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b/TFG_Sixto_José_Martínez_Gan.tex index 84060a0..03ee3ba 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -739,9 +739,9 @@ Si $S=A \setminus \left\lbrace 0 \right\rbrace$, denotaremos la localización co Esto nos va a permitir completar la demostración de un resultado anterior. -\mbox{}\vspace{-1.5cm} +\mbox{}\vspace{-1.7cm} \begin{wrapfigure}[8]{r}{0.3\textwidth} - % + \footnotesize \begin{equation*} \begin{tikzcd} M_\mathrm{Q} \times \left( \dual{M} \right)_\mathrm{Q} \arrow[r, "\Phi^{M_\mathrm{Q}}"] & \cuerpofracciones{\Kx} & \\ @@ -749,7 +749,7 @@ Esto nos va a permitir completar la demostración de un resultado anterior. 0 \hspace{17pt} \phantom{. \times } 0\arrow[u, shift left=6mm] \arrow[u, shift right=6mm] & 0 \arrow[u] & \end{tikzcd} \end{equation*} -\vspace{-1cm} +\vspace{-0.8cm} \caption{} \label{Diagrama Dual Reflexivo} \end{wrapfigure} From 170f117a346015995fb5451eed223656e3c4f76c Mon Sep 17 00:00:00 2001 From: Sixto Date: Wed, 11 Jun 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z?9wv!YuP-Q4~?(@7FzMayYy6SL$g(|*wXXQD#kW6({F16cD~tiSOIJ{G8Wowf3>9_ zn$j6;QnOZA2kU{=Hfw`2bO0OIY%_EL`_pW#rFXuqtFW?W+ko82+JSv%)(Km9<9bbp zXz4W_h9l4m`rsaeJ~$31;3V|JDd2xVrf;p@_-SCevon_7)2wG1w)9r3S|e6*?;~oB zTH#~&sdd&0_g+xzoE2VqTCFiFe;)N}ja%{DYPBY;_|dPZHEG4ib#5M6>4nC~ikCmF zu37Ovx73QQR9&rBVx{gzwNfjczo}M*l|G?c<^?N#=FxDKivxlCD+6D6D)7ajz?a?# zeEG4!SMLlw{8r%WmjjPI6L>Og7E9l34SefL;M<=JJl7R?{HVV6jYbmqmRKj=r3L<#3T(r^;N1dNeyC1AcYDj jK+#H9+rq)uYNG!EvVLy;m#o?d9||xwGYTaoMNdWwN&O-w diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 03ee3ba..99006a3 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -263,6 +263,8 @@ urlcolor=black]{hyperref} \chapter{Resumen} \todo{Configuraciones de hiperplanos, justificar. Conjetura de Terao.} + + El presente trabajo se enmarca dentro del contexto de la Conjetura de Terao para configuraciones de hiperplanos. En este trabajo estudiaremos varios aspectos del módulo de derivaciones polinómicas de un polinomio. Comenzamos dando las pinceladas básicas de la teoría de módulos para comprender con qué tipo de objeto estamos trabajando. A continuación, exploramos algunas de las dificultades que presenta este tipo de objeto con respecto a otros similares como son el espacio vectorial, en particular la determinación de la pertenencia a este y la existencia de una base. Solucionamos el primero de estos problemas introduciendo dos herramientas fundamentales de esta teoría: las bases de \Groebner{} y las sicigias. Por último, para solucionar el segundo problema, damos una detallada demostración del Criterio de Saito, un resultado que nos caracteriza para qué módulos existe una base. \selectlanguage{english} diff --git a/bibliografia.bib b/bibliografia.bib index 17b0b70..0251ece 100644 --- a/bibliografia.bib +++ b/bibliografia.bib @@ -65,15 +65,19 @@ url = {https://repository.dl.itc.u-tokyo.ac.jp/records/39646}, urldate = {2025-05-28} } -@misc{VALLES, +@article {VALLES, shorthand = {Faenzi}, - title={Saito criterion and its avatars}, - author={Daniele Faenzi and Marcos Jardim and Jean Vallès}, - year={2024}, - eprint={2402.08305}, - archivePrefix={arXiv}, - primaryClass={math.AG}, - url={https://arxiv.org/abs/2402.08305}, + AUTHOR = {Faenzi, Daniele and Jardim, Marcos and Vall\`es, Jean}, + TITLE = {Saito criterion and its avatars}, + JOURNAL = {Rend. Circ. Mat. Palermo (2)}, + FJOURNAL = {Rendiconti del Circolo Matematico di Palermo. Second Series}, + VOLUME = {73}, + YEAR = {2024}, + NUMBER = {6}, + PAGES = {2233--2243}, + ISSN = {0009-725X,1973-4409}, + DOI = {10.1007/s12215-024-01061-z}, + URL = {https://doi.org/10.1007/s12215-024-01061-z}, } @online{STACKS, shorthand = {Stacks}, From e087ba1572f3417d82607d571fad76b926bf8dea Mon Sep 17 00:00:00 2001 From: Sixto Date: Wed, 11 Jun 2025 17:38:59 +0200 Subject: [PATCH 3/5] =?UTF-8?q?Res=C3=BAmenes.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 580309 -> 582125 bytes TFG_Sixto_José_Martínez_Gan.tex | 21 +++++++++++++++------ 2 files changed, 15 insertions(+), 6 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index e36398add33a9d7a49747610c84fe52150aa0156..aea5c0438680688171b639010ce8b5a36c01b426 100644 GIT binary patch delta 208730 zcmZsiLzFIDtfq^nY}>YN+qP|=!dJF!+qP}nw(ZmZ?K|wAWUXY+b_RLgtT!tyBsW7O 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elemento que utilizamos para reducir en la división. A partir de aquí, mostraremos cómo construir nuevos conjuntos que nos permitan utilizar esta nueva división, las bases de \Groebner, y utilizaremos estas para obtener otro objeto que generaliza a las soluciones de sistemas lineales, la sicigia. Con todo esto, ya seremos capaces de calcular el módulo de derivaciones. + + Por último, en la tercera parte, daremos una detallada demostración del Criterio de Saito, el resultado que nos da un criterio para garantizar si un conjunto de derivaciones de este módulo forma una base de este. Con lo cual damos fin al trabajo habiendo no solo contextualizado estos objetos sino también presentado dos resultados fundamentales en esta teoría. \selectlanguage{english} \chapter{Abstract} \section*{English} - In this paper we are going to study several aspects of the module of polynomial derivations of a polynomial. We start by stating the basic features of module theory in order to understand which kind of object we are working with. Afterwards, we explore some of the difficulties this kind of object presents with respect to others similar to it such as the vector space, in particular the determination of whether an element belongs to it and the existence of a basis. We solve the first of those problems by introducing two fundamental tools in this theory: \Groebner{} bases and syzygies. Lastly, so as to solve the second problem, we give a detailed proof of Saito's Criterion, a result that characterizes which modules admit a basis. - + 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. + + In the second part, our objective will be to tackle the problem of determining whether any given element belongs to a module of polynomials. To do so, we will explore with some greater detail the behavior in simpler cases where this is still feasible without introducing any new tools. This will lead us to abstract the division algorithm to this new space we are in, get to know its properties, and understand how they are strongly tied to the choice of the principal term of an element that we use for reduction in division. From here, we will demonstrate how to construct new sets that allow us to apply this new division, Gröbner bases, and we will use them to obtain another object that generalizes the solutions of linear systems: the syzygy. With all this, we will finally be equipped to compute the module of derivations. + + Finally, in the third part, we will give a detailed proof of Saito’s Criterion, the result that offers a criterion to guarantee whether a set of derivations of this module forms a basis of it. With that we conclude our work having not only contextualized these objects but also presented two fundamental results in this theory. + \selectlanguage{spanish} \clearemptydoublepage From 7301db970ccb69aadd3589cc477fe37ce10aaf85 Mon Sep 17 00:00:00 2001 From: Sixto Date: Wed, 11 Jun 2025 18:47:19 +0200 Subject: [PATCH 4/5] =?UTF-8?q?Carnicer=C3=ADa=20de=20espacios.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 582125 -> 579065 bytes TFG_Sixto_José_Martínez_Gan.tex | 127 +++++++++++++------------------- 2 files changed, 53 insertions(+), 74 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index aea5c0438680688171b639010ce8b5a36c01b426..ad7ece5a575aa9ba6c97b3d8d8fd4a4bf2effc22 100644 GIT binary patch delta 183877 zcmV)DK*7K5!z1~tBd~>60x&g`VFW0DeHqJ=qq=oJU*TOu)wC}!1ATYNt;|%iO;@tW z%|ZsXtqcazg6Z^pea=fl*i2TKqoebF;`DGiq=&DcQhEOA<gidig8d%nn(4d`_R~W_oyks?uXt 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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, $\derivaciones{f} = \bigcap_{i = 1}^r \derivaciones{f_i}$. 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a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 9df446b..e8b4f6f 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -1267,7 +1267,6 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} Ahora bien, por ser $\conjunto{a_1, a_2, \dots, a_k}$ una base, necesariamente tendremos que $yp_i = xq_i$ para todo $i$. Como $\mcd{x, y} = 1$, es necesario que $p_i = xp_i'$. Esto nos lleva a que \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{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. @@ -1281,19 +1280,16 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \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. - - \section{Demostración del Criterio de Saito} - - A lo largo de esta demostración, utilizaremos fuertemente el hecho de que \derivaciones{f} es un \modulo{\Kx} reflexivo. Esto se debe a que podemos encontrar ciertas relaciones entre los módulos reflexivos y los módulos libres. Por un lado, tenemos la siguiente relación: - \begin{gather*} - \text{$M$ módulo libre} \xRightarrow{\text{\referenciamodomatematico{lema}{Lema Libre implica Proyectivo}}} \text{$M$ módulo proyectivo} \xRightarrow{\phantom{fd}} \text{$M$ módulo reflexivo} \\ - \text{$M$ módulo libre} \ \xLeftarrow[\substack{\text{Teorema de Quillen-Suslin} \\ \text{\cite{ALGEBRA} }}]{\text{si $M$ es un \modulo{\Kx}}} \text{$M$ módulo proyectivo} \ \xLeftarrow{\text{si $M$ es un \modulo{\anillodepolinomios{K}{x, y}}}} \text{$M$ módulo reflexivo} - \end{gather*} - Gracias a estas implicaciones podemos asegurar que todo módulo libre es reflexivo. Sin embargo, solo podemos asegurar que los módulos reflexivos sean libres cuando son \modulo{\anillodepolinomios{K}{x, y}}, es decir, cuando admitan la estructura de módulo asociada a un anillo de polinomios en dos variables. Esto, por lo tanto, nos garantiza que \derivaciones{f} es siempre libre cuando $f \in \anillodepolinomios{K}{x,y}$. Más que solucionar nuestro problema, estos resultados nos sirven para ilustrar la gran dificultad que supone garantizar que un módulo reflexivo sea libre. + % A lo largo de esta demostración, utilizaremos fuertemente el hecho de que \derivaciones{f} es un \modulo{\Kx} reflexivo. Esto se debe a que podemos encontrar ciertas relaciones entre los módulos reflexivos y los módulos libres. Por un lado, tenemos la siguiente relación: + % \begin{gather*} + % \text{$M$ módulo libre} \xRightarrow{\text{\referenciamodomatematico{lema}{Lema Libre implica Proyectivo}}} \text{$M$ módulo proyectivo} \xRightarrow{\phantom{fd}} \text{$M$ módulo reflexivo} \\ + % \text{$M$ módulo libre} \ \xLeftarrow[\substack{\text{Teorema de Quillen-Suslin} \\ \text{\cite{ALGEBRA} }}]{\text{si $M$ es un \modulo{\Kx}}} \text{$M$ módulo proyectivo} \ \xLeftarrow{\text{si $M$ es un \modulo{\anillodepolinomios{K}{x, y}}}} \text{$M$ módulo reflexivo} + % \end{gather*} + % Gracias a estas implicaciones podemos asegurar que todo módulo libre es reflexivo. Sin embargo, solo podemos asegurar que los módulos reflexivos sean libres cuando son \modulo{\anillodepolinomios{K}{x, y}}, es decir, cuando admitan la estructura de módulo asociada a un anillo de polinomios en dos variables. Esto, por lo tanto, nos garantiza que \derivaciones{f} es siempre libre cuando $f \in \anillodepolinomios{K}{x,y}$. Más que solucionar nuestro problema, estos resultados nos sirven para ilustrar la gran dificultad que supone garantizar que un módulo reflexivo sea libre. - \subsection{Comienzo de la demostración} - Puede encontrarse en \cite[corolario 1.7]{SAITO} una demostración de que \derivaciones{f} es un módulo reflexivo de rango $n$. Siguiendo el esquema, tenemos que \derivaciones{f} es libre de torsión, pero necesitaríamos que tuviese rango $1$ para continuar hacia delante. Lo que haremos será estudiar el módulo $\bigwedge^n \derivaciones{f}$ para reducir este rango y poder continuar. + \section{Comienzo de la demostración} + Puede encontrarse en \cite[corolario 1.7]{SAITO} una demostración de que \derivaciones{f} es un módulo reflexivo de rango $n$. Siguiendo el esquema siguiente, tenemos que \derivaciones{f} es libre de torsión, pero necesitaríamos que tuviese rango $1$ para continuar hacia delante. Lo que haremos será estudiar el módulo $\bigwedge^n \derivaciones{f}$ para reducir este rango y poder continuar. Esta cadena de implicaciones que recogemos aquí nos será muy útil a lo largo de esta demostración: \begin{gather} \label{Esquema Demostración} @@ -1313,7 +1309,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \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 $\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$} + \section{\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 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}. @@ -1332,9 +1328,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \end{equation*} Vemos\footnote{De nuevo, recordemos que $\det\Phi$ no es el determinante de una matriz, sino una convención que hemos explicado antes. Por otro lado, $\det\left( \Phi \circ \iota \right)$ sí que es el determinante de una matriz ya que tanto $N$ como \derivaciones{\Kx} son módulos libres.} por ella que $\det\left( \Phi \circ \iota \right) = \det\Phi \det\iota = \det A$. Como $\iota$ es un monomorfismo, puede representarse como una matriz que tiene inversa en $\mathrm{Mat}\!\left( n \times n, \Kx\right)$ y por la regla de Cramer\footnote{Por si no la recordamos, esta dice que toda matriz $C$ inversible cumple que $C^{-1} = \frac{1}{\det C} \mathrm{adj}\!\left( C \right)$, con $\mathrm{adj}\!\left( C \right)$ la matriz de adjuntos.} para la matriz inversa, $\det i$ debe ser unidad de \Kx. - \subsubsection{El valor del determinante} - - 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, + Nos dedicaremos ahora a obtener el valor del determinante. 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*} \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*} @@ -1342,7 +1336,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \begin{equation*} \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 $\det\Phi$: + Esto quiere decir que $\det\Phi \vert f\left( \diffp{f}{x_i} \right)^{n-1}$ para todo $i$. Esto también ocurre con su máximo común divisor: \begin{equation*} \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*} @@ -1361,7 +1355,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \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*} + \vspace{-0.4cm}\begin{align*} \delta\!\left( f \right) &= \delta\!\left( f_1 f_2 \cdots f_r \right) \\ pf_1 f_2 \cdots f_r &= \delta\!\left( f_i \right) f_{-i} + f_i \delta\!\left( f_{-i} \right) \\ pf_1 f_2 \cdots f_r - f_i \delta\!\left( f_{-i} \right) &= \delta\!\left( f_i \right) f_{-i} \\ @@ -1386,7 +1380,7 @@ a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k}\rar[mapsto]&a_1f_1 + a_2 \det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = \det A = \det \left( \Phi \circ \iota \right) = \det\Phi \det\iota = u'f \det\iota = uf, \quad \text{con $u$ una unidad de \Kx.} \end{equation*} - \subsection{$\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ implica que \derivaciones{f} es libre} + \section{$\det\!\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ implica que \derivaciones{f} es libre} Ahora asumamos que existen $\delta_1, \delta_2, \dots, \delta_n$ de forma que $\det\left( \delta_1, \delta_2, \dots, \delta_n \right) = uf$ con $u$ una unidad de \Kx{} y volvamos al esquema de \ref{Secuencia Demostración Saito}. A partir de este, sabemos que tenemos \begin{equation*}