From 2eaa7db9fdeace9aaa19f3365888d215498dcfd4 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sat, 21 Jan 2023 13:03:00 +0100 Subject: [PATCH] feat(algebrario): aggiunge il teorema fond. dei pol. sym. --- ...ci e Teorema fondamentale dell'Algebra.tex | 250 ++++++++++++++++++ Aritmetica/aritmetica.pdf | Bin 873898 -> 911428 bytes Aritmetica/aritmetica.tex | 6 + Aritmetica/evan.sty | 1 + 4 files changed, 257 insertions(+) create mode 100644 Aritmetica/11. Polinomi simmetrici e Teorema fondamentale dell'Algebra.tex diff --git a/Aritmetica/11. Polinomi simmetrici e Teorema fondamentale dell'Algebra.tex b/Aritmetica/11. Polinomi simmetrici e Teorema fondamentale dell'Algebra.tex new file mode 100644 index 0000000..6d1dc64 --- /dev/null +++ b/Aritmetica/11. Polinomi simmetrici e Teorema fondamentale dell'Algebra.tex @@ -0,0 +1,250 @@ +\section{Polinomi simmetrici} + +\subsection{Definizione e prime proprietà} + +Sia $\KK$ un campo. Dati $\sigma \in S_n$ e un polinomio $f \in \KK[x_1, \ldots, x_n]$, +si definisce il seguente polinomio: + +\[ (\sigma \cdot f) (x_1, \ldots, x_n) = f(x_{\sigma(1)}, \ldots, x_{\sigma(n)}), \] + +\vskip 0.1in + +ossia il polinomio ottenuto permutando le variabili $x_i$ secondo $\sigma$. + +\begin{definition} + Si definisce $\Sym[X_n]$ su $K$ come il sottoanello di $\KK[x_1, \ldots, x_n]$ dei + \textbf{polinomi simmetrici}, ossia di quei polinomi tali che + $\sigma \cdot f = f$, $\forall \sigma \in S_n$. +\end{definition} + +\begin{definition} + Sia $d \in \NN$ tale che $0 \leq d \leq n$. Si definisce \textbf{polinomio simmetrico elementare} su $\Sym[X_n]$ ogni polinomio + della seguente forma: + + \[ e_d(x_1, \ldots, x_n) = \sum_{1 \leq i_1 < \cdots < i_d \leq n} \underbrace{x_{i_1} \cdots x_{i_n}}_{d\text{ volte}}, \] + + \vskip 0.1in + + dove si pone $e_0(x_1, \ldots, x_n) := 1$ +\end{definition} + +\begin{remark*} + Qualora siano noti al contesto le variabili su cui è definito $\Sym[X_n]$ si + può omettere la parentesi di $e_d$, scrivendo pertanto semplicemente + $e_d$. +\end{remark*} + +\begin{remark*} + Sia $p(x) = a_n x^n + \ldots + a_0$ un polinomio in $\KK[x]$. Siano $\lambda_1$, + ..., $\lambda_n$ le sue radici nel suo campo di spezzamento. Allora vale + che: + + \[ a_{n-i} = (-1)^i \, a_n \, e_i(\lambda_1, \ldots, \lambda_n). \] +\end{remark*} + +\begin{definition} + Sia $\alpha = (\alpha_1, \ldots, \alpha_n) \in \NN^n$, si definisce: + + \[ x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}, \quad + \card{\alpha} = \sum_{i=1}^n \alpha_i. \] +\end{definition} + +\begin{remark*} + Ogni monomio nelle variabili $x_1$, ..., $x_n$ può essere rappresentato + nella forma $x^\alpha$, ponendo $\alpha_i$ uguale al numero di volte + in cui la variabile $x_i$ compare nel monomio. +\end{remark*} + +\begin{definition} + Si definisce \textit{degree lexicographic order} (\textbf{deglex}) la seguente + relazione di ordine sui monomi monici di un polinomio: + + \[ x^\alpha > x^\beta \defiff \card{\alpha} > \card{\beta} \text{ oppure } \\ + \card{\alpha} = \card{\beta} \text{ e } \alpha > \beta \text{ secondo il LO,} \] + + \vskip 0.1in + + dove con LO si indica il \textit{lexicographic order}. +\end{definition} + +\begin{proposition} + Il \textit{deglex} è una relazione di ordine totale. +\end{proposition} + +\begin{proof} + [TODO] +\end{proof} + +\begin{proposition} + \label{prop:moltiplicazione_disuguaglianza_deglex} + Vale la seguente equivalenza: + + \[ x^\alpha x^\gamma > x^\beta x^\gamma \iff x^\alpha > x^\beta. \] +\end{proposition} + +\begin{proof} + Si dimostrano le due implicazioni separamente. \\ + + \ ($\implies$)\; Se $\card{\alpha} + \card{\gamma} > \card{\beta} + \card{\gamma}$, allora + anche $\card{\alpha} > \card{\beta}$, e dunque + $x^\alpha > x^\beta$. Altrimenti, esiste un $i \in \NN$ tale + per cui $\alpha_i + \gamma_i > \beta_i + \gamma_i$ e $\alpha_j + \gamma_j + = \beta_j + \gamma_j$ $\forall j < i$. Allora + anche $\alpha_j = \beta_j$ $\forall j < i$ e + $\alpha_i > \beta_i$. Dunque, per il LO, $\alpha + > \beta$, e quindi $x^\alpha > x^\beta$. \\ + + \ ($\,\,\Longleftarrow\,\;$)\; Se $\card{\alpha} > \card{\beta}$, allora + anche $\card{\alpha} + \card{\gamma} > \card{\beta} + \card{\gamma}$, e dunque + $x^\alpha x^\gamma > x^\beta x^\gamma$. Altrimenti, esiste un $i \in \NN$ tale + per cui $\alpha_i > \beta_i$ e $\alpha_j = \beta_j$ $\forall j < i$. Allora + anche $\alpha_j + \gamma_j = \beta_j + \gamma_j$ $\forall j < i$ e + $\alpha_i + \gamma_i > \beta_i + \gamma_i$. Dunque, per il LO, $\alpha + \gamma + > \beta + \gamma$, e quindi $x^\alpha x^\gamma > x^\beta x^\gamma$. +\end{proof} + +\begin{proposition} + \label{prop:numero_finito_soluzioni_deglex} + + Sia $\alpha \in \NN^n$. Allora esiste un numero finito di $\beta \in \NN^n$ + tale che $x^\alpha > x^\beta$. +\end{proposition} + +\begin{proof} + Siano fissati gli $\alpha_i$. Se $x^\alpha > x^\beta$, + allora vale sicuramente l'equazione: + + \[ \alpha_1 + \ldots + \alpha_n > \beta_1 + \ldots + \beta_n, \] + + \vskip 0.1in + + che ammette un numero finito di soluzioni. +\end{proof} + +\begin{definition} + Si definisce \textbf{leading term} di un polinomio in + $x_1$, ..., $x_n$ il termine $cx^\alpha$ tale che + $x^\alpha > x^\beta$, per ogni altro monomio $x^\beta$ + del polinomio. +\end{definition} + +\begin{proposition} + \label{prop:leading_term_prodotto} + + Siano $f$ e $g \in \KK[x_1, \ldots, x_n]$. + Il \textit{leading term} di $fg$ è il + prodotto dei \textit{leading term} di $f$ e di $g$. +\end{proposition} + +\begin{proof} + Siano $x^\alpha$ e $x^\beta$ i rispettivi \textit{leading term} + di $f$ e di $g$. Sia inoltre $x^\gamma$ il \textit{leading term} + di $fg$. Si assuma che $x^\gamma \neq x^\alpha x^\beta$. \\ + + Poiché ogni monomio del prodotto di $fg$ è un prodotto di due + monomi di $f$ e di $g$, $x^\gamma$ potrà scriversi come + prodotto di $x^\delta x^\zeta$, dove $x^\delta$ è un monomio + di $f$ e $x^\zeta$ è un monomio di $g$. \\ + + Poiché $x^\alpha$ è il \textit{leading term} di $f$, vale + la seguente disuguaglianza: + + \[ x^\alpha > x^\delta, \] + + \vskip 0.1in + + da cui, dalla \propref{prop:moltiplicazione_disuguaglianza_deglex}, si + ricava che: + + \[ x^\alpha x^\zeta > x^\delta x^\zeta. \] + + \vskip 0.1in + + Analogamente vale la seguente altra disuguaglianza: + + \[ x^\beta > x^\zeta, \] + + \vskip 0.1in + + da cui si ottiene che: + + \[ x^\alpha x^\beta > x^\alpha x^\zeta. \] + + \vskip 0.1in + + Combinando le due disuguaglianze si ottiene infine che: + + \[ x^\alpha x^\beta > x^\delta x^\zeta, \] + + \vskip 0.1in + + che è assurdo, dal momento che $x^\delta x^\zeta = x^\gamma$ è il \textit{leading + term} di $fg$, \Lightning{}. Quindi $x^\gamma = x^\alpha x^\beta$. +\end{proof} + +\begin{lemma} + \label{lem:leading_term_simmetrico_disuguaglianza} + Sia $c x^\alpha$ il \textit{leading term} + di $f \in \Sym[X_n]$, con $\alpha = (\alpha_1, \ldots, \alpha_n) \in \NN^n$. + Allora $\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_n$. +\end{lemma} + +\begin{proof} + Si dimostra la tesi contronominalmente. \\ + + Sia $c x^\beta$ un monomio di $f$ con $\beta = (\beta_1, \ldots, \beta_n)$ tale che esistano $i < j + \mid \beta_i < \beta_j$. Si consideri $\gamma \in \NN^n$ come + la tupla riordinata in modo decrescente di $\beta$ e sia + $\sigma \in S_n$ tale che $\gamma = (\beta_{\sigma(1)}, + \ldots, \beta_{\sigma(n)})$. \\ + + Poiché $f$ è un polinomio simmetrico, $\sigma \cdot f = f$. Quindi + $f$ ammette un monomio della forma $c x^\gamma$. Dal momento + che $\gamma > \beta$ per il LO, $x^\gamma > x^\beta$. Quindi + $c x^\beta$ non è il \textit{leading term} di $f$. +\end{proof} + +\begin{theorem}[\textit{Teorema fondamentale dei polinomi simmetrici}] + Sia $\KK$ un campo. Vale il seguente isomorfismo: + + \[ \Sym[X_n] \cong \KK\left[e_1, \ldots, e_n\right]. \] +\end{theorem} + +\begin{proof} + Sia $c x^\alpha$ il \textit{leading term} di $f$, con + $\alpha = (\alpha_1, \ldots, \alpha_n) \in \NN^n$. + Per il \lemref{lem:leading_term_simmetrico_disuguaglianza}, + $\alpha_i - \alpha_{i+1} \geq 0$ $\forall 1 \leq i < n$. \\ + + Si definisca dunque $\beta \in \NN^n$ in modo tale + che $\beta_i = \alpha_i - \alpha_{i+1} \geq 0$ $\forall 1 \leq i < n$ + e $\beta_n = \alpha_n$. \\ + + Si consideri il monomio $e_1^{\beta_1} e_2^{\beta_2} \ldots e_n^{\beta_n}$: + il suo \textit{leading term}, per la \propref{prop:leading_term_prodotto}, + è il prodotto dei \textit{leading term} dei suoi fattori, + ossia $x_1^{\alpha_1} \cdots x_n^{\alpha_n} = x^\alpha$. \\ + + Si consideri adesso come polinomio $f - c e_1^{\beta_1} e_2^{\beta_2} \ldots e_n^{\beta_n}$, + e si reiteri l'algoritmo fino a quando il risultato non è zero. Che l'algoritmo + termini è garantito dalla \propref{prop:numero_finito_soluzioni_deglex}, da cui + si desume che vi è numero finito di \textit{leading term} possibili una + volta tolto ad ogni iterazione il termine $c e_1^{\beta_1} e_2^{\beta_2} \ldots e_n^{\beta_n}$. \\ + + Infine si sarà ottenuto una rappresentazione di $f$ come combinazione di + $e_1$, ..., $e_n$. Questa rappresentazione è unica perché + i termini $e_1^{\beta_1} e_2^{\beta_2} \ldots e_n^{\beta_n}$ sono + linearmente indipendenti, dal momento che i loro + \textit{leading term} sono distinti. \\ + + Si costruisca dunque l'omomorfismo $\Pi : \Sym[X_n] \to \KK\left[e_1, \ldots, e_n\right]$ + che associa ad ogni polinomio simmetrico la sua rappresentazione in + $\KK\left[e_1, \ldots, e_n\right]$. \\ + + Si verifica che $\Pi$ è un omomorfismo. Poiché tale omomorfismo è iniettivo + e surgettivo, è un isomorfismo, da cui la tesi. +\end{proof} + +\subsection{Teorema fondamentale dell'Algebra} + + diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index b35d2f9602b32fcdd2d2c6b6d8c2545c5dc7d6f7..4cfbf562fcc8326f352c6c64fe078bc1e4a773b1 100644 GIT binary patch delta 454881 zcmZ^Lby%Fq@->6IJHg!<+%>pE2=4A090s@GHaJ0py9Jlv4#7P@a1RzBd?dSjf4h6X z`w!1sRo&IqJ@d}f=k#gH%DFyI@x=Rs1eE*O&OEV^;==uDQ6ilgV=zr$?a2E)%d?dxNSexbleHnjubQB3PqXY35q?}5MXI3P(_k@B_lnRMjnPq-Sn$}pIxS;2-;>`n zN@v$>hqVOH1*LzfA|W%asW_VliO(V#Q#au(q|sB0T%H=XkUXPs{gm1V*DeCEnk6Uz zoB%>jHZpduKg<97v|gr?KnbV?fB~uidax}u02u_SL0Np}<~S#8W+ITXl)g3`yJwy# zh#twSA1WHG;y&8tkLJen)n=6ak{XT0>`Tx$z0R{$(e=jg_GU{F{p=^t(iU0?YUrz~ zPEWPa*T$@gRG;ZCtKP1|J?l5kj8fgXU0=n4)AHtR7+%YyhuF>`fxNEy$$FKmlO`p* zOG!a^{MG*M?#}OS76`)3hx*2u^XZa9d`7RA!j*?~dptyZ=nCb$pDQ)1gu*20utO(H zZy%N(ILw-3mRBr4_aVEJ%lJGC20hLG{0XNiO1|~#bi~LtFL{h{)AxB5w7$IZL4~}9 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Polinomi simmetrici e Teorema fondamentale dell'Algebra} + +\newpage +\thispagestyle{empty} +~\newpage + \section{Riferimenti bibliografici} \printbibliography[heading=none] diff --git a/Aritmetica/evan.sty b/Aritmetica/evan.sty index 3aaa4f5..2584dc6 100644 --- a/Aritmetica/evan.sty +++ b/Aritmetica/evan.sty @@ -175,6 +175,7 @@ \DeclareMathOperator{\MCD}{MCD} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\mcm}{mcm} +\DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\tr}{tr} \let\oldemptyset\emptyset

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