poner otra ecuación de definición \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 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. + 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} (véase \cite[Teorema 1.7.3 y 2.5.9]{SINGULARLIBRO} para la demostración), 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 \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}. - \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 $A^k$ 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} + \vspace{-0.5cm}\begin{equation} \label{Definición Funcón Sicigias} \begin{tikzcd}[row sep=0pt,/tikz/column 1/.append style={anchor=base east},/tikz/column 2/.append style={anchor=base west}] A^k\rar["\sigma"]&M\\ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2f_2 + \dots + a_kf_k \end{tikzcd} +\vspace{-0.3cm} \end{equation} Con la definición, ya podemos dar por respondida la primera pregunta que nos hacíamos en la introducción: ¿qué es \derivaciones{f}? @@ -1218,10 +1198,8 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{definicion} Vemos que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse prácticamente como un módulo 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};} - &\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}. + \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 = 0, p \in \Kx}; \\ + &\simeq \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*} 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} @@ -1233,11 +1211,12 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{cases} \quad \text{donde $>'$ es un orden monomial.} \end{equation} Es decir, un orden que mira primero la posición en el orden usual. Sea $M$ un \submodulo{\Kx} de $\Kx^n$ y $G$ una base de \Groebner{} de este con respecto al orden $>$. Entonces, para el módulo restringido a las componentes a partir de la \nolbreaks{$s$-ésima}, es decir, - \begin{equation*} - M_{s} := \Big\lbrace\left.\vectorhorizontalcuatro{m_s}{m_{s+1}}{\dots}{m_n} \right\rvert (\underbrace{0, 0, \dots, 0}_{s-1}, m_s, m_{s+1}, \dots, m_n) \in M \Big\rbrace, + \vspace{-0.4cm}\begin{equation*} + M_{s} := \Big\lbrace\left.\vectorhorizontalcuatro{m_s}{m_{s+1}}{\dots}{m_n} \right\rvert (\overbrace{0, 0, \dots, 0}^{s-1}, m_s, m_{s+1}, \dots, m_n) \in M \Big\rbrace, \end{equation*} el conjunto $G_s$ es una base de \Groebner{} con respecto al orden $>$. \end{lema} + \vspace{-0.3cm} \begin{proof} Tomemos un $f_s = \vectorhorizontalcuatro{f_s}{f_{s+1}}{\dots}{f_n}$ en $M_s$. Por definición, este tiene un elemento asociado en $M$, $f = \left( 0, 0, \dots, 0, f_s, f_{s+1}, \dots, f_m \right)$. Al ser $G$ una base de \Groebner{} de $M$, tendremos que habrá algún $g \in G$ de forma que $\LM{g} \vert \LM{f}$. Ahora bien, por la condición que satisface el orden, sabemos que todos los términos de $g$ deben encontrarse en la \nolbreaks{$s$-ésima} componente o por debajo, lo cual nos garantiza que $g = \left( 0, 0, \dots, 0, g_s, g_{s+1}, \dots, g_n \right)$. Gracias a esto, sabemos que también se cumplirá que $\LM{g_s} \vert \LM{f_s}$ y, como $G_s \subseteq M_s$, esto es equivalente a que $\LM{M_s} = \engendrado{\LM{G}}$ lo cual nos garantiza que $G_s$ es base de \Groebner{} con respecto a este orden. \end{proof} @@ -1247,7 +1226,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{teorema} \label{Teorema Calculo Sicigias} Sean $f_1, f_2, \dots, f_k$ elementos de $\Kx^r$ y sean $f_1^+, f_2^+, \dots, f_k^+$ los correspondientes elementos en $\Kx^{r+k}$ pero en este caso por el principio, es decir, si $f_i = \vectorhorizontalcuatro{a_1}{a_2}{\dots}{a_r}$ entonces \begin{equation*} - f_i^+ = (a_1, a_2, \dots, a_r, \underbrace{0, 0, \dots, 0}_{k}). + f_i^+ = (a_1, a_2, \dots, a_r, 0, 0, \dots, 0) \in \Kx^{r+k}. \end{equation*} Entonces, si $G$ es una base de \Groebner{} de $M := \engendrado{f_1^+ + \vectorbasecanonica{r + 1}, f_2^+ + \vectorbasecanonica{r + 2}, \dots, f_k^+ + \vectorbasecanonica{r + k}}$ con respecto a un orden $>$ que satisfaga la condición \eqref{Condición Orden Eliminación}, se tiene que ---siguiendo la notación introducida en el Lema~\ref{Lema Restricción Groebner}--- \begin{equation*} @@ -1255,6 +1234,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} En particular, $G_{r+1}$ es una base de \Groebner{} de \syz{f_1, f_2, \dots, f_k} con respecto al orden $>$. \end{teorema} + \vspace{-0.3cm} \begin{proof} Por el lema anterior sabemos que $G_{r+1}$ es una base de \Groebner{} de $M_{r+1}$. Queda pues comprobar que $M_{r+1} = \syz{f_1, f_2, \dots, f_k}$. Supongamos\footnote{Para que el lector sepa en todo momento cuándo un elemento está en $\Kx^{r+k}$ o $\Kx^{k}$, utilizamos el subíndice en el elemento dándole consistencia con la notación utilizada en el lema anterior. Pero este no debe ser interpretado de forma distinta que si pusiéramos $m'$ o $\overline{m}$; es decir, el subíndice sobre un elemento no debe verse como el resultado de una <>.} que $m_{r+1} = \left(\mu_{r+1}, \mu_{r+2}, \dots, \mu_{r+k}\right) \in M_{r+1}$, entonces existe un elemento asociado $m = \left( 0, 0, \dots, 0, \mu_{r+1}, \mu_{r+2}, \dots, \mu_{r+k}\right) \in M$. Podemos expresar $m$ de dos formas, una en términos de los vectores de la base canónica y otra en términos de los $f_i^+ + \vectorbasecanonica{r + i}$ que generan $M$: \begin{align*} @@ -1319,19 +1299,18 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{gather} \label{Esquema Demostración} \left.\begin{aligned} \left.\begin{aligned} - \text{$M$ módulo reflexivo} \xRightarrow{\text{\cite[\href{https://stacks.math.columbia.edu/tag/0AV0}{Tag 0AV0}]{STACKS}}} \text{$M$ libre de torsión}& \\ + \text{$M$ módulo reflexivo} \implies \text{$M$ libre de torsión}& \\ \text{$M$ de rango 1}& \end{aligned}\right\rbrace \implies \text{$M$ es un ideal de $\Kx$}& \\ \text{$M$ es reflexivo} \end{aligned}\right\rbrace \implies \text{$M$ es un módulo libre} \end{gather} - 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 \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*} + \vspace{-0.2cm}\begin{equation*} + \textstyle \bigwedge^n \derivaciones{f} \xrightarrow{\bigwedge^n \Phi} \bigwedge^n \derivaciones{\Kx}. + \vspace{-0.15cm}\end{equation*} 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 $\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$} @@ -1357,15 +1336,15 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 Por un lado, fijamos $f\partial_i \in \derivaciones{f}$; y por otro, definimos $\dertheta{i}{j} := \diffp{f}{x_i}\partial_j - \diffp{f}{x_j}\partial_i \in \derivaciones{f}$. Así, \begin{equation*} - \dertheta{i}{1} \wedge \dertheta{i}{2} \wedge \dots \wedge \dertheta{i}{i-1} \wedge f\partial_i \wedge \dertheta{i}{i+1} \wedge \dots \wedge \dertheta{i}{n-1} \wedge \dertheta{i}{n} \in \bigwedge^n \derivaciones{f}. + \textstyle \dertheta{i}{1} \wedge \dertheta{i}{2} \wedge \dots \wedge \dertheta{i}{i-1} \wedge f\partial_i \wedge \dertheta{i}{i+1} \wedge \dots \wedge \dertheta{i}{n-1} \wedge \dertheta{i}{n} \in \bigwedge^n \derivaciones{f}. \end{equation*} Pero, simplificando todos los productos exteriores nulos nos queda \begin{equation*} - f\left( \diffp{f}{x_i} \right)^{n-1} \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n \in \bigwedge^n \derivaciones{f} = \engendrado{\det\Phi \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}. + \textstyle f\left( \diffp{f}{x_i} \right)^{n-1} \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n \in \bigwedge^n \derivaciones{f} = \engendrado{\det\Phi \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}. \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{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.. + \textstyle \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 $f$. @@ -1379,6 +1358,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{lema} Sea $f$ un polinomio libre de cuadrados de \Kx{} que puede ser descompuesto en factores irreducibles como $f = f_1f_2\cdots f_r$. Entonces, $\derivaciones{f} = \bigcap_{i = 1}^r \derivaciones{f_i}$. \end{lema} + \vspace{-0.3cm} \begin{proof} Sea $\delta \in \derivaciones{f}$ y llamemos $f_{-i} := f_1 f_2 \cdots f_{i-1} f_{i+1} \cdots f_{r-1} f_r$. Por la regla del producto tenemos que \begin{align*} @@ -1392,9 +1372,8 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{align*} \delta\!\left( f_1f_2\cdots f_{k+1} \right) &= \delta\!\left( f_{k+1} \right) f_{-\left( k+1 \right)} + f_{k+1}\delta\!\left( f_{-\left( k+1 \right)} \right) \\ &= pf_{k+1}f_{-\left( k+1 \right)} + f_{k+1}\delta\!\left( f_{-\left( k+1 \right)} \right), \quad \text{donde $p \in \Kx$,} \\ - &= pf + f_{k+1}\delta\!\left( f_{-\left( k+1 \right)} \right); - \intertext{por la hipótesis de inducción, $\delta\!\left( f_{-\left( k+1 \right)} \right) = qf_{-\left( k+1 \right)}$ con $q \in \Kx$;} - &= pf + f_{k+1}qf_{-\left( k+1 \right)} \\ + &= pf + f_{k+1}\delta\!\left( f_{-\left( k+1 \right)} \right) \\ + &\overset{*}{=} pf + f_{k+1}qf_{-\left( k+1 \right)} \\ &= \left( p + q \right)f. \end{align*} Por lo tanto, concluimos $\bigcap_{i = 1}^r \derivaciones{f_i} \subseteq \derivaciones{f}$ y, por ende, $\derivaciones{f} = \bigcap_{i = 1}^r \derivaciones{f_i}$.