From ec32f6c844c8a9b39a75dec744a0a957f4434ec9 Mon Sep 17 00:00:00 2001 From: Sixto Date: Mon, 9 Jun 2025 12:33:58 +0200 Subject: [PATCH] =?UTF-8?q?Explicaci=C3=B3n=20m=C3=A1s=20detallada=20de=20?= =?UTF-8?q?la=20divisi=C3=B3n=20compuesta=20y=20cambio=20de=20los=20tachon?= =?UTF-8?q?es=20horizontales=20por=20diagonales.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- TFG_Sixto_José_Martínez_Gan.pdf | Bin 470808 -> 475276 bytes TFG_Sixto_José_Martínez_Gan.tex | 36 +++++++++++++++++++------------- 2 files changed, 22 insertions(+), 14 deletions(-) diff --git a/TFG_Sixto_José_Martínez_Gan.pdf b/TFG_Sixto_José_Martínez_Gan.pdf index c71117e169aa88cec6ff792783c6343eab1c9ca6..bc84acbe19ffc0efe31f0abb6b56966fb9768e92 100644 GIT binary patch delta 46671 zcmZs>Q*`GI5AWN5ZQHipPHmf0+njc)X@Aw!wr$&<+P2N<)Nc3l?sM+Wxl1mRwX$-N z&o@chHepl%QJt*1ID~-j{;%U~Ri4%L2Ljic{v4v-EF!h2%N=GH%@?B?DP$}{#Eh!p 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zK>?9owYSOAj>yt#VZD|n8?AC_?y5kx&BB!j`Yh89KDy(L2Dyv3|I@WuEX;P*LtqeO zDz=5#CE+8uzy|Xk_K14il>GcM&*%F0AP`PpiY4@azq}p*b9aEZyR{7>H!nXwCkG-8 Kjf|Qs;(r0Sf8Z+s diff --git a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index fe14700..7dd31f4 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -31,6 +31,7 @@ \usepackage[utf8]{inputenc} % para usar el teclado normalmente \usepackage[spanish]{babel} % selecciona idioma \usepackage[T1]{fontenc} % gestión de fuentes con acentos +\usepackage{ulem} \usepackage{graphicx} @@ -80,7 +81,7 @@ \belowrulesep=-0.2pt \newcommand{\cocientecaja}[1]{\hspace{-0.21cm}\underline{\phantom{a}\ensuremath{#1}}} -\usepackage{ulem} +\usepackage{cancel} %\usepackage{lmodern} % fuente Times \usepackage{tikz} % para los dibujos @@ -637,9 +638,9 @@ urlcolor=black]{hyperref} \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 1\textsuperscript{a} iteración: & \cancel{$2x^2y\edos$} & \cancel{$-6x^2y\etres\phantom{)}$} & +$x^2\euno$ & $-4xy\etres\phantom{)}$ & & & \\ & & \tiny $\uparrow$ 1\textsuperscript{a} iteración & \tiny $\uparrow$ 2\textsuperscript{a} iteración & $-(x^2\euno$ & & $+2y\etres)$ & & \\ \cmidrule{5-7} - & & & & \tiny 3\textsuperscript{a} iteración: & \sout{$-4xy\etres\phantom{)}$} & \sout{$-2y\etres$} & & \\ + & & & & \tiny 3\textsuperscript{a} iteración: & \cancel{$-4xy\etres\phantom{)}$} & \cancel{$-2y\etres$} & & \\ & & & & & \tiny $\uparrow$ 3\textsuperscript{a} iteración & \tiny $\uparrow$ 4\textsuperscript{a} iteración & & \\ \end{block} \end{blockarray} @@ -1194,24 +1195,31 @@ urlcolor=black]{hyperref} \end{equation*} Calculamos ahora una base de Gr\"obner a partir de estos elementos. Para empezar con el mejor pie posible, lo que haremos será convertir este conjunto en un conjunto reducido, es decir, que buscaremos garantizar que ninguno de los términos de $g_i$ sea divisible por algún \LM{g_j} donde $j \neq i$. En este caso, vemos que $g_1$, $g_2$ y $g_3$ no tienen ningún termino divisible, sin embargo $g_4'$ sí, así que lo reducimos: \begin{equation} \label{Ejemplo División Compuesta} - \begin{blockarray}{rrrrrcl} - \begin{block}{rrrrrc|l} - x^2z\euno & +y^3\euno & & & +\ecinco & & \cocientecaja{2xz\euno + \edos} \\ - -(x^2z\euno & & +\frac{1}{2}x\edos) & & & & \frac{1}{2}x \\ + \begin{blockarray}{rrrrrclcl} + \begin{block}{rrrrrc|lc|l} + x^2z\euno & +y^3\euno & & & +\ecinco & & \cocientecaja{2xz\euno + \edos \ \ \left( g_1 \right)} & & \cocientecaja{x^2\euno + \ecuatro \ \ \left( g_3 \right)}\\ + -(x^2z\euno & & +\frac{1}{2}x\edos) & & & & \frac{1}{2}x & & \\ \end{block} - \begin{block}{rrrrrcl} + \begin{block}{rrrrrclcl} \cmidrule{1-5} - & y^3\euno & -\frac{1}{2}x\edos\phantom{)} & & +\ecinco & & \\ + & y^3\euno & -\frac{1}{2}x\edos\phantom{)} & & +\ecinco & & & & \\ \end{block} - \begin{block}{rrrrrc|l} - & -(y^3\euno & & +\frac{1}{3}y\etres) & & & \cocientecaja{3y^2\euno + \etres} \\ \cmidrule{1-5} - & & -\frac{1}{2}x\edos & -\frac{1}{3}y\etres\phantom{)} & +\ecinco & & \frac{1}{3}y \\ + \begin{block}{rrrrrc|lcl} + & -(y^3\euno & & +\frac{1}{3}y\etres) & & & \cocientecaja{3y^2\euno + \etres \ \ \left( g_2 \right)} & & & \\ \cmidrule{1-5} + & \text{\tiny 1\textsuperscript{a} iteración:} & \cancel{-\frac{1}{2}x\edos}\phantom{)} & \cancel{-\frac{1}{3}y\etres}\phantom{)} & \cancel{+\ecinco} & & \frac{1}{3}y & & & \\ + & & \uparrow \text{\tiny 1\textsuperscript{a} iteración} & \uparrow \text{\tiny 2\textsuperscript{a} iteración} & \text{\tiny 3\textsuperscript{a} iteración} \uparrow & & & & & \\ \end{block} \end{blockarray} \end{equation} - Añadimos <> conforme nos vayan haciendo falta ya que no solo dividimos por un elemento sino por varios a la vez, por lo que puede ser que para anular un término hagamos uso de uno, para anular otro usemos un elemento distinto e incluso puede que necesitemos volver a utilizar algún elemento de nuevo. Pero en esencia, el procedimiento de división es el mismo pero ahora podemos <> de caja en caja. + Por ser esta la primera división con más de un divisor que hacemos, la desglosaremos algo más siguiendo la notación del algoritmo en \ref{Algoritmo de la División Compuesta}. + \begin{description} + \item[1\textsuperscript{a} iteración: ] $\LT{D} = x^2z\euno$ es divisible por $\LM{g_1}$ y $\LM{g_3}$. Escogemos, por pura preferencia personal, $g_1$ y reducimos ---la primera resta---. Después de reducir, $\LT{D} = y^3\euno$ es divisible por $\LM{g_2}$ así que reducimos por este. Tras esta reducción, $\LT{D} = -\frac{1}{2}x\edos$, que no es divisible por ninguno de los monomios líderes de $g_1$, $g_2$ y $g_3$. Se lo sustraemos a $D$ y lo añadimos al resto. Esto deja $D = -\frac{1}{3}y\etres + \ecinco$, $c_1 = \frac{1}{2}x$, $c_2 = \frac{1}{3}y$, $c_3 = 0$ y $r = -\frac{1}{2}x\edos$. + \item[2\textsuperscript{a} iteración: ] $\LT{D} = -\frac{1}{3}y\etres$ no es divisible por ninguno de los términos líderes de los divisores. Así pues, no reducimos y sustraemos este término de $D$ para pasarlo a $r$. Esto deja $D = \ecinco$, $r = -\frac{1}{2}x\edos - \frac{1}{3}y\etres$ y los cocientes como antes. + \item[3\textsuperscript{a} iteración: ] $\LT{D} = \ecinco$ y nos encontramos con la misma situación que antes. Al sustraer este de $D$, $D$ pasa a ser $0$, por lo que el algoritmo termina. Nos queda $r = -\frac{1}{2}x\edos - \frac{1}{3}y\etres + \ecinco$. + \end{description} + De aquí en adelante, no escribiremos todas las <> desde el principio, sino que las iremos añadiendo conforme las necesitemos. Tampoco seguiremos tachando ni indicando hasta dónde alcanza cada iteración para descargar las divisiones. - El resto es irreducible por los demás elementos, así que lo usaremos como elemento $g_4$. Multiplicar por un escalar un elemento no afecta al resultado, por lo tanto, intentamos obtener la expresión más sencilla para este y así + El resto de la anterior división es irreducible por los demás elementos, así que lo usaremos como elemento $g_4$. Multiplicar por un escalar un elemento no afecta al resultado, por lo tanto, intentamos obtener la expresión más sencilla para este y así \begin{equation*} g_4 := -6\left( -\frac{1}{2}x\edos - \frac{1}{3}y\etres +\ecinco\right) = \vectorvertical{0 \\ 3x \\ 2y \\ 0 \\ -6}. \end{equation*}