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b/TFG_Sixto_José_Martínez_Gan.tex @@ -44,6 +44,21 @@ \usepackage{diffcoeff} \usepackage{nolbreaks} \usepackage{nicefrac} + +\usepackage{algpseudocodex} +\algrenewcommand\algorithmicwhile{\textbf{mientras}} +\algrenewcommand\algorithmicrepeat{\textbf{repite}} +\algrenewcommand\algorithmicdo{\textbf{haz}} +\algrenewcommand\algorithmicuntil{\textbf{hasta que}} + +\usepackage{blkarray} +\usepackage{booktabs} +\aboverulesep=-0.2pt +\belowrulesep=-0.2pt +\newcommand{\cocientecaja}[1]{\hspace{-0.21cm}\underline{\phantom{a}\ensuremath{#1}}} + +\usepackage{ulem} + %\usepackage{lmodern} % fuente Times \usepackage{tikz} % para los dibujos \usetikzlibrary{babel} % para evitar conflictos entre TikZ y babel @@ -121,6 +136,8 @@ urlcolor=black]{hyperref} \newcommand{\mayorquemodular}[2]{\mayorque{\left(\formatonombreorden{#1}, \formatonombreorden{#2} \right)}} \newcommand{\mayorqueposlex}{\mayorquemodular{POS}{LEX}} \newcommand{\mayorquelexpos}{\mayorquemodular{LEX}{POS}} +\newcommand{\mayorqueposdeglex}{\mayorquemodular{POS}{GRAD-LEX}} +\newcommand{\mayorquedeglexpos}{\mayorquemodular{GRAD-LEX}{POS}} \newcommand{\LT}[1]{\ensuremath{\mathrm{LT}\left(#1\right)}} \newcommand{\LM}[1]{\ensuremath{\mathrm{LM}\left(#1\right)}} @@ -352,6 +369,12 @@ urlcolor=black]{hyperref} Para indicar que una aplicación $\phi : M \rightarrow N$ es inyectiva, escribiremos $0 \rightarrow M \xrightarrow{\phi} N$; y para indicar que es sobreyectiva, $M \xrightarrow{\phi} N \rightarrow 0$. \end{definicion} + + Con las secuencias exactas, podemos resumir nuestro problema en una sola expresión. Recordemos que estamos estudiando el \modulo{\Kx} \derivaciones{f}, que es un \submodulo{\Kx} de \derivaciones{\Kx}. Por otro lado, cada derivación $\delta$, cuando se aplica a $f$ nos da lugar a un elemento en el \submodulo{\Kx} $J := \engendrado{\partial_1 f, \dots, \partial_n f}$. Finalmente, a nosotros los elementos que nos interesan son aquellos que sean múltiplos de $f$, es decir, que pertenezcan al \submodulo{\Kx} \engendrado{f}; para poder identificarlo, haremos el cociente de $J$ por \engendrado{f} para que así todas las expresiones de $J$ que sean múltiplos de $f$ vayan al cero. Todo esto lo podemos expresar como sigue: + \begin{equation*} + 0 \rightarrow \derivaciones{f} \xrightarrow{\iota} \derivaciones{\Kx} \xrightarrow{\kappa} \Kx^n \xrightarrow{\Phi} J \xrightarrow{\pi} \frac{J}{J \cap \engendrado{f}} \rightarrow 0. + \end{equation*} + La función $\iota$ es una inclusión, la función $\kappa$ es el homomorfismo que envía $\partial_i$ a \vectorbasecanonica{i}, $\Phi$ envía una derivación $\delta$ a su evaluación en $f$ ---$\delta\!\left( f \right)$--- y $\pi$ es simplemente la aplicación canónica. \chapter{Bases de Gröbner y Syzygias} A partir de este momento nos fijaremos en $\Kx^n$ ya que sus elementos son los involucrados en el módulo de derivaciones. En particular nos fijaremos en sus submódulos. @@ -473,13 +496,47 @@ urlcolor=black]{hyperref} Con este aparataje ya tenemos una noción de cuáles son los términos <> dentro de un polinomio, lo cual nos hace empezar a vislumbrar cómo dividir uno por otro. Lo primero es entender cuándo pueden dividirse dos monomios de $\Kx^r$. Según la definición \ref{Definición Divisibilidad Módulos}, para $x^\mathbf{\alpha}\vectorbasecanonica{i}$ y $x^\mathbf{\beta}\vectorbasecanonica{j}$ dos monomios de \Kx{} tenemos que $x^\mathbf{\alpha}\vectorbasecanonica{i}$ divide a $x^\mathbf{\beta}\vectorbasecanonica{j}$ si $i = j$ y $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. Ahora sí, podemos dar un algoritmo de la división en $\Kx^r$. Este consiste en imitar el algoritmo usual para dividir un polinomio por otro en una variable; pero en este caso, en vez de guiarnos por el grado para encontrar los términos <> del cociente y el dividendo, utilizamos el término líder. \begin{definicion}[algoritmo de la división] - Supongamos que tenemos dos polinomios $f$ y $g$ en \Kx. Para dividir $f$ entre $g$ aplicaremos un algoritmo que consiste en replicar el algoritmo que ya conocemos para los polinomios en una variable. Sin embargo, ahora los términos que guían los coeficientes que vamos poniendo son $\LT{f}$ y $\LT{g}$: - - - \todo[inline]{Escribir el algoritmo con la librería que toca.} + Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^n$ junto con un orden monomial sobre este. Para dividir $f$ entre $g$ aplicaremos un algoritmo que consiste en replicar el algoritmo que ya conocemos para los polinomios en una variable. Sin embargo, ahora los términos que guían los coeficientes que vamos poniendo son $\LT{f}$ y $\LT{g}$: + \begin{algorithmic}[1] + \State Definimos $D \gets f$, $d \gets g$, $c \gets 0$ y $r \gets 0$ como las variables que representan el dividendo, divisor, cociente y resto, respectivamente. + \Repeat + \While{$\LM{d} \vert \LM{D}$} + \State Sea $m \in \Kx$ de forma que $\LM{D} = m\LM{d}$. + \State $D \gets D - m d$, $c \gets c + m$ + \EndWhile + \State $D \gets D - \LM{D}$, $r \gets r + \LM{D}$ + \Until{$D = 0$} + \end{algorithmic} + Al finalizar este algoritmo, tendremos que $f = c g + r$ donde ninguno de los términos de $r$ es divisible por \LM{g}. \end{definicion} + La única novedad de este algoritmo con respecto al que ya conocemos de polinomios en una sola variable es la instrucción de la línea 6. En ella, le sustraemos a $D$ su término líder y repetimos todo el proceso otra vez con el resto de términos de $D$. Esto lo hacemos para reducir todo lo posible el resto, garantizando que ninguno de sus términos sea divisible por \LM{g}. De no hacer esto, solo podríamos garantizar que, al final de la ejecución, $\LT{D}$ no es divisible por $\LT{g}$. \begin{ejemplo} - contenidos... + Sean $f := 3x^4\euno + 2x^3\euno + 2x^2y\edos + x^2\euno$ y $g := x^2\euno + 2y\etres$ dos elementos de $\anillodepolinomios{\reales}{x, y}^3$ con el orden monomial \mayorquedeglexpos. Aunque hayamos expresado en la definición todo de una forma muy <>, en esencia, lo que hacemos es bastante similar a una división usual: + \begin{center} + \begin{blockarray}{rrrrrrrcl} + \begin{block}{rrrrrrrc|l} + $3x^4\euno$ & +$2x^3\euno$ & +$2x^2y\edos$ & & +$x^2\euno$ & & & & \cocientecaja{x^2\euno + 2y\etres} \\ + $-(3x^4\euno$ & & & $+6x^2y\etres)$ & & & & & $3x^2 + 2x + 1$ \\ + \end{block} + \begin{block}{rrrrrrrcl} + \cmidrule{1-5} + & +$2x^3\euno$ & +$2x^2y\edos$ & $-6x^2y\etres\phantom{)}$ & +$x^2\euno$ & & & & \\ + & $-(2x^3\euno$ & & & & $+4xy\etres)$ & & & \\ \cmidrule{2-6} + & \tiny 1\textsuperscript{a} iteración: & \sout{$2x^2y\edos$} & \sout{$-6x^2y\etres\phantom{)}$} & +$x^2\euno$ & $-4xy\etres\phantom{)}$ & & & \\ + & & \tiny $\uparrow$ 2\textsuperscript{a} iteración & \tiny $\uparrow$ 3\textsuperscript{a} iteración & $-(x^2\euno$ & & $+2y\etres)$ & & \\ \cmidrule{5-7} + & & & & \tiny 4\textsuperscript{a} iteración: & \sout{$-4xy\etres\phantom{)}$} & \sout{$-2y\etres$} & & \\ + & & & & & \tiny $\uparrow$ 5\textsuperscript{a} iteración & \tiny $\uparrow$ 6\textsuperscript{a} iteración & & \\ + \end{block} + \end{blockarray} + \end{center} + + La primera iteración ejecuta el bucle interno dos veces dejando $D = 2x^2y\edos - 6x^2y\etres + x^2\euno - 4xy\etres$. Después, la segunda iteración se encuentra con un términos líder que no es divisible por $\LT{g} = x^2\euno$: $2x^2y\edos$; así que lo elimina del dividendo, que pasa a ser $D = - 6x^2y\etres + x^2\euno - 4xy\etres$ y lo añade al resto, que pasa a ser $r = 2x^2y\edos$. En la tercera iteración ocurre lo mismo, así que procedemos de igual manera eliminando el término líder no divisible por \LT{g} y añadiéndolo al resto. En la cuarta iteración, $\LT{D} = x^2\euno$ sí es divisible por \LT{g}, por lo que procedemos como en la primera iteración. Por último, en las dos últimas iteraciones, retiramos términos de nuevo del dividendo. + + El resultado puede leerse como + \begin{equation*} + f = \left( 3x^2 + 2x + 1 \right)g + \underbrace{2x^2y\edos - 6x^2y\etres - 4xy\etres - 2y\etres} + \end{equation*} + donde ninguno de los términos en la llave es divisible por \LT{g}. \end{ejemplo} Para completar estas herramientas, introducimos las bases de Gröbner.