From e64323fae3cd9f1890118d7c7241fa1fd6792f6e Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Mon, 12 Apr 2021 21:45:59 +0200
Subject: Polinomios

---
 ealg/n1.lyx | 2398 +++++++++++++++++++++++++++++++++++++++++++++++++++++++----
 1 file changed, 2234 insertions(+), 164 deletions(-)

(limited to 'ealg')

diff --git a/ealg/n1.lyx b/ealg/n1.lyx
index e50b9eb..54cf78e 100644
--- a/ealg/n1.lyx
+++ b/ealg/n1.lyx
@@ -134,6 +134,17 @@ variable
 \end_inset
 
 , con las operaciones
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+{
+\end_layout
+
+\end_inset
+
+
 \begin_inset Formula 
 \begin{align*}
 (a_{n})_{n}+(b_{n})_{n} & :=(a_{n}+b_{n})_{n}; & (a_{n})_{n}(b_{n})_{n} & :=\left(\sum_{k=0}^{n}a_{k}b_{n-k}\right)_{n}.
@@ -142,6 +153,17 @@ variable
 \end_inset
 
 
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -237,7 +259,7 @@ coeficiente principal
 \series bold
 mónico
 \series default
- si su coeficiente princial es 1.
+ si su coeficiente principal es 1.
  El polinomio 0 tiene grado 
 \begin_inset Formula $-\infty$
 \end_inset
@@ -291,7 +313,7 @@ Si
 \begin_inset Formula $\text{gr}(P+Q)\leq\max\{\text{gr}(P),\text{gr}(Q)\}$
 \end_inset
 
-, con desigualdad estricta si y sólo si 
+, con desigualdad estricta si y solo si 
 \begin_inset Formula $\text{gr}(P)=\text{gr}(Q)$
 \end_inset
 
@@ -306,7 +328,7 @@ Si
 \begin_inset Formula $\text{gr}(PQ)\leq\text{gr}(P)+\text{gr}(Q)$
 \end_inset
 
-, con igualdad si y sólo si 
+, con igualdad si y solo si 
 \begin_inset Formula $pq\neq0$
 \end_inset
 
@@ -317,7 +339,7 @@ Si
 \begin_inset Formula $A[X]$
 \end_inset
 
- [...] es un dominio si y sólo si lo es 
+ [...] es un dominio si y solo si lo es 
 \begin_inset Formula $A$
 \end_inset
 
@@ -446,39 +468,7 @@ A/\ker f\cong\text{Im}f.
 \end_layout
 
 \begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-eremember
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-sremember{GyA}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
+[...] 
 \series bold
 Propiedad universal del anillo de polinomios
 \series default
@@ -752,7 +742,7 @@ status open
 
 
 \backslash
-sremember
+sremember{GyA}
 \end_layout
 
 \end_inset
@@ -866,39 +856,7 @@ dominio euclídeo
 \end_layout
 
 \begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-eremember
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-sremember{GyA}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Sean 
+[...] Sean 
 \begin_inset Formula $f,g\in A[X]$
 \end_inset
 
@@ -1002,7 +960,7 @@ Teorema de Ruffini
 \begin_inset Formula $X-a$
 \end_inset
 
- si y sólo si 
+ si y solo si 
 \begin_inset Formula $f(a)=0$
 \end_inset
 
@@ -1047,7 +1005,7 @@ multiplicidad
 \begin_inset Formula $f$
 \end_inset
 
- si y sólo si 
+ si y solo si 
 \begin_inset Formula $m\geq1$
 \end_inset
 
@@ -1129,9 +1087,6 @@ Si
 \end_inset
 
 .
-\end_layout
-
-\begin_layout Standard
 \begin_inset ERT
 status open
 
@@ -1139,7 +1094,7 @@ status open
 
 
 \backslash
-eremeber
+eremember
 \end_layout
 
 \end_inset
@@ -1222,7 +1177,24 @@ derivada
 \end_inset
 
 .
- Dados 
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+Dados 
 \begin_inset Formula $a,b\in A$
 \end_inset
 
@@ -1283,7 +1255,7 @@ Si
 \begin_inset Formula $p\in A[X]$
 \end_inset
 
- si y sólo si 
+ si y solo si 
 \begin_inset Formula $p(a)=p'(a)=0$
 \end_inset
 
@@ -1330,12 +1302,24 @@ identidad de Bézout
 , que existe [...].
 \end_layout
 
+\begin_layout Standard
+[...] Si 
+\begin_inset Formula $1\in(S)$
+\end_inset
+
+, 
+\begin_inset Formula $\text{mcd}S=1$
+\end_inset
+
+.
+\end_layout
+
 \begin_layout Standard
 [...] 
 \begin_inset Formula $A[X]$
 \end_inset
 
- es un dominio euclídeo si y sólo si es un DIP, si y sólo si 
+ es un dominio euclídeo si y solo si es un DIP, si y solo si 
 \begin_inset Formula $A$
 \end_inset
 
@@ -1359,11 +1343,20 @@ eremember
 \end_layout
 
 \begin_layout Standard
-Sean 
-\begin_inset Formula $K$
+Llamamos 
+\begin_inset Formula $\text{car}A$
 \end_inset
 
- un cuerpo y 
+ a la característica del anillo 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Sean 
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ cuerpos y 
 \begin_inset Formula $f\in K[X]$
 \end_inset
 
@@ -1375,7 +1368,7 @@ Si
 \begin_inset Formula $\text{mcd}\{f,f'\}=1$
 \end_inset
 
- entonces 
+, entonces 
 \begin_inset Formula $f$
 \end_inset
 
@@ -1422,11 +1415,7 @@ Existen
 \end_deeper
 \begin_layout Enumerate
 Si 
-\begin_inset Formula $K\subseteq L$
-\end_inset
-
- son cuerpos y 
-\begin_inset Formula $f\in K[X]$
+\begin_inset Formula $f$
 \end_inset
 
  es irreducible en 
@@ -1445,7 +1434,7 @@ Si
 \begin_inset Formula $L$
 \end_inset
 
- si y sólo si 
+ si y solo si 
 \begin_inset Formula $f'=0$
 \end_inset
 
@@ -1491,133 +1480,368 @@ Si fuera
 \end_inset
 
 .
- Ahora bien,
-\begin_inset Note Note
-status open
+ Entonces existe una identidad de Bézout 
+\begin_inset Formula $pf+qf'=1$
+\end_inset
 
-\begin_layout Plain Layout
-pasar a que lo es en 
-\begin_inset Formula $L[X]$
+ con 
+\begin_inset Formula $p,q\in K[X]\subseteq L[X]$
 \end_inset
 
- por identidades de Bézout y cosas de DIP y caracterización de mcd por ideal
- principal
-\end_layout
+, de modo que 
+\begin_inset Formula $1\in(\{f,f'\})_{L[X]}$
+\end_inset
 
+ y 
+\begin_inset Formula $\text{mcd}\{f,f'\}=1$
 \end_inset
 
+ en 
+\begin_inset Formula $L[X]$
+\end_inset
 
-\end_layout
+, y por el apartado anterior, 
+\begin_inset Formula $f$
+\end_inset
 
-\end_deeper
-\begin_layout Section
-Divisibilidad
+ no tiene raíces múltiples en 
+\begin_inset Formula $L\#$
+\end_inset
+
+.
 \end_layout
 
-\begin_layout Standard
-\begin_inset ERT
+\begin_layout Enumerate
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
 
 
-\backslash
-sremember{GyA}
 \end_layout
 
 \end_inset
 
-
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $D$
+Sea 
+\begin_inset Formula $\alpha\in L$
 \end_inset
 
- un dominio y 
-\begin_inset Formula $p\in D$
+ raíz de 
+\begin_inset Formula $f$
 \end_inset
 
- [...] 
-\begin_inset Formula $p$
+, entonces 
+\begin_inset Formula $f(\alpha)=f'(\alpha)=0$
 \end_inset
 
- es irreducible en 
-\begin_inset Formula $D$
+ y 
+\begin_inset Formula $\alpha$
 \end_inset
 
- si y sólo si lo es en 
-\begin_inset Formula $D[X]$
+ es raíz múltiple de 
+\begin_inset Formula $f$
 \end_inset
 
 .
 \end_layout
 
-\begin_layout Standard
-[...] Como 
-\series bold
-teorema
-\series default
-, 
-\begin_inset Formula $D$
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{car}K=0$
 \end_inset
 
- es un DFU si y sólo si lo es 
-\begin_inset Formula $D[X]$
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ no tiene raíces múltiples en 
+\begin_inset Formula $L$
 \end_inset
 
 .
 \end_layout
 
+\begin_deeper
 \begin_layout Standard
-[...] Si 
-\begin_inset Formula $D$
+Como 
+\begin_inset Formula $f$
 \end_inset
 
- es un DFU [...][para 
-\begin_inset Formula $p\in D[X]$
+ no es cero ni unidad, 
+\begin_inset Formula $n:=\text{gr}f>0$
 \end_inset
 
-], 
-\begin_inset Formula $c(p):=\{x:x=\text{mcd}_{k\geq0}p_{k}\}$
+, y como el coeficiente principal de 
+\begin_inset Formula $f'$
 \end_inset
 
-, y [...] si 
-\begin_inset Formula $c(p)=aD^{*}$
+ es 
+\begin_inset Formula $nf_{n}\neq0$
 \end_inset
 
-, 
-\begin_inset Formula $a$
+, entonces 
+\begin_inset Formula $f'\neq0$
 \end_inset
 
- es el 
-\series bold
-contenido
-\series default
- de 
-\begin_inset Formula $p$
+ y, por lo anterior, si 
+\begin_inset Formula $f$
 \end_inset
 
- (
-\begin_inset Formula $a=c(p)$
+ tiene raíces en 
+\begin_inset Formula $L$
 \end_inset
 
-).
- [...] 
-\begin_inset Formula $p$
+, estas no son múltiples.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p:=\text{car}K\neq0$
 \end_inset
 
- es 
-\series bold
-primitivo
-\series default
- si 
-\begin_inset Formula $c(p)=1$
+, 
+\begin_inset Formula $f'=0$
 \end_inset
 
-, esto es, si [...]
-\begin_inset Formula $\text{mcd}_{k}p_{k}=1$
+ si y solo si 
+\begin_inset Formula $f\in K[X^{p}]$
+\end_inset
+
+.
+ En particular, si 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene raíces múltiples en 
+\begin_inset Formula $L$
+\end_inset
+
+ si y solo si 
+\begin_inset Formula $f\in K[X^{p}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $i\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $f_{i}\neq0$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $(f')_{i-1}=if_{i}=0$
+\end_inset
+
+, luego 
+\begin_inset Formula $p\mid i$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $f$
+\end_inset
+
+ es de la forma 
+\begin_inset Formula $f=:\sum_{j}b_{j}X^{jp}$
+\end_inset
+
+ y, sea 
+\begin_inset Formula $g:=\sum_{j}b_{j}X^{j}$
+\end_inset
+
+, 
+\begin_inset Formula $f(X)=g(X^{p})$
+\end_inset
+
+, luego 
+\begin_inset Formula $f\in K[X^{p}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $g=\sum_{j}b_{j}X^{j}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f(X)=g(X^{p})$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f=\sum_{j}b_{j}X^{jp}$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+f'=\sum_{j}b_{j}jpX^{jp-1}=0.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Divisibilidad
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $D$
+\end_inset
+
+ un dominio y 
+\begin_inset Formula $p\in D$
+\end_inset
+
+ [...] 
+\begin_inset Formula $p$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D$
+\end_inset
+
+ si y solo si lo es en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Como 
+\series bold
+teorema
+\series default
+, 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU si y solo si lo es 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU [...][para 
+\begin_inset Formula $p\in D[X]$
+\end_inset
+
+], 
+\begin_inset Formula $c(p):=\{x:x=\text{mcd}_{k\geq0}p_{k}\}$
+\end_inset
+
+, y [...] si 
+\begin_inset Formula $c(p)=aD^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ es el 
+\series bold
+contenido
+\series default
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+ (
+\begin_inset Formula $a=c(p)$
+\end_inset
+
+).
+ [...] 
+\begin_inset Formula $p$
+\end_inset
+
+ es 
+\series bold
+primitivo
+\series default
+ si 
+\begin_inset Formula $c(p)=1$
+\end_inset
+
+, esto es, si [...]
+\begin_inset Formula $\text{mcd}_{k}p_{k}=1$
 \end_inset
 
 .
@@ -1674,7 +1898,7 @@ Si
 \begin_inset Formula $K[X]$
 \end_inset
 
- si y sólo si no tiene raíces en 
+ si y solo si no tiene raíces en 
 \begin_inset Formula $K$
 \end_inset
 
@@ -1818,16 +2042,1862 @@ eremember
 \end_layout
 
 \begin_layout Standard
-\begin_inset Note Note
-status open
+La irreducibilidad se conserva por automorfismos de dominios, por lo que
+ si 
+\begin_inset Formula $D$
+\end_inset
 
-\begin_layout Plain Layout
-1.24,1.25
+ es un dominio, 
+\begin_inset Formula $a\in D^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in D$
+\end_inset
+
+, 
+\begin_inset Formula $f\in D[X]$
+\end_inset
+
+ es irreducible si y solo si lo es 
+\begin_inset Formula $f(aX+b)$
+\end_inset
+
+.
 \end_layout
 
+\begin_layout Standard
+Si 
+\begin_inset Formula $p\in\mathbb{Z}$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $f(X):=\frac{X^{p}-1}{X-1}=X^{p-1}+X^{p-2}+\dots+X+1$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $\mathbb{Q}[X]$
+\end_inset
+
+ y en 
+\begin_inset Formula $\mathbb{Z}[X]$
+\end_inset
+
+.
+ En efecto, aplicando el automorfismo 
+\begin_inset Formula $X\mapsto X+1$
+\end_inset
+
+ a 
+\begin_inset Formula $(X-1)f(X)=X^{p}-1$
+\end_inset
+
+, queda 
+\begin_inset Formula 
+\[
+Xf(X+1)=(X+1)^{p}-1=X^{p}+\binom{p}{1}X^{p-1}+\dots+\binom{p}{p-1}X,
+\]
+
+\end_inset
+
+luego 
+\begin_inset Formula 
+\[
+f(X+1)=X^{p-1}+\binom{p}{1}X^{p-2}+\dots+\binom{p}{p-1},
+\]
+
+\end_inset
+
+que es irreducible por Eisenstein porque 
+\begin_inset Formula $p$
+\end_inset
+
+ divide a 
+\begin_inset Formula $\binom{p}{i}$
 \end_inset
 
+ para 
+\begin_inset Formula $i\in\{1,\dots,p-1\}$
+\end_inset
+
+ y 
+\begin_inset Formula $p^{2}$
+\end_inset
+
+ no divide a 
+\begin_inset Formula $\binom{p}{p-1}=\binom{p}{1}=p$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Factorización en 
+\begin_inset Formula $\mathbb{C}[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{R}[X]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{FVC}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema fundamental del álgebra:
+\series default
+ 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ es algebraicamente cerrado, esto es, todo polinomio complejo de grado 
+\begin_inset Formula $n$
+\end_inset
+
+ es la forma 
+\begin_inset Formula $p(x)=\alpha\prod_{k=1}^{n}(x-a_{k})$
+\end_inset
+
+ con 
+\begin_inset Formula $\alpha,a_{1},\dots,a_{n}\in\mathbb{C}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Equivalentemente, todo polinomio complejo no constante tiene raíces en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+, y los polinomios irreducibles en 
+\begin_inset Formula $\mathbb{C}[X]$
+\end_inset
+
+ son los de grado 1.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\alpha\in\mathbb{C}$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $f\in\mathbb{R}[X]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\overline{\alpha}$
+\end_inset
+
+ también lo es, y ambas tienen la misma multiplicidad.
+ 
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene grado 
+\begin_inset Formula $n$
+\end_inset
+
+, como 
+\begin_inset Formula $f_{i}=\overline{f_{i}}$
+\end_inset
+
+ para cada 
+\begin_inset Formula $i\in\{0,\dots,n\}$
+\end_inset
+
+ por ser 
+\begin_inset Formula $f_{i}\in\mathbb{R}$
+\end_inset
+
+, se tiene 
+\begin_inset Formula 
+\[
+0=a_{0}+a_{1}\alpha+\dots+a_{n}\alpha^{n}=\overline{a_{0}+a_{1}\alpha+\dots+a_{n}\alpha^{n}}=a_{0}+a_{1}\overline{\alpha}+\dots+a_{n}\overline{\alpha}^{n}.
+\]
+
+\end_inset
+
+Además, si 
+\begin_inset Formula $(X-\alpha)^{m}\mid f$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f=(X-\alpha)^{m}g$
+\end_inset
+
+ para cierto 
+\begin_inset Formula $g\in\mathbb{C}[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $f=\overline{f}=(X-\overline{\alpha})^{m}\overline{g}$
+\end_inset
+
+, luego 
+\begin_inset Formula $(X-\overline{\alpha})^{m}\mid f$
+\end_inset
+
+, y el recíproco es análogo.
+\end_layout
+
+\begin_layout Standard
+Los irreducibles de 
+\begin_inset Formula $\mathbb{R}[X]$
+\end_inset
+
+ son los de grado 1 y los de grado 2 sin raíces reales.
+ Además, todo polinomio en 
+\begin_inset Formula $\mathbb{R}[X]$
+\end_inset
+
+ se puede expresar de forma única (salvo orden) como 
+\begin_inset Formula 
+\[
+a\prod_{i=1}^{r}(X-c_{i})^{k_{i}}\prod_{i=1}^{s}(X^{2}-2\text{Re}\alpha_{i})X+\alpha_{i}\overline{\alpha}_{i})^{m_{i}}
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $r,s\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $c_{1},\dots,c_{r}\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha_{1},\dots,\alpha_{s}\in\mathbb{C}\setminus\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $k_{1},\dots,k_{r},m_{1},\dots,m_{s}\in\mathbb{N}^{*}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $p\in\mathbb{R}[X]$
+\end_inset
+
+, si 
+\begin_inset Formula $\text{gr}p=1$
+\end_inset
+
+, 
+\begin_inset Formula $p$
+\end_inset
+
+ es irreducible, y si 
+\begin_inset Formula $\text{gr}p=2$
+\end_inset
+
+, lo es en 
+\begin_inset Formula $\mathbb{R}[X]$
+\end_inset
+
+ si y solo si tiene raíces reales.
+ Si 
+\begin_inset Formula $\text{gr}p\geq3$
+\end_inset
+
+, por el teorema fundamental del álgebra, 
+\begin_inset Formula $p$
+\end_inset
+
+ tiene una raíz 
+\begin_inset Formula $\alpha\in\mathbb{C}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $p$
+\end_inset
+
+ es divisible por 
+\begin_inset Formula $X-\alpha\in\mathbb{R}[X]$
+\end_inset
+
+, y en otro caso, 
+\begin_inset Formula $\overline{\alpha}$
+\end_inset
+
+ también es raíz, luego 
+\begin_inset Formula $p$
+\end_inset
+
+ es divisible por 
+\begin_inset Formula 
+\[
+(X-\alpha)(X-\overline{\alpha})=X^{2}-(\alpha+\overline{\alpha})X+\alpha\overline{\alpha}=X^{2}-2\text{Re}\alpha X+|\alpha|^{2}\in\mathbb{R}[X].
+\]
+
+\end_inset
+
+La segunda parte se obtiene por inducción.
+\end_layout
+
+\begin_layout Section
+Polinomios en varias variables
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados un anillo conmutativo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+, definimos el 
+\series bold
+anillo de polinomios
+\series default
+ en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas [o variables] con coeficientes en 
+\begin_inset Formula $A$
+\end_inset
+
+ como 
+\begin_inset Formula 
+\[
+A[X_{1},\dots,X_{n}]:=A[X_{1},\dots,X_{n-1}][X_{n}].
+\]
+
+\end_inset
+
+ Llamamos [...] 
+\series bold
+polinomios en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas
+\series default
+ a los elementos de 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Todo 
+\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ se escribe de forma única como suma de monomios de distinto tipo,
+\begin_inset Formula 
+\[
+p:=\sum_{i\in\mathbb{N}^{n}}p_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}},
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $p_{i}=0$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $i\in\mathbb{N}^{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] 
+\series bold
+PUAP en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas:
+\series default
+ Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo conmutativo, 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $u:A\to A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ la inclusión:
+\end_layout
+
+\begin_layout Enumerate
+Dados un homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, existe un único homomorfismo de anillos 
+\begin_inset Formula $\tilde{f}:A[X_{1},\dots,X_{n}]\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}\circ u=f$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}(X_{k})=b_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+.
+ 
+\begin_inset Formula 
+\[
+\left[\tilde{f}\left(\sum_{i\in\mathbb{N}^{n}}p_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}\right)=\sum_{i\in\mathbb{N}^{n}}f(p_{i})b_{1}^{i_{1}}\cdots b_{n}^{i_{n}}.\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Dados dos anillos conmutativos 
+\begin_inset Formula $A\subseteq B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, el 
+\series bold
+homomorfismo de 
+\series default
+[...][
+\series bold
+evaluación
+\series default
+] 
+\begin_inset Formula $S:A[X_{1},\dots,X_{n}]\to B$
+\end_inset
+
+ viene dado por 
+\begin_inset Formula $p(b_{1},\dots,b_{n}):=S(p):=\sum_{i\in\mathbb{N}^{n}}p_{i}b_{1}^{i_{1}}\cdots b_{n}^{i_{n}}$
+\end_inset
+
+[, y 
+\begin_inset Formula $S(p)$
+\end_inset
+
+ es la 
+\series bold
+evaluación
+\series default
+ o 
+\series bold
+valor
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $b:=(b_{1},\dots,b_{n})$
+\end_inset
+
+].
+ [La imagen de 
+\begin_inset Formula $S$
+\end_inset
+
+] es el subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+ generado por 
+\begin_inset Formula $A\cup\{b_{1},\dots,b_{n}\}$
+\end_inset
+
+ [o 
+\series bold
+engendrado por 
+\begin_inset Formula $b_{1},\dots,b_{n}$
+\end_inset
+
+ sobre 
+\begin_inset Formula $A$
+\end_inset
+
+
+\series default
+], 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]$
+\end_inset
+
+ [...].
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Entonces 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+ son 
+\series bold
+algebraicamente independientes
+\series default
+ sobre 
+\begin_inset Formula $A$
+\end_inset
+
+ si 
+\begin_inset Formula $\ker S=0$
+\end_inset
+
+, y son 
+\series bold
+algebraicamente dependientes
+\series default
+ en otro caso, es decir, si 
+\begin_inset Formula $b$
+\end_inset
+
+ es cero de un polinomio no nulo, en cuyo caso 
+\begin_inset Formula $\ker S$
+\end_inset
+
+ es el 
+\series bold
+ideal de las relaciones algebraicas
+\series default
+ de 
+\begin_inset Formula $b_{1},\dots,b_{n}$
+\end_inset
+
+ sobre 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Por el primer teorema de isomorfía, 
+\begin_inset Formula 
+\[
+A[b_{1},\dots,b_{n}]\cong\frac{A[X_{1},\dots,X_{n}]}{\ker S},
+\]
+
+\end_inset
+
+ y en particular, si 
+\begin_inset Formula $b_{1},\dots,b_{n}$
+\end_inset
+
+ son algebraicamente independientes, 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]\cong A[X_{1},\dots,X_{n}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Por ejemplo, 
+\begin_inset Formula $b_{1}:=1/\pi$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{2}:=1+\sqrt{\pi}$
+\end_inset
+
+ son algebraicamente dependientes, pues satisfaces 
+\begin_inset Formula $(b_{2}-1)^{2}=1/b_{1}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $b_{1}b_{2}^{2}-2b_{1}b_{2}+b_{1}-1=0$
+\end_inset
+
+, y 
+\begin_inset Formula $(b_{1},b_{2})$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $X_{1}X_{2}^{2}-2X_{1}X_{2}+X_{1}-1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $B$
+\end_inset
+
+ es un dominio, llamamos 
+\begin_inset Formula $A(b_{1},\dots,b_{n})$
+\end_inset
+
+ al cuerpo de fracciones del dominio 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]$
+\end_inset
+
+, que en general no está contenido en 
+\begin_inset Formula $B$
+\end_inset
+
+, pero sí en su cuerpo de fracciones 
+\begin_inset Formula $K$
+\end_inset
+
+, y de hecho es el menor subcuerpo de 
+\begin_inset Formula $K$
+\end_inset
+
+ que contiene a 
+\begin_inset Formula $A\cup\{b_{1},\dots,b_{n}\}$
+\end_inset
+
+.
+ En efecto, 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]$
+\end_inset
+
+ es el menor anillo que contiene a 
+\begin_inset Formula $A\cup\{b_{1},\dots,b_{n}\}$
+\end_inset
+
+, que es un dominio por ser un subanillo del dominio 
+\begin_inset Formula $B$
+\end_inset
+
+, pero todo subcuerpo de 
+\begin_inset Formula $K$
+\end_inset
+
+ que contenga a 
+\begin_inset Formula $A\cup\{b_{1},\dots,b_{n}\}$
+\end_inset
+
+ y por tanto a 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]$
+\end_inset
+
+ debe contener a los cocientes de elementos de 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]$
+\end_inset
+
+ y por tanto a 
+\begin_inset Formula $A(b_{1},\dots,b_{n})$
+\end_inset
+
+, que es un subcuerpo de 
+\begin_inset Formula $B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+2.
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ una permutación de 
+\begin_inset Formula $\mathbb{N}_{n}$
+\end_inset
+
+ con inversa 
+\begin_inset Formula $\tau:=\sigma^{-1}$
+\end_inset
+
+, tomando 
+\begin_inset Formula $B=A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{k}=X_{\sigma(k)}$
+\end_inset
+
+ en el punto anterior obtenemos un automorfismo 
+\begin_inset Formula $\hat{\sigma}$
+\end_inset
+
+ en 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ con inversa 
+\begin_inset Formula $\hat{\tau}$
+\end_inset
+
+ que permuta las indeterminadas.
+ [Llamamos 
+\begin_inset Formula $f^{\sigma}:=\hat{\sigma}(f)$
+\end_inset
+
+.]
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+3.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $A[X_{1},\dots,X_{n},Y_{1},\dots,Y_{m}]\cong A[X_{1},\dots,X_{n}][Y_{1},\dots,Y_{m}]\cong A[Y_{1},\dots,Y_{m}][X_{1},\dots,X_{n}]$
+\end_inset
+
+, por lo que en la práctica no distinguimos entre estos anillos.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+4.
+\end_layout
+
+\end_inset
+
+Todo homomorfismo de anillos conmutativos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ induce un homomorfismo 
+\begin_inset Formula $\hat{f}:A[X_{1},\dots,X_{n}]\to B[X_{1},\dots,X_{n}]$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\hat{f}(p):=\sum_{i\in\mathbb{N}^{n}}f(p_{i})X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+grado
+\series default
+ de un monomio 
+\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+ a 
+\begin_inset Formula $i_{1}+\dots+i_{n}$
+\end_inset
+
+, y grado de 
+\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]\setminus0$
+\end_inset
+
+, 
+\begin_inset Formula $\text{gr}(p)$
+\end_inset
+
+, al mayor de los grados de los monomios no nulos en la expresión por monomios
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\text{gr}(p+q)\leq\max\{\text{gr}(p),\text{gr}(q)\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(pq)\leq\text{gr}(p)+\text{gr}(q)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un polinomio es 
+\series bold
+homogéneo
+\series default
+ de grado 
+\begin_inset Formula $n$
+\end_inset
+
+ si es suma de monomios de grado 
+\begin_inset Formula $n$
+\end_inset
+
+.
+ Todo polinomio se escribe de modo único como suma de polinomios homogéneos
+ de distintos grados, [las 
+\series bold
+componentes homogéneas
+\series default
+ del polinomio.]
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Polinomios simétricos
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $f\in A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es 
+\series bold
+simétrico
+\series default
+ si 
+\begin_inset Formula $f^{\sigma}=f$
+\end_inset
+
+ para todo 
+\begin_inset Formula $\sigma\in{\cal S}_{n}$
+\end_inset
+
+, si y solo si todas sus componentes homogéneas son simétricas.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $f=f_{0}+\dots+f_{k}$
+\end_inset
+
+ la descomposición de 
+\begin_inset Formula $f$
+\end_inset
+
+ por componentes homogéneas con 
+\begin_inset Formula $\text{gr}f_{i}=i$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $\sigma\in{\cal S}_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $f_{0}+\dots+f_{k}=(f_{0}+\dots+f_{k})^{\sigma}=f_{0}^{\sigma}+\dots+f_{k}^{\sigma}$
+\end_inset
+
+ usando en la última igualdad que 
+\begin_inset Formula $f\mapsto f^{\sigma}$
+\end_inset
+
+ es un automorfismo, pero cada 
+\begin_inset Formula $f_{i}^{\sigma}$
+\end_inset
+
+ es homogéneo de grado 
+\begin_inset Formula $i$
+\end_inset
+
+, y por la unicidad de las descomposiciones, cada 
+\begin_inset Formula $f_{i}^{\sigma}=f_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $f^{\sigma}=(f_{0}+\dots+f_{k})^{\sigma}=f_{0}^{\sigma}+\dots+f_{k}^{\sigma}=f_{0}+\dots+f_{k}=f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $F\in A[X_{1},\dots,X_{n}][T]$
+\end_inset
+
+ dado por 
+\begin_inset Formula 
+\[
+F:=(T-X_{1})\cdots(T-X_{n})=:\sum_{k=0}^{n}(-1)^{k}s_{k}(X_{1},\dots,X_{n})T^{n-k},
+\]
+
+\end_inset
+
+llamamos 
+\series bold
+polinomios simétricos elementales
+\series default
+ en las indeterminadas 
+\begin_inset Formula $X_{1},\dots,X_{n}$
+\end_inset
+
+ a los polinomios 
+\begin_inset Formula $s_{k}\in A[X_{1},\dots,X_{n}]$
+\end_inset
+
+, que vienen dados por
+\begin_inset Formula 
+\[
+s_{k}(X_{1},\dots,X_{n})=\sum_{1\leq i_{1}<\dots<i_{k}\leq n}\prod_{j=1}^{k}X_{i_{j}}
+\]
+
+\end_inset
+
+y son simétricos.
+ En particular 
+\begin_inset Formula $s_{0}=1$
+\end_inset
+
+, 
+\begin_inset Formula $s_{1}=X_{1}+\dots+X_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $s_{n}=X_{1}\cdots X_{n}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\tilde{s}_{1}(X_{1},\dots,X_{n-1}),\dots,\tilde{s}_{n-1}(X_{1},\dots,X_{n-1})$
+\end_inset
+
+ son los polinomios simétricos elementales en las variables 
+\begin_inset Formula $X_{1},\dots,X_{n-1}$
+\end_inset
+
+, entonces, para 
+\begin_inset Formula $i\in\{1,\dots,n-1\}$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+\tilde{s}_{i}(X_{1},\dots,X_{n-1})=s_{i}(X_{1},\dots,X_{n-1},0).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema fundamental de los polinomios simétricos:
+\series default
+ Sea 
+\begin_inset Formula $S[X_{1},\dots,X_{n}]$
+\end_inset
+
+ el subanillo de los polinomios simétricos de 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+, el homomorfismo de evaluación 
+\begin_inset Formula $\varphi:A[X_{1},\dots,X_{n}]\to S[X_{1},\dots,X_{n}]$
+\end_inset
+
+ con 
+\begin_inset Formula $\varphi(X_{i})=s_{i}$
+\end_inset
+
+ es un isomorfismo, es decir, todo polinomio simétrico se escribe de forma
+ única como expresión polinómica en los polinomios simétricos elementales.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Fórmulas de Cardano-Vieta:
+\series default
+ Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+, 
+\begin_inset Formula $f\in A[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}\in B[X]$
+\end_inset
+
+ con 
+\begin_inset Formula $f=(X-\alpha_{1})\cdots(X-\alpha_{n})$
+\end_inset
+
+, entonces, para 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\sum_{1\leq i_{1}<\dots<i_{k}}\prod_{j=1}^{k}\alpha_{i_{j}}=(-1)^{k}f_{k}.
+\]
+
+\end_inset
+
+En efecto, sea 
+\begin_inset Formula 
+\[
+F(X_{1},\dots,X_{n},T):=(T-X_{1})\cdots(T-X_{n})=T^{n}+s_{1}(X_{1},\dots,X_{n})T^{n-1}+\dots+s_{n}(X_{1},\dots,X_{n}),
+\]
+
+\end_inset
+
+entonces
+\begin_inset Formula 
+\[
+f(X)=(X-\alpha_{1})\cdots(X-\alpha_{n})=f(\alpha_{1},\dots,\alpha_{n},X)=X^{n}+s_{1}(\alpha_{1},\dots,\alpha_{n})X^{n-1}+\dots+s_{n}(\alpha_{1},\dots,\alpha_{n}).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+orden lexicográfico
+\series default
+ en 
+\begin_inset Formula $\mathbb{N}^{n}$
+\end_inset
+
+ es el buen orden dado por 
+\begin_inset Formula $(i_{1},\dots,i_{n})<(j_{1},\dots,j_{n})$
+\end_inset
+
+ si y solo si existe 
+\begin_inset Formula $k$
+\end_inset
+
+ con 
+\begin_inset Formula $i_{k}\neq j_{k}$
+\end_inset
+
+ y, para el menor de esos 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $i_{k}<j_{k}$
+\end_inset
+
+.
+ Sea
+\begin_inset Formula 
+\[
+f=\sum_{i\in\mathbb{N}^{n}}a_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}},
+\]
+
+\end_inset
+
+llamamos 
+\series bold
+término superior
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ al término 
+\begin_inset Formula $a_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{i}\neq0$
+\end_inset
+
+ y máximo 
+\begin_inset Formula $i$
+\end_inset
+
+ en orden lexicográfico.
+ Para cada 
+\begin_inset Formula $k$
+\end_inset
+
+, el término superior de 
+\begin_inset Formula $s_{k}$
+\end_inset
+
+ es 
+\begin_inset Formula $X_{1}X_{2}\cdots X_{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un dominio y 
+\begin_inset Formula $f_{1},\dots,f_{r}\in D[X_{1},\dots,X_{n}]$
+\end_inset
+
+, el término superior de 
+\begin_inset Formula $f_{1}\cdots f_{r}$
+\end_inset
+
+ es el producto de los términos superiores de los factores.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Basta ver esto para 
+\begin_inset Formula $r=2$
+\end_inset
+
+ y el resultado se sigue por inducción.
+ Sean 
+\begin_inset Formula $f:=\sum_{i\in\mathbb{N}^{n}}a_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+ y 
+\begin_inset Formula $g:=\sum_{i\in\mathbb{N}^{n}}b_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+, entonces 
+\begin_inset Formula 
+\[
+fg=\left(\sum_{i\in\mathbb{N}^{n}}a_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}\right)\left(\sum_{i\in\mathbb{N}^{n}}b_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}\right)=\sum_{i,j\in\mathbb{N}^{n}}a_{i}b_{j}X_{1}^{i_{1}+j_{1}}\cdots X_{n}^{i_{n}+j_{n}}.
+\]
+
+\end_inset
+
+Queremos ver que, para 
+\begin_inset Formula $i,j,k,l\in\mathbb{N}^{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $i\leq k$
+\end_inset
+
+ y 
+\begin_inset Formula $j\leq l$
+\end_inset
+
+, 
+\begin_inset Formula $i+j\leq k+l$
+\end_inset
+
+, con desigualdad estricta si 
+\begin_inset Formula $(i,j)\neq(k,l)$
+\end_inset
+
+.
+ Para ello si 
+\begin_inset Formula $(i,j)=(k,l)$
+\end_inset
+
+ entonces 
+\begin_inset Formula $i+j=k+l$
+\end_inset
+
+.
+ Si solo una de las dos desigualdades es estricta, por ejemplo 
+\begin_inset Formula $i=k$
+\end_inset
+
+ y 
+\begin_inset Formula $j<l$
+\end_inset
+
+, sea 
+\begin_inset Formula $r$
+\end_inset
+
+ el menor índice con 
+\begin_inset Formula $j_{r}\neq l_{r}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $i_{p}+j_{p}=k_{p}+l_{p}$
+\end_inset
+
+ para 
+\begin_inset Formula $p<r$
+\end_inset
+
+ e 
+\begin_inset Formula $i_{r}+j_{r}<k_{r}+l_{r}$
+\end_inset
+
+.
+ Finalmente, si ambas desigualdades son estrictas, sea 
+\begin_inset Formula $r$
+\end_inset
+
+ el menor índice con 
+\begin_inset Formula $i_{r}\neq k_{r}$
+\end_inset
+
+ o 
+\begin_inset Formula $j_{r}\neq k_{r}$
+\end_inset
+
+, necesariamente 
+\begin_inset Formula $i_{r}\leq k_{r}$
+\end_inset
+
+, 
+\begin_inset Formula $j_{r}\leq k_{r}$
+\end_inset
+
+ y una de las desigualdades es estricta, luego 
+\begin_inset Formula $i_{p}+j_{p}=k_{p}+l_{p}$
+\end_inset
+
+ para 
+\begin_inset Formula $p<r$
+\end_inset
+
+ e 
+\begin_inset Formula $i_{r}+j_{r}<k_{r}+l_{r}$
+\end_inset
+
+.
+ Con esto, sean 
+\begin_inset Formula $A:=\{i\in\mathbb{N}^{n}:a_{i}\neq0\}$
+\end_inset
+
+, 
+\begin_inset Formula $B:=\{j\in\mathbb{N}^{n}:b_{j}\neq0\}$
+\end_inset
+
+, 
+\begin_inset Formula $i^{*}:=\max A$
+\end_inset
+
+ y 
+\begin_inset Formula $j^{*}:=\max B$
+\end_inset
+
+, para 
+\begin_inset Formula $(i,j)\in A\times B$
+\end_inset
+
+, como 
+\begin_inset Formula $i\leq i^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $j\leq j^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $i+j\leq i^{*}+j^{*}$
+\end_inset
+
+, luego el término superior de 
+\begin_inset Formula $fg$
+\end_inset
+
+ es el 
+\begin_inset Formula $i^{*}+j^{*}$
+\end_inset
+
+ en caso de que este sea no nulo.
+ Si además 
+\begin_inset Formula $(i,j)\neq(i^{*},j^{*})$
+\end_inset
+
+, 
+\begin_inset Formula $i+j<i^{*}+j^{*}$
+\end_inset
+
+, luego dicho término viene dado solo por 
+\begin_inset Formula $a_{i^{*}}b_{j^{*}}X_{1}^{i_{1}^{*}+j_{1}^{*}}\cdots X_{n}^{i_{n}^{*}+j_{n}^{*}}=(a_{i^{*}}X_{1}^{i_{1}^{*}}\cdots X_{n}^{i_{n}^{*}})(b_{j^{*}}X_{1}^{j_{1}^{*}}\cdots X_{n}^{j_{n}^{*}})$
+\end_inset
+
+, y efectivamente 
+\begin_inset Formula $a_{i^{*}}b_{j^{*}}\neq0$
+\end_inset
+
+ por estar en un dominio.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es simétrico con término superior 
+\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $i_{1}\geq i_{2}\geq\dots\geq i_{n}$
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues si hubiera algún 
+\begin_inset Formula $i_{k-1}<i_{k}$
+\end_inset
+
+, la transposición de índices 
+\begin_inset Formula $(k-1\,k)$
+\end_inset
+
+ nos daría un monomio mayor que también estaría en 
+\begin_inset Formula $f\#$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float algorithm
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+Entrada{Dominio $(D,+,
+\backslash
+cdot)$ y polinomio $f
+\backslash
+in D[X_1,
+\backslash
+dots,X_n]$ simétrico.}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+Salida{Polinomio $g
+\backslash
+in D[X_1,
+\backslash
+dots,X_n]$ con $g(s_1,
+\backslash
+dots,s_n)=f$.}
+\end_layout
+
+\begin_layout Plain Layout
+
+$g
+\backslash
+gets0$
+\backslash
+;
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+Mientras{$f
+\backslash
+neq0$}{
+\end_layout
+
+\begin_layout Plain Layout
+
+	Obtener el término superior
+\end_layout
+
+\begin_layout Plain Layout
+
+		$M=aX_1^{i_1}
+\backslash
+cdots X_n^{i_n}$ de $f$
+\backslash
+;
+\end_layout
+
+\begin_layout Plain Layout
+
+	$p
+\backslash
+gets aX_1^{i_1-i_2}X_2^{i_2-i_3}
+\backslash
+cdots
+\end_layout
+
+\begin_layout Plain Layout
+
+					X_{n-1}^{i_{n-1}-i_n}X_n^{i_n}$
+\backslash
+;
+\end_layout
+
+\begin_layout Plain Layout
+
+	$f
+\backslash
+gets f-p(s_1,
+\backslash
+dots,s_n)$
+\backslash
+;
+\end_layout
+
+\begin_layout Plain Layout
+
+	$g
+\backslash
+gets g+p$
+\backslash
+;
+\end_layout
+
+\begin_layout Plain Layout
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "alg:domain-esp"
+
+\end_inset
+
+Método para obtener la descomposición de un polinomio simétrico en un dominio
+ como expresión polinómica de polinomios simétricos elementales.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El algoritmo 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "alg:domain-esp"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ permite descomponer un polinomio simétrico en un dominio como expresión
+ polinómica de polinomios simétricos elementales.
+ 
+\series bold
+Demostración:
+\series default
+ Lo vemos primero para 
+\begin_inset Formula $f\in D[X_{1},\dots,X_{n}]$
+\end_inset
+
+ homogéneo de grado 
+\begin_inset Formula $d$
+\end_inset
+
+.
+ Tras tomar el término superior 
+\begin_inset Formula $M$
+\end_inset
+
+, como 
+\begin_inset Formula $i_{1}\geq i_{2}\geq\dots\geq i_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $p$
+\end_inset
+
+ es un polinomio, y como el término superior del producto es el producto
+ de los términos superiores, el término superior de 
+\begin_inset Formula $p(s_{1},\dots,s_{n})=as_{1}^{i_{1}-i_{2}}s_{2}^{i_{2}-i_{3}}\cdots s_{n-1}^{i_{n-1}-i_{n}}s_{n}^{i_{n}}$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+aX_{1}^{i_{1}-i_{2}}(X_{1}X_{2})^{i_{2}-i_{3}}\cdots(X_{1}\cdots X_{n-1})^{i_{n-1}-i_{n}}(X_{1}\cdots X_{n})^{i_{n}}=aX_{1}^{i_{1}}X_{2}^{i_{2}}\cdots X_{n}^{i_{n}}=M.
+\]
+
+\end_inset
+
+Así, al restar 
+\begin_inset Formula $f-p$
+\end_inset
+
+, los términos superiores se cancelan, y como 
+\begin_inset Formula $p(s_{1},\dots,s_{n})$
+\end_inset
+
+ es simétrico y homogéneo de grado 
+\begin_inset Formula $(i_{1}-i_{2})+2(i_{2}-i_{3})+\dots+(n-1)(i_{n-1}-i_{n})+ni_{n}=i_{1}+\dots+i_{n}=d$
+\end_inset
+
+, 
+\begin_inset Formula $f-p$
+\end_inset
+
+ es 0 o un polinomio simétrico homogéneo de grado 
+\begin_inset Formula $d$
+\end_inset
+
+ con término superior menor que el de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Como hay una cantidad finita de tuplas 
+\begin_inset Formula $(i_{1},\dots,i_{n})$
+\end_inset
+
+ con 
+\begin_inset Formula $i_{1}+\dots+i_{n}=d$
+\end_inset
+
+, en algún momento 
+\begin_inset Formula $f$
+\end_inset
+
+ se hace 0.
+ Sea ahora 
+\begin_inset Formula $f\in D[X_{1},\dots,X_{n}]$
+\end_inset
+
+ un polinomio cualquiera y 
+\begin_inset Formula $f=f_{0}+\dots+f_{k}$
+\end_inset
+
+ su descomposición en componentes homogéneas, como el término superior de
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ será el término superior de algún 
+\begin_inset Formula $f_{i}\neq0$
+\end_inset
+
+, el algoritmo va disminuyendo el término superior de los distintos 
+\begin_inset Formula $f_{i}$
+\end_inset
+
+ hasta que todos se hagan nulos.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f\in D[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es un polinomio homogéneo con término superior 
+\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+, la expresión será una 
+\begin_inset Formula $D$
+\end_inset
+
+-combinación lineal de polinomios del tipo 
+\begin_inset Formula $s_{1}^{j_{1}-j_{2}}s_{2}^{j_{2}-j_{3}}\cdots$
+\end_inset
+
+ con 
+\begin_inset Formula $j_{1}+\dots+j_{n}=i_{1}+\dots+i_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(j_{1},\dots,j_{n})\leq(i_{1},\dots,i_{n})$
+\end_inset
+
+, donde el coeficiente correspondiente a 
+\begin_inset Formula $(i_{1},\dots,i_{n})$
+\end_inset
+
+ será 
+\begin_inset Formula $a$
+\end_inset
+
+ y, para casos sencillos, podemos determinar el resto dando valores 
+\begin_inset Quotes cld
+\end_inset
+
+fáciles
+\begin_inset Quotes crd
+\end_inset
 
+ a las indeterminadas y resolviendo las ecuaciones lineales en los coeficientes
+ obtenidas.
 \end_layout
 
 \end_body
-- 
cgit v1.2.3