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z71})a!b;hheU(zuKkCSy$MRR)FL#QI;ap$;mMgi+f8vB)qKQ+_pb=6+3YGZf)7ZyV z<7}e^boSBb0WPS{AzmJns=F%M^>Ni-TciNGWhT~f8}vvQhViSs#0`n;!wuBz@O4H_ z;kbb)UG;pAb**gBdX^d7f=Qzyf>fBzy_6D~;qD-SQ=aUdWa`@i3iFp*z z5;%FLi3JmtYMCVX8{+T-R}hGA8)_!8jSw3JtOppP;9I_!L|7SG&E-k{rJXd=+L5K1 zG8yYe(s0+Udm0EjRILL#W{Sr=S|S7;su%yF(+HXzQaz0qcY6?5juuaO0K6P0ZJRm> zA+>`ZPpu|-KFE;{_ms`#mE%kWOeFDgj#k8`222%7kW~@Wa2EtsT5QYnAlZ@2Mfr zDmdo)Y{wpen}`;(Ui5r%FpXzwva>o=vO37Ks5nOA{q&JF##vLRWyyARVm~*HxJMlP z0F(nYnpvMuxf1Pi#PBy$r09;Zj{Iz{2uE?$MhUn_@tl)kzKr6lMJbCTF5Tp%DN%b^ TbOZuGA$|co78ZF81-$> a través de cualquier epimorfismo $e$ a un homomorfismo $\phi$ y queda ilustrado por el diagrama de la derecha. \begin{ejemplo} @@ -515,7 +532,7 @@ urlcolor=black]{hyperref} Todo módulo libre es proyectivo. \end{lema} \begin{proof} - Sea $P$ un \modulo{A} libre con base $\left\lbrace p_1, p_2, \dots, p_k \right\rbrace$ y tomemos el resto de premisas como en la definición \ref{Definición Módulo Proyectivo}. Al ser $P$ un módulo libre, cualquier homomorfismo de \modulo{A}s $\phi: P \rightarrow N$ está unívocamente determinado por las imágenes de la base de $P$. Sean $m_1, m_2, \dots, m_k$ elementos de $M$ de forma que $e\!\left( m_i \right) = h\!\left( p_i \right)$, cuya existencia está garantizada por ser $e$ un epimorfismo. Definiendo $\phi\!\left(p_i \right) := m_i$ tenemos que $e\!\left( \phi\!\left( p_i \right) \right) = h\!\left( p_i \right)$ por lo que $e \circ \phi = h$. + Sea $P$ un \modulo{A} libre con base $\left\lbrace p_1, p_2, \dots, p_k \right\rbrace$ y tomemos el resto de premisas como en la Definición~\ref{Definición Módulo Proyectivo}. Al ser $P$ un módulo libre, cualquier homomorfismo de \modulo{A}s $\phi: P \rightarrow N$ está unívocamente determinado por las imágenes de la base de $P$. Sean $m_1, m_2, \dots, m_k$ elementos de $M$ de forma que $e\!\left( m_i \right) = h\!\left( p_i \right)$, cuya existencia está garantizada por ser $e$ un epimorfismo. Definiendo $\phi\!\left(p_i \right) := m_i$ tenemos que $e\!\left( \phi\!\left( p_i \right) \right) = h\!\left( p_i \right)$ por lo que $e \circ \phi = h$. \end{proof} Una introducción a los módulos proyectivos puede encontrarse en \cite[sección 1.2]{LIBROPROYECTIVOS}. @@ -527,6 +544,7 @@ urlcolor=black]{hyperref} Veamos cómo se afronta este problema a distintos niveles. \begin{ejemplo} \label{Ejemplo Pertenencia Ideal Una Variable} +\todo{Es una tontería, pondría los racionales en lugar de los reales, también para otros ejemplos.} Consideremos el \modulo{\reales} \anillodepolinomios{\reales}{x} ---solo una variable--- y a \nolbreaks{$f := x^{3} - 6x^{2} + 11x - 6$} y \nolbreaks{$g := x^{3} - 9x^{2} + 26x - 24$} dos elementos de este. Queremos determinar si un polinomio cualquiera, por ejemplo \nolbreaks{$p = 3x^7 + 4x^5 - 3x^2 + x + 1$}, pertenece al submódulo ---o en este caso, ideal--- $\engendrado{f, g}$. Como \anillodepolinomios{\reales}{x} es un dominio euclídeo, es dominio de ideales principales y, por lo tanto, \begin{equation*} \engendrado{f, g} = \engendrado{\mcd{f, g}} = \engendrado{x^2 - 5x + 6}. @@ -537,15 +555,15 @@ urlcolor=black]{hyperref} \end{equation*} Por el Teorema de la División, sabemos que este resto es el único polinomio de \anillodepolinomios{\reales}{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} - Este ejemplo también ilustra un cambio de perspectiva que será muy importante a lo largo de toda esta sección: la división deja de ser importante por los valores exactos que arroja y su relevancia es más bien cualitativa. Con esto nos referimos a que, como tal, los valores concretos del cociente y el resto no tienen ninguna relevancia a la hora de determinar si $p$ pertenece o no a \engendrado{f, g}. Conocidas las propiedades de este algoritmo de la división en \anillodepolinomios{\reales}{x}, solo nos interesa si el resto es $0$ o no, puesto que es lo que determina si $p$ pertenece o no a \engendrado{f, g}. En base a esto, nuestra motivación para la división debe cambiar de ser <> a un más abstracto <>. + Este ejemplo también ilustra un cambio de perspectiva que será muy importante a lo largo de toda esta sección: la división deja de ser importante por los valores exactos que arroja y su relevancia es más bien cualitativa. Con esto nos referimos a que, como tal, los valores concretos del cociente y el resto no tienen ninguna relevancia a la hora de determinar si $p$ pertenece o no a \engendrado{f, g}. Conocidas las propiedades de este algoritmo de la división en \anillodepolinomios{\reales}{x}, solo nos interesa si el resto es $0$ o no, puesto que es lo que determina si $p$ pertenece o no a \engendrado{f, g}. En base a esto, nuestra motivación para la división debe cambiar de ser <> a un más abstracto <>. \todo{En realidad con este lenguaje, $r$ es importante ya que determina la clase de $p$ en el cociente.} Visto así, el Ejemplo \ref{Ejemplo Pertenencia Ideal Una Variable} supone una situación idílica: por ser un dominio de ideales principales, sabemos que vamos a poder representar a cualquier conjunto generador con un solo elemento; por lo tanto, sobre cualquier conjunto generador siempre habrá definida una división, que consistirá en dividir como ya conocemos por el elemento generador. Además, por el Teorema de la División, conocemos que el representante $r$ que satisfaga $\grado r < \grado d$ es único, lo cual nos permite discernir si $p$ está en \engendrado{f, g} viendo si $r = 0$ o no. En espacios más complicados, muchas de estas buenas propiedades desaparecen e impiden extender el proceso anterior. - \begin{description} - \item[Más variables: ] Consideremos ahora el anillo de polinomios $\anillodepolinomios{\reales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Si bien este espacio no tiene por qué ser un dominio de ideales principales, al solo haber un generador podemos obviar de momento esa parte. Siguiendo la línea del ejemplo \ref{Ejemplo Pertenencia Ideal Una Variable}, para ver si $p \in \anillodepolinomios{\reales}{x, y, z}$ deberíamos dividir por $f$ y ver si el resto es $0$ o no. Pero llegados a este punto, ¿cómo deberíamos dividir? Podemos intentar imitar el algoritmo usual de la división, pero este necesita siempre una noción de orden entre los términos del divisor y también del dividendo. En este caso, ¿deberíamos tomar $4x^2y$ como <> de $f$ o igual sería mejor tomar a uno de mayor grado como $\frac{5}{2}xz^3$? Pero $\frac{5}{2}xz^3$ y $x^2yz$ tienen el mismo grado, ¿a cuál coger, entonces? La elección de este <> en dividendo y divisor cambiará el funcionamiento del algoritmo, así que debemos cerciorarnos de cuáles de estas opciones son válidas y también revisar si el algoritmo tradicional adaptado de esta manera sirve su propósito. - \item[Más generadores: ] Sin cambiar de espacio, consideremos ahora el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. En este caso, el que \anillodepolinomios{\reales}{x, y, z} no sea un dominio de ideales principales es un problema. ¿Cómo podemos dividir siquiera? Otra dificultad a la que nos enfrentamos aquí una vez hayamos entendido cómo dividir será saber si nuestro sistema de generadores es compatible con la división ya que, por ejemplo, a $x^2 + y^2 + z^2$ se le puede restar $\left( x+y \right)^2$ e $I$ podría expresarse como $\engendrado{x+y, xyz, -2xy + z^2}$. ¿Sirven ambas representaciones? - \item[Más <>: ] Por último, también debemos tener en cuenta que por lo general trabajaremos en $\Kx^r$, lo cual hará que los elementos tengan más de una componente. ¿Cómo integrar las componentes en las divisiones? + \begin{description}[wide, leftmargin=0pt,itemindent=6pt] + \item[Más variables:] Consideremos ahora el anillo de polinomios $\anillodepolinomios{\reales}{x, y, z}$, un elemento $f: = 4x^2y+ 12xy^2 + \frac{5}{2}xz^3 + x^2yz$ y el ideal $I := \engendrado{f}$. Si bien este espacio no tiene por qué ser un dominio de ideales principales, al solo haber un generador podemos obviar de momento esa parte. Siguiendo la línea del Ejemplo~\ref{Ejemplo Pertenencia Ideal Una Variable}, para ver si $p \in \anillodepolinomios{\reales}{x, y, z}$ deberíamos dividir por $f$ y ver si el resto es $0$ o no. Pero llegados a este punto, ¿cómo deberíamos dividir? Podemos intentar imitar el algoritmo usual de la división, pero este necesita siempre una noción de orden entre los términos del divisor y también del dividendo. En este caso, ¿deberíamos tomar $4x^2y$ como <> de $f$ o igual sería mejor tomar a uno de mayor grado como $\frac{5}{2}xz^3$? Pero $\frac{5}{2}xz^3$ y $x^2yz$ tienen el mismo grado, ¿cuál escoger, entonces? La elección de este <> en dividendo y divisor cambiará el funcionamiento del algoritmo, así que debemos cerciorarnos de cuáles de estas opciones son válidas y también revisar si el algoritmo tradicional adaptado de esta manera sirve su propósito. + \item[Más generadores:] Sin cambiar de espacio, consideremos ahora el ideal $I := \engendrado{x + y, xyz, x^2 + y^2 + z^2}$. En este caso, el que \anillodepolinomios{\reales}{x, y, z} no sea un dominio de ideales principales es un problema. ¿Cómo podemos dividir siquiera? Otra dificultad a la que nos enfrentamos aquí una vez hayamos entendido cómo dividir será saber si nuestro sistema de generadores es compatible con la división ya que, por ejemplo, a $x^2 + y^2 + z^2$ se le puede restar $\left( x+y \right)^2$ e $I$ podría expresarse como $\engendrado{x+y, xyz, -2xy + z^2}$. ¿Sirven ambas representaciones? + \item[Más <>:] Por último, también debemos tener en cuenta que por lo general trabajaremos en $\Kx^r$, lo cual hará que los elementos tengan más de una componente. ¿Cómo integrar las componentes en las divisiones? \end{description} 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. @@ -557,7 +575,7 @@ urlcolor=black]{hyperref} \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. \begin{definicion} - Un orden monomial $>$ es un orden total sobre los monomios en $n$ variables que respeta la multiplicación, es decir, que satisface + 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*} @@ -585,10 +603,10 @@ urlcolor=black]{hyperref} 2xy + y^2 \\ x^3 \\ y \end{psmallmatrix}$ puede escribirse como $2xy\euno + y^2\euno + x^3\edos + y\etres$. \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 monomios de \Kx. + 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 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 + 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 $x^\mathbf{\alpha} > x^\mathbf{\beta} \implies x^\mathbf{\alpha}\vectorbasecanonica{i} \mayorquemodulos x^\mathbf{\beta}\vectorbasecanonica{i}$ @@ -603,8 +621,9 @@ urlcolor=black]{hyperref} i = j \ \text{y} \ x^\mathbf{\alpha} \mayorquelex x^\mathbf{\beta}. \end{cases} \end{equation} - Sin embargo, podríamos haber tomado la decisión de que primero se comparasen $x^\mathbf{\alpha}$ y $x^\mathbf{\beta}$ y, en caso de empate, se hubiesen mirado las posiciones. Ambas posibilidades dan lugar a órdenes monomiales de módulos distintos: %Poner otra ecuación de definición con el otro orden. - Vemos pues que es necesario aclarar si la posición de los monomios debe ser evaluada antes o después, así que denotamos a estos órdenes como \mayorquemodular{POS}{LEX} para el correspondiente a la definición \eqref{Definicion Orden Lexicográfico Módulos} y \mayorquemodular{LEX}{POS} para aquel que primero compara según el orden lexicográfico y después según la posición. Por ver un caso concreto, $xy\edos \mayorquelexpos y^3\euno$ pero $y^3\euno \mayorqueposlex xy\edos$. + Sin embargo, podríamos haber tomado la decisión de que primero se comparasen $x^\mathbf{\alpha}$ y $x^\mathbf{\beta}$ y, en caso de empate, se hubiesen mirado las posiciones. Ambas posibilidades dan lugar a +distintos órdenes monomiales de módulos: %Poner otra ecuación de definición con el otro orden. + Vemos pues que es necesario aclarar si la posición de los monomios debe ser evaluada antes o después, así que denotamos a estos órdenes como \mayorquemodular{POS}{LEX} para el correspondiente a \eqref{Definicion Orden Lexicográfico Módulos} y \mayorquemodular{LEX}{POS} para aquel que primero compara según el orden lexicográfico y después según la posición. Por ver un caso concreto, $xy\edos \mayorquelexpos y^3\euno$ pero $y^3\euno \mayorqueposlex xy\edos$. Esta generalización que hemos hecho no está limitada a \mayorquelex, también se podría realizar con otros órdenes como \mayorquedeglex. Incluso se podrían permutar las posiciones para obtener todavía más órdenes. \end{ejemplo} @@ -613,18 +632,18 @@ urlcolor=black]{hyperref} \begin{definicion} Sea $p := c_1x^\mathbf{s_1}\vectorbasecanonica{i_1} + c_2x^\mathbf{s_2}\vectorbasecanonica{i_2} + \dots + c_mx^\mathbf{s_m}\vectorbasecanonica{i_m}$ un polinomio de $\Kx^r$ y $>$ un orden monomial de módulos. Sea $k \in \left\lbrace 1, \dots, m \right\rbrace$ de forma que $x^\mathbf{s_k}$ sea el máximo del conjunto de monomios de $p$ según el orden $>$, entonces \begin{itemize} - \item el monomio líder de $p$ se denomina $\LM{p}:= x^\mathbf{s_k}\vectorbasecanonica{i_k}$; - \item el coeficiente líder de $p$ es el coeficiente que acompaña al monomio líder en $p$ y se define como $\LC{p} := c_k$; - \item el término líder de $p$ es $\LT{p} := \LC{p}\LM{p}$. + \item el \emph{monomio líder} de $p$ se denomina $\LM{p}:= x^\mathbf{s_k}\vectorbasecanonica{i_k}$; + \item el \emph{coeficiente líder} de $p$ es el coeficiente que acompaña al monomio líder en $p$ y se define como $\LC{p} := c_k$; + \item el \emph{término líder} de $p$ es $\LT{p} := \LC{p}\LM{p}$. \end{itemize} \end{definicion} - Necesitamos también concretar la noción de divisibilidad para poder integrar las distintas componentes que puedan tener los elementos dentro de la división. 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 $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. + Necesitamos también concretar la noción de divisibilidad para poder integrar las distintas componentes que puedan tener los elementos dentro de la división. 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 $x^\mathbf{\alpha} \vert x^\mathbf{\beta}$. Con todo esto, ya podemos definir un algoritmo de la división de un elemento por otro en $\Kx^r$. En esencia, imitaremos al algoritmo tradicional solo que, en este caso, nos guiaremos por los monomios líderes y no por el grado. \begin{definicion} \label{Algoritmo de la División Simple} \todo{diferenciar vectores y polinomios. Por ejemplo $f$ es un vector pero cociente $c$ es un polinomio.} - Supongamos que tenemos dos polinomios $f$ y $g$ en $\Kx^n$ junto con un orden monomial sobre este. + El \emph{algoritmo de división simple} se aplica a dos elementos $f$ y $g$ de $\Kx^n$ junto con un orden monomial sobre este. \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 @@ -657,7 +676,7 @@ urlcolor=black]{hyperref} \end{block} \end{blockarray} \end{center} - \begin{description} + \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 $r = 2x^2y\edos - 6x^2y\etres -4xy\etres$. @@ -672,7 +691,7 @@ urlcolor=black]{hyperref} 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. \begin{definicion} \label{Algoritmo de la División Compuesta} - Sean $f$ un elemento y $N = \engendrado{g_1, \dots, g_k}$ un \submodulo{\Kx} de $\Kx^r$. Definimos el algoritmo de la división de $f$ entre $N$ como sigue: + Sean $f$ un elemento y $N = \engendrado{g_1, \dots, g_k}$ un \submodulo{\Kx} de $\Kx^r$. Definimos el \emph{algoritmo de división compuesta} de $f$ entre $N$ como sigue: \begin{algorithmic}[1] \State Definimos $D \gets f$, $d_i \gets g_i$, $c_i \gets 0$ y $r \gets 0$ como las variables que representan el dividendo, los divisores, los cocientes y el resto, respectivamente. \Repeat @@ -687,25 +706,25 @@ urlcolor=black]{hyperref} \end{definicion} Un ejemplo como el de \eqref{Ejemplo División Compuesta} muestra cómo funciona este algoritmo. Ahora, haría falta generalizar todavía más para ser capaces también de dividir por un número infinito de divisores, pero los siguientes resultado nos permitirán ver que esto no es necesario. \begin{teorema}[de la base de Hilbert] - Sea $A$ un anillo noetheriano. Entonces el anillo de polinomios \anillodepolinomios{A}{x_1, \dots, x_n} es también noetheriano. + Sea $A$ un anillo noetheriano. Entonces el anillo de polinomios \anillodepolinomios{A}{x_1, \dots, x_n} es también noetheriano. \todo{Di algo más.} \end{teorema} \begin{proposicion} \label{Proposición Kx^r noetheriano}%Proposición 2.1.29 de Singular. Sea $A$ un anillo noetheriano y $M$ un \modulo{A} finitamente generado. Entonces, $M$ es un \modulo{A} noetheriano. \end{proposicion} - Como $K$ es un cuerpo, todos sus ideales son principales, así que es noetheriano. Por el teorema de la base de Hilbert, \Kx{} es noetheriano también. Y por último, por la proposición \ref{Proposición Kx^r noetheriano}, $\Kx^r$ es igualmente noetheriano. Esto nos asegura que es suficiente con tener un algoritmo de la división para un número finito de divisores. + Como $K$ es un cuerpo, todos sus ideales son principales, así que es noetheriano. Por el teorema de la base de Hilbert, \Kx{} es noetheriano también. Y por último, por la Proposición~\ref{Proposición Kx^r noetheriano}, $\Kx^r$ es igualmente noetheriano. Esto nos asegura que es suficiente con tener un algoritmo de la división para un número finito de divisores. - Con respecto a los problemas que se presentaban al principio, ya hemos resuelto varios: tenemos un orden para los monomios que nos permite tener un convenio de cuáles son los términos principales de los polinomios, hemos integrado las distintas componentes de un vector de $\Kx^r$ al concepto de monomio y sabemos cómo dividir por ellas, y también sabemos cómo afrontar la división con más de un divisor. Sin embargo, todavía podemos encontrarnos con casos como el siguiente: + Con respecto a los problemas que se presentaban al principio, ya hemos resuelto varios: tenemos un orden para los monomios que nos permite tener un convenio de cuáles son los términos principales de los polinomios, hemos integrado las distintas componentes de un vector de $\Kx^r$ al concepto de monomio y sabemos cómo dividir por ellas, y también sabemos cómo afrontar la división con más de un divisor. Sin embargo, todavía podemos encontrarnos con casos como el siguiente. \begin{ejemplo} \label{Ejemplo Fallo División} Supongamos que $N := \engendrado{x\euno + y\edos, y\euno + x\edos}$ es un \submodulo{\anillodepolinomios{\reales}{x,y}} de $\anillodepolinomios{\reales}{x, y}^2$ y queremos saber si $f := x^2\edos - y^2\edos$ pertenece a $N$. Al aplicar el algoritmo de la división siguiendo el orden \mayorqueposlex{} de \ref{Algoritmo de la División Compuesta} nos encontramos que ninguno de los dos generadores de $N$ puede reducir a $f$, por lo tanto el resto es el propio $f$. Como el resto es distinto de $0$, estaríamos tentados a decir que $f$ no pertenece a $N$, pero démonos cuenta de que $f = -y\left( x\euno + y\edos \right) + x\left( y\euno + x\edos \right)$ y, por ende, sí que pertenece. \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 \submodulo{\Kx} de los monomios líderes de $G$ lo denotaremos por + 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*} \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 $\engendrado{\LM{x\euno + y\edos}, \LM{y\euno + x\edos}}$. + ¿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 $\engendrado{\LM{x\euno + y\edos}, \LM{y\euno + x\edos}}$. Esto supone un duro golpe para nuestros algoritmos de la división, ya que cuando para un \submodulo{\Kx} $N = \engendrado{g_1, \dots, g_k}$ tengamos que $\LM{N} \neq \engendrado{\LM{g_1}, \dots, \LM{g_k}}$, aunque el resto sea distinto de $0$, no podremos garantizar que el elemento esté fuera de $N$. Por supuesto, si el resto es $0$ sí que podemos garantizar que sí está. Para solucionar este problema, necesitamos encontrar una representación de los submódulos que sí sea compatible con el algoritmo de la división. Es decir, que cuando dividamos por los generadores y el resto sea distinto de $0$, podamos decir que el elemento no pertenece. @@ -714,12 +733,12 @@ urlcolor=black]{hyperref} Para conseguir esta mejor representación, simplemente exigimos la condición que discutíamos en la sección anterior. \begin{definicion} - Sea $N$ un \submodulo{\Kx} de $\Kx^r$ con un orden monomial de módulos $>$ y $G$ un subconjunto de $N$. Este subconjunto $G$ es una base de Gröbner de $N$ si $N = \engendrado{G}$ y $\engendrado{\LM{G}} = \LM{\engendrado{G}}$. + Sea $N$ un \submodulo{\Kx} de $\Kx^r$ con un orden monomial de módulos $>$ y $G$ un subconjunto de $N$. Este subconjunto $G$ es una \emph{base de Gröbner} de $N$ si $N = \engendrado{G}$ y $\engendrado{\LM{G}} = \LM{\engendrado{G}}$. \end{definicion} Para verificar si esta clase de conjuntos generadores soluciona de una vez por todas estos problemas, necesitamos unos resultados intermedios. \begin{lema}[de Dickson] - Sea \nolbreaks{$N \subseteq \naturales^n$} e $I := \engendrado{x^\mathbf{\alpha} \ \vert \ \mathbf{\alpha} \in N}$ un ideal monomial. Entonces existen $\mathbf{\alpha}_1, \mathbf{\alpha}_2, \dots, \mathbf{\alpha}_k$ en $N$ de forma que \nolbreaks{$I = \engendrado{x^{\mathbf{\alpha}_1}, \dots, x^{\mathbf{\alpha}_k}}$}. + Sea \nolbreaks{$N \subseteq \naturales^n$} e $I := \engendrado{x^\mathbf{\alpha} \ \vert \ \mathbf{\alpha} \in N}$ un ideal monomial. Entonces existen $\mathbf{\alpha}_1, \dots, \mathbf{\alpha}_k$ en $N$ de forma que \nolbreaks{$I = \engendrado{x^{\mathbf{\alpha}_1}, \dots, x^{\mathbf{\alpha}_k}}$}. \end{lema} Aunque el lema de Dickson habla de ideales monomiales dentro del anillo de polinomios \Kx, extenderlo a submódulos es inmediato. Si pensamos en un \submodulo{\Kx} $N$ de $\Kx^r$ generado por monomios, es decir $N := \engendrado{x^\mathbf{\alpha}\vectorbasecanonica{i_\mathbf{\alpha}} \vert \mathbf{\alpha} \in J}$ donde $J \subset \naturales^n$, podemos expresarlo como \begin{align*} @@ -729,13 +748,13 @@ urlcolor=black]{hyperref} &= \engendrado{x^{\mathbf{\alpha}_1}\euno, x^{\mathbf{\alpha}_2}\euno, \dots, x^{\mathbf{\alpha}_{k_1}}\euno, x^{\mathbf{\beta}_1}\edos, x^{\mathbf{\beta}_2}\edos, \dots, x^{\mathbf{\beta}_{k_2}}\edos, \dots, x^{\mathbf{\gamma}_1}\vectorbasecanonica{r}, x^{\mathbf{\gamma}_2}\vectorbasecanonica{r}, \dots, x^{\mathbf{\gamma}_{k_r}}\vectorbasecanonica{r}}. \end{align*} - Este resultado nos permite ver una condición más fuerte sobre los \submodulo{\Kx}s monomiales de $\Kx^r$: no solo son finitamente generados, como ya sabíamos gracias a la proposición \ref{Proposición Kx^r noetheriano}; sino que además admiten conjuntos generadores finitos formados por algunos monomios del conjunto generador original. + Este resultado nos permite ver una condición más fuerte sobre los \submodulo{\Kx}s monomiales de $\Kx^r$: no solo son finitamente generados, como ya sabíamos gracias a la Proposición~\ref{Proposición Kx^r noetheriano}, sino que además admiten conjuntos generadores finitos formados por algunos monomios del conjunto generador original. \begin{lema} Sea $N$ un \submodulo{\Kx} monomial de $\Kx^r$ de forma que $N = \engendrado{x^{\mathbf{\alpha}_1}\vectorbasecanonica{i_1}, x^{\mathbf{\alpha}_2}\vectorbasecanonica{i_2}, \dots, x^{\mathbf{\alpha}_k}\vectorbasecanonica{i_k}}$. Un monomio $x^\mathbf{\beta}\vectorbasecanonica{i}$ pertenece a $N$ si y solo si es un múltiplo de alguno de los monomios generadores. En otras palabras, existe $j \in \left\lbrace 1, \dots, k \right\rbrace$ de forma que $x^{\mathbf{\alpha}_j}\vectorbasecanonica{i_j} \vert x^\mathbf{\beta}\vectorbasecanonica{i}$. \end{lema} - Este pequeño resultado nos da una forma bastante rápida de determinar si un monomio pertenece o no a un \submodulo{\Kx} monomial de $\Kx^r$. Démonos cuenta de que esto es justo lo que hacemos cuando intentamos reducir el término líder de $f$ en el algoritmo de la división de \ref{Algoritmo de la División Compuesta}. Con todo esto, ya damos la respuesta final a la pregunta que abre el capítulo. + Este pequeño resultado nos da una forma bastante rápida de determinar si un monomio pertenece o no a un \submodulo{\Kx} monomial de $\Kx^r$. Démonos cuenta de que esto es justo lo que hacemos cuando intentamos reducir el término líder de $f$ en el algoritmo de la división de la Definición~\ref{Algoritmo de la División Compuesta}. Con todo esto, ya damos la respuesta final a la pregunta que abre el capítulo. \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 $0$. @@ -763,7 +782,7 @@ urlcolor=black]{hyperref} En esencia, lo que buscaremos hacer es evaluar las parejas de polinomios en el conjunto generador a través de esta expresión, si nos descubren un monomio líder nuevo, lo incorporamos a la base y, si no, seguimos. \begin{definicion} - Sea $G := \left\lbrace g_1, \dots, g_k \right\rbrace$ y $N := \engendrado{G}$. El algoritmo que utilizaremos para obtener una base de Gröbner para $N$, llamado el Algoritmo de Buchberger, se define como sigue: + Sea $G := \left\lbrace g_1, \dots, g_k \right\rbrace$ y $N := \engendrado{G}$. El algoritmo que utilizaremos para obtener una base de Gröbner para $N$, llamado el \emph{Algoritmo de Buchberger}, se define como sigue: \begin{algorithmic}[1] \State Definimos $d_i \gets g_i$ para $i \in \left\lbrace 1, \dots, k \right\rbrace$ y $t \gets k$. \While{hayan $i$ y $j$ en $\left\lbrace 1, \dots, t \right\rbrace$ de forma que el resto de $\spoly{g_i}{g_i}$ entre $d_1, \dots, d_t$ no sea $0$} @@ -775,12 +794,12 @@ urlcolor=black]{hyperref} \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}, 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 el apéndice \ref{Apéndice Demostración del Criterio de Buchberger}. + Véase el apéndice \ref{Apéndice Demostración del Criterio de Buchberger}.\todo{Creo que es mejor gastar energía en otras cosas.} \end{proof} Un ejemplo de cálculo de base de \Groebner{} puede encontrarse en apéndice \ref{Sección A mano}. @@ -794,16 +813,17 @@ urlcolor=black]{hyperref} \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} - \begin{aligned} - \sigma : A^k &\longrightarrow M \\ - a_1\euno + a_2\edos + \dots + a_k\vectorbasecanonica{k} &\longmapsto a_1f_1 + a_2f_2 + \dots + a_kf_k - \end{aligned} +\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} \end{equation} \begin{definicion} - Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Denotaremos por $\syz{f_1, f_2, \dots, f_k}$ al \submodulo{A} de $A^k$ formado por todas las sicigias de $f_1, f_2, \dots, f_k$. + Sean $A$ un anillo, $M$ un \modulo{A} y $f_1, f_2, \dots, f_k$ elementos de $M$. Denotaremos por $\syz{f_1, f_2, \dots, f_k}$ \emph{al \submodulo{A} de $A^k$ formado por todas las sicigias de} $f_1, f_2, \dots, f_k$. \end{definicion} +\todo{Dale un poco más de bombo a este resultado} Véase que \derivaciones{f}, tal y como está definido en \eqref{Definicion Modulo Derivaciones de f} puede expresarse en términos 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}; \\ @@ -830,14 +850,14 @@ urlcolor=black]{hyperref} 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} - Démonos cuenta de que esto nos permite calcular bases de \Groebner{} a $M_s$ con respecto a cualquier orden: escogemos un orden que satisfaga la condición que impone el lema \ref{Lema Restricción Groebner} con el cual $G_s$. Pero démonos cuenta de que $G_s$ es un conjunto generador de $M_s$ independientemente del orden escogido, así que cambiamos el orden y calculamos la base de \Groebner{} a partir de $G_s$ con respecto a este nuevo orden. + Démonos cuenta de que esto nos permite calcular bases de \Groebner{} a $M_s$ con respecto a cualquier orden: escogemos un orden que satisfaga la condición que impone el Lema~\ref{Lema Restricción Groebner} con el cual $G_s$. Pero démonos cuenta de que $G_s$ es un conjunto generador de $M_s$ independientemente del orden escogido, así que cambiamos el orden y calculamos la base de \Groebner{} a partir de $G_s$ con respecto a este nuevo orden. \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}). \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}--- + 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*} \syz{f_1, f_2, \dots, f_k} = \engendrado{G_{r+1}}. \end{equation*} @@ -853,7 +873,7 @@ urlcolor=black]{hyperref} \begin{equation*} \mu_1f_1 + \mu_2f_2 + \dots + \mu_kf_k = b_1f_1 + b_2f_2 + \dots + b_kf_k = 0. \end{equation*} - Por ende, $m_{r+1} \in \syz{f_1, f_2, \dots, f_k}$. En el otro sentido, si tenemos $m_{r+1} := \left( \mu_1, \mu_2, \dots, \mu_k \right) \in \syz{f_1, f_2, \dots, f_k}$, entonces existe + Por ende, $m_{r+1} \in \syz{f_1, f_2, \dots, f_k}$. En el otro sentido, si tenemos que $m_{r+1} := \left( \mu_1, \mu_2, \dots, \mu_k \right)$ pertenece a $\syz{f_1, f_2, \dots, f_k}$, entonces existe \begin{equation*} m := \mu_{r+1}\left( f_1^+ + \vectorbasecanonica{r+1} \right) + \mu_{r+2}\left( f_2^+ + \vectorbasecanonica{r+2} \right) + \dots + \mu_{r+k}\left( f_k^+ + \vectorbasecanonica{r+k} \right) \in M \end{equation*} @@ -1050,7 +1070,7 @@ urlcolor=black]{hyperref} \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. - 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 $k^{-1} \in \Kx$. + El Lema~\ref{Lema Rango Producto Exterior Módulos} nos garantiza que $\bigwedge^n \derivaciones{f}$ es un \modulo{\Kx} reflexivo de rango $1$. Siguiendo el esquema en \ref{Esquema Demostración}, tenemos que $\bigwedge^n \derivaciones{f}$ es también libre de torsión y, por lo tanto, $\bigwedge^n \derivaciones{f}$ es isomorfo a un ideal de \Kx. Por último, aplicando la última de las implicaciones, tenemos que $\bigwedge^n \derivaciones{f}$ es un \modulo{\Kx} libre y además de rango $1$. Sumado a esto tenemos que $\bigwedge^n \derivaciones{f}$ es un \submodulo{\Kx} de $\bigwedge^n \derivaciones{\Kx} = \engendrado{\partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$, por lo tanto, $\derivaciones{f} = \engendrado{p \partial_1 \wedge \partial_2 \wedge \dots \wedge \partial_n}$ para algún $p$. Nótese que este $p$ está bien definido salvo por alguna unidad de \Kx, es decir, tanto nos daría poner $p$ que $kp$ siempre y cuando exista $k^{-1} \in \Kx$. Ahora bien, recordemos la sucesión exacta de \ref{Secuencia Exacta Definición Módulo Derivaciones}. Podemos aplicar productos exteriores a los elementos, pero esto hace que la sucesión pueda dejar de ser exacta. Sin embargo, habrá un homomorfismo \begin{equation*} @@ -1091,7 +1111,7 @@ urlcolor=black]{hyperref} \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.. \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 $f$. + 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 $f$. En el caso de que $f$ ya sea irreducible, $\det\Phi$ solo puede ser $1$ o $f$. Supongamos que es $1$, entonces como hemos dicho algo más arriba, $\det A = 1 \cdot \det\iota$, que es una unidad de \Kx. Nuevamente por la regla de Cramer, tendremos que $A^{-1}$ es una matriz con elementos en \Kx. Pero esto implicaría que la expresión \ref{Relación Matricial Derivadas} es bidireccional y que podemos expresar \begin{equation*} @@ -1404,7 +1424,7 @@ x2z+y3,3z,-3y2,-2z,0, \end{lstlisting} \chapter{Demostración del Criterio de Buchberger} \label{Apéndice Demostración del Criterio de Buchberger} - Si $G$ es una base de \Groebner, entonces, como $\spoly{f}{g} \in \engendrado{G}$, por el teorema \ref{Teorema Buen Funcionamiento División Groebner} tenemos que el resto de su división entre los elementos de $G$ será $0$. + Si $G$ es una base de \Groebner, entonces, como $\spoly{f}{g} \in \engendrado{G}$, por el Teorema~\ref{Teorema Buen Funcionamiento División Groebner} tenemos que el resto de su división entre los elementos de $G$ será $0$. Como $\Kx^n$ es noetheriano, podemos asumir desde el principio que $G = \conjunto{g_1, g_2, \dots, g_k}$ \todo[inline]{Terminar}

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