From b36db2de4ea5fac39b58fe7ea1f2a44f83eb77e6 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 14 Jul 2023 17:15:55 +0200 Subject: [PATCH] feat: aggiunge gli articoli dal vecchio sito di Poisson --- .../anelli_ideali_quozienti.pdf | Bin 0 -> 255257 bytes .../anelli_ideali_quozienti.tex | 418 + .../anelli_euclidei_pid_ufd.pdf | Bin 0 -> 361588 bytes .../anelli_euclidei_pid_ufd.tex | 830 + .../bibliography.bib | 20 + .../proprietà_fondamentali_zi_zx_zpx_qx.pdf | Bin 0 -> 307281 bytes .../proprietà_fondamentali_zi_zx_zpx_qx.tex | 709 + ...one dei polinomi simmetrici negli e_i.html | 13884 +++++++++++++++ ...ne dei polinomi simmetrici negli e_i.ipynb | 735 + .../spazio_vett_sottinsiemi.pdf | Bin 0 -> 222295 bytes .../spazio_vett_sottinsiemi.tex | 153 + .../spazio_duale.pdf | Bin 0 -> 253623 bytes .../spazio_duale.tex | 231 + ...azione della forma canonica di Jordan.html | 13943 ++++++++++++++++ ...zione della forma canonica di Jordan.ipynb | 840 + LICENSE | 156 + README.md | 12 + 17 files changed, 31931 insertions(+) create mode 100644 Algebra/Articoli/1. Anelli, ideali e quozienti/anelli_ideali_quozienti.pdf create mode 100644 Algebra/Articoli/1. Anelli, ideali e quozienti/anelli_ideali_quozienti.tex create mode 100644 Algebra/Articoli/2. Anelli euclidei, PID e UFD/anelli_euclidei_pid_ufd.pdf create mode 100644 Algebra/Articoli/2. Anelli euclidei, PID e UFD/anelli_euclidei_pid_ufd.tex create mode 100644 Algebra/Articoli/2. Anelli euclidei, PID e UFD/bibliography.bib create mode 100644 Algebra/Articoli/3. Proprietà fondamentali di Z[i], di Z_p[x], di Z[x] e di Q[x]/proprietà_fondamentali_zi_zx_zpx_qx.pdf create mode 100644 Algebra/Articoli/3. Proprietà fondamentali di Z[i], di Z_p[x], di Z[x] e di Q[x]/proprietà_fondamentali_zi_zx_zpx_qx.tex create mode 100644 Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.html create mode 100644 Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.ipynb create mode 100644 Geometria/Articoli/1. Lo spazio vettoriale dei sottinsiemi/spazio_vett_sottinsiemi.pdf create mode 100644 Geometria/Articoli/1. Lo spazio vettoriale dei sottinsiemi/spazio_vett_sottinsiemi.tex create mode 100644 Geometria/Articoli/2. V e il suo duale a confronto/spazio_duale.pdf create mode 100644 Geometria/Articoli/2. V e il suo duale a confronto/spazio_duale.tex create mode 100644 Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.html create mode 100644 Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.ipynb create mode 100644 LICENSE create mode 100644 README.md diff --git a/Algebra/Articoli/1. Anelli, ideali e quozienti/anelli_ideali_quozienti.pdf b/Algebra/Articoli/1. 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Anelli, ideali e quozienti/anelli_ideali_quozienti.tex b/Algebra/Articoli/1. Anelli, ideali e quozienti/anelli_ideali_quozienti.tex new file mode 100644 index 0000000..5191cf2 --- /dev/null +++ b/Algebra/Articoli/1. Anelli, ideali e quozienti/anelli_ideali_quozienti.tex @@ -0,0 +1,418 @@ +\documentclass[a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage[italian]{babel} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsopn} +\usepackage{mathtools} +\usepackage{marvosym} +\usepackage{xpatch} + +\usepackage[colorlinks=false,bookmarksopen=true]{hyperref} + +\usepackage{bookmark} + +\newcommand{\dual}[1]{#1^{*}} + +\title{Anelli, ideali e quozienti} +\author{Gabriel Antonio Videtta} +\date{\today} + +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Ker}{Ker} +\DeclareMathOperator{\Imm}{Imm} + +\setlength\parindent{0pt} + +\let\oldforall\forall +\let\forall\undefined +\DeclareMathOperator{\forall}{\oldforall} + +\let\oldexists\exists +\let\exists\undefined +\DeclareMathOperator{\exists}{\oldexists} + +\let\oldnexists\nexists +\let\nexists\undefined +\DeclareMathOperator{\nexists}{\oldnexists} + +\let\oldland\land +\let\land\undefined +\DeclareMathOperator{\land}{\oldland} + +\let\oldlor\lor +\let\lor\undefined +\DeclareMathOperator{\lor}{\oldlor} + +\let\oldlnot\lnot +\let\lnot\undefined +\DeclareMathOperator{\lnot}{\oldlnot} + +\let\oldcirc\circ +\let\circ\undefined +\DeclareMathOperator{\circ}{\oldcirc} + +\DeclareMathOperator{\existsone}{\exists !} + +\let\oldemptyset\emptyset +\let\emptyset\varnothing + +\begin{document} + +\maketitle + +\newcommand{\nsg}{\mathrel{\unlhd}} + +\newcommand{\BB}{\mathcal{B}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\FF}{\mathbb{F}_2} +\newcommand{\HH}{\mathbb{H}} +\newcommand{\NN}{\mathbb{N}} +\newcommand{\ZZ}{\mathbb{Z}} +\newcommand{\QQ}{\mathbb{Q}} +\newcommand{\RR}{\mathbb{R}} +\newcommand{\KK}{\mathbb{K}} +\newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} + +\newcommand{\ii}{\mathbf{i}} +\newcommand{\jj}{\mathbf{j}} +\newcommand{\kk}{\mathbf{k}} + +\newcommand{\MM}[2]{\mathcal{M}_{#1 \times #2}\left(\KK\right)} +\newcommand{\M}[1]{\mathcal{M}_{#1}\left(\KK\right)} +\newcommand{\Mbb}[3]{\mathcal{M}^{#1}_{#2} \left( #3 \right)} +\newcommand{\Mb}[2]{\mathcal{M}^{#1}_{#2}} + +\theoremstyle{definition} +\newtheorem{definition}{Definizione}[section] + +\renewcommand{\vec}[1]{\underline{#1}} + +\newtheorem{example}{Esempio}[section] +\newtheorem{exercise}{Esercizio}[section] +\newtheorem{theorem}{Teorema}[section] +\newtheorem{proposition}{Proposizione}[section] +\newtheorem{corollary}{Corollario}[section] +\newtheorem*{note}{Osservazione} + +\tableofcontents + +\section{Anelli e prime proprietà} + +\begin{definition} + Si definisce \textbf{anello}\footnote{In realtà, si parla in questo caso di anello \textit{con unità}, in cui vale l'assioma di esistenza di un'identità + moltiplicativa. In questo documento si identificherà con "anello" solamente un anello con unità.} una struttura algebrica + costruita su un insieme $A$ e due operazioni binarie $+$ + e $\cdot$\footnote{D'ora in avanti il punto verrà omesso.} avente le seguenti proprietà: + + \begin{itemize} + \item $\left(A,\, +\right)$ è un \textit{gruppo abeliano}, alla cui + identità, detta \textit{identità additiva}, ci si riferisce con il simbolo $0$, + \item $\forall a, b, c \in A$, $(ab)c = a(bc)$, + \item $\forall a, b, c \in A$, $(a+b)c=ac+bc$, + \item $\forall a, b, c \in A$, $a(b+c)=ab+ac$, + \item $\exists 1 \in A \mid \forall a \in A$, $1a=a=a1$, e tale $1$ viene + detto \textit{identità moltiplicativa}. + \end{itemize} +\end{definition} + +Come accade per i gruppi, gli anelli soddisfano alcune proprietà algebriche +particolari, tra le quali si citano le più importanti: + +\begin{proposition} + $\forall a \in A$, $0a=0=a0$. +\end{proposition} + +\begin{proof} + $0a=(0+0)a=0a+0a \implies 0a=0$. Analogamente $a0=a(0+0)=a0+a0 \implies a0=0$. +\end{proof} + +\begin{proposition} + $\forall a \in A$, $-(-a)=a$. +\end{proposition} + +\begin{proof} + $-(-a)-a=0 \,\land\, a-a=0 \implies -(-a)=a$, per la proprietà di unicità + dell'inverso in un gruppo\footnote{In questo caso, il gruppo additivo dell'anello.}. +\end{proof} + +\begin{proposition} + \label{prop:inverso_inverso} + $a(-b)=(-a)b=-(ab)$. +\end{proposition} + +\begin{proof} + $a(-b)+ab=a(b-b)=a0=0 \implies a(-b)=-(ab)$, per la proprietà di unicità dell'inverso in un gruppo. Analogamente $(-a)b+ab=(a-a)b=0b=0 \implies + (-a)b=-(ab)$. +\end{proof} + +\begin{corollary} + $(-1)a=a(-1)=-a$. +\end{corollary} + +\begin{proposition} + $(-a)(-b)=ab$. +\end{proposition} + +\begin{proof} + $(-a)(-b)=-(a(-b))=-(-(ab))=ab$, per la \textit{Proposizione \ref{prop:inverso_inverso}}. +\end{proof} + +Si enuncia invece adesso la nozione di \textbf{sottoanello}, in tutto e per +tutto analoga a quella di \textit{sottogruppo}. + +\begin{definition} + Si definisce sottoanello rispetto all'anello $A$ un anello $B$ avente le + seguenti proprietà: + + \begin{itemize} + \item $B \subseteq A$, + \item $0, 1 \in B$, + \item $\forall a, b \in B,$ $a + b \in B \,\land\, ab \in B$. + \end{itemize} +\end{definition} + +\begin{definition} + Un sottoanello $B$ rispetto ad $A$ si dice \textbf{proprio} se + $B \neq A$. +\end{definition} + +\begin{definition} + Un anello si dice \textbf{commutativo} se $\forall a$, $b \in A$, $ab=ba$. +\end{definition} + +\begin{example} + Un facile esempio di anello commutativo è $\ZZ/n\ZZ$. +\end{example} + +\begin{definition} + Un elemento $a$ di un anello $A$ si dice \textbf{invertibile} se + $\exists b \in A \mid ab = ba = 1$. +\end{definition} + +\begin{definition} + Dato un anello $A$, si definisce $A^*$ come l'insieme degli elementi + invertibili di $A$, che a sua volta forma un \textit{gruppo moltiplicativo}. +\end{definition} + +\begin{definition} + Un anello $A$ si dice \textbf{corpo} se $\forall a \neq 0 \in A$, $\exists b \in A \mid ab=ba=1$, + ossia se $A \setminus \{0\} = A^*$. +\end{definition} + +\begin{example} + L'esempio più rilevante di corpo è quello dei \textit{quaternioni} $\HH$, definiti + nel seguente modo: + + \[\HH = \{a+b\ii+c\jj+d\kk \mid a,\, b,\, c,\, d \in \RR\},\] + + dove: + + \[\ii^2 = \jj^2 = \kk^2 = -1, \quad \ii\jj = \kk,\, \jj\kk = \ii,\, \kk\ii = \jj. \] + + Infatti ogni elemento non nullo di $\HH$ possiede un inverso moltiplicativo: + + \[\left(a+b\ii+c\jj+d\kk\right)^{-1} = \frac{a-b\ii-c\jj-d\kk}{a^2+b^2+c^2+d^2},\] + + mentre la moltiplicazione non è commutativa. + +\end{example} + +\begin{definition} + Un anello commutativo che è anche un corpo si dice \textbf{campo}. +\end{definition} + +\begin{example} + Alcuni campi, tra i più importanti, sono $\QQ$, $\RR$, $\CC$ e $\ZZ/p\ZZ$ con + $p$ primo. +\end{example} + +\begin{definition} + Un elemento $a \neq 0$ appartenente a un anello $A$ si dice \textbf{divisore di zero} se + $\exists b \neq 0 \in A \mid ab = 0$ o $ba = 0$. +\end{definition} + +\begin{example} + $2$ è un divisore di zero in $\ZZ/6\ZZ$, infatti $2 \cdot 3 \equiv 0 \pmod 6.$ +\end{example} + +\begin{definition} + Un anello commutativo in cui non sono presenti divisori di zero si dice \textbf{dominio d'integrità}, + o più semplicemente \textit{dominio}. +\end{definition} + +\begin{proposition}[\textit{Legge di annullamento del prodotto}] + Sia $D$ un dominio. Allora $ab=0 \implies a=0 \,\lor\, b=0$. +\end{proposition} + +\begin{proof} + Siano $a$, $b \in D \mid ab = 0$. Se $a=0$, la condizione è soddisfatta. + Se invece $a \neq 0$, $b$ deve essere per forza nullo, altrimenti si + sarebbe trovato un divisore di $0$, e $D$ non sarebbe un dominio, \Lightning. +\end{proof} + +\begin{example} + L'anello dei polinomi su un campo, $\KK[x]$, è un dominio. +\end{example} + +\section{Omomorfismi di anelli e ideali} + +\begin{definition} + Un \textbf{omomorfismo di anelli}\footnote{La specificazione "di anelli" è d'ora in avanti omessa.} è una mappa $\phi : A \to B$ -- con + $A$ e $B$ anelli -- soddisfacente alcune particolari proprietà: + + \begin{itemize} + \item $\phi$ è un \textit{omomorfismo di gruppi} rispetto all'addizione + di $A$ e di $B$, ossia $\forall a, b \in A, \, \phi(a+b)=\phi(a)+\phi(b)$, + \item $\phi(ab)=\phi(a)\phi(b)$, + \item $\phi(1_A)=1_B$. + \end{itemize} +\end{definition} + +\begin{definition} + Se $\phi : A \to B$ è un omomorfismo iniettivo, si dice che + $\phi$ è un \textbf{monomorfismo}. +\end{definition} + +\begin{definition} + Se $\phi : A \to B$ è un omomorfismo suriettivo, si dice che + $\phi$ è un \textbf{epimorfismo}. +\end{definition} + +\begin{definition} + Se $\phi : A \to B$ è un omomorfismo bigettivo\footnote{Ovvero se è sia un monomorfismo che un epimorfismo.}, si dice che + $\phi$ è un \textbf{isomorfismo}. +\end{definition} + +Prima di enunciare l'analogo del \textit{Primo teorema d'isomorfismo} dei gruppi +in relazione agli anelli, si rifletta su un esempio di omomorfismo: + +\begin{example} + Sia $\phi : \ZZ \to \ZZ, k \mapsto 2k$ un omomorfismo. Esso è un monomorfismo, + infatti $\phi(x)=\phi(y) \implies 2x=2y \implies x=y$. Pertanto $\Ker \phi = \{0\}$. Sebbene $\Ker \phi < \ZZ$, esso \textbf{non è un sottoanello}\footnote{Infatti $1 \notin \Ker \phi$.}. +\end{example} + +Dunque, con lo scopo di definire meglio le proprietà di un \textit{kernel}, +così come si introdotto il concetto di \textit{sottogruppo normale} per i gruppi, si introduce ora il concetto di \textbf{ideale}. + +\begin{definition} + Si definisce ideale rispetto all'anello $A$ un insieme $I$ avente le seguenti proprietà: + + \begin{itemize} + \item $I \leq A$, + \item $\forall a \in A$, $\forall b \in I$, $ab \in I$ e $ba \in I$. + \end{itemize} +\end{definition} + +\begin{example} + \label{exmpl:polinomi} + Sia $I$ l'insieme dei polinomi di $\RR[x]$ tali che $2$ ne sia radice. Esso + altro non è che un ideale, infatti $0 \in I \,\land\, \forall f(x), g(x) \in I, (f+g)(2)=0$ (i.e. $I<\RR[x]$) e $\forall f(x) \in A, \, g(x) \in I, \, (fg)(2) = 0$. +\end{example} + +\begin{proposition} + Sia $I$ un ideale di $A$. $1 \in I \implies I = A$. +\end{proposition} + +\begin{proof} + Per le proprietà dell'ideale $I$, $\forall a \in A$, $a1 = a \in I \implies + A \subseteq I$. Dal momento che anche $I \subseteq A$, si deduce che $I = A$. +\end{proof} + +\begin{proposition} + Sia $\phi : A \to B$ un omomorfismo. $\Ker \phi$ è allora un ideale di $A$. +\end{proposition} + +\begin{proof} + Poiché $\phi$ è anche un omomorfismo tra gruppi, si deduce che $\Ker \phi \leq A$. + Inoltre $\forall a \in A$, $\forall b \in \Ker \phi$, $\phi(ab)=\phi(a)\phi(b)=\phi(a)0=0 \implies ab \in I$. +\end{proof} + +\begin{proposition} + Sia $\phi : A \to B$ un omomorfismo. $\Imm \phi$ è allora un sottoanello di $B$. +\end{proposition} + +\begin{proof} + Chiaramente $0, 1 \in \Imm \phi$, dal momento che $\phi(0) = 0,\, \phi(1)=1$. Inoltre, dalla teoria dei gruppi, si ricorda anche che $\Imm \phi \leq B$. + Infine, $\forall \phi(a),\, \phi(b) \in \Imm \phi, \, \phi(a)\phi(b) = \phi(ab) \in \Imm \phi$. +\end{proof} + +\begin{definition} + Si definisce con la notazione $(a)$ l'ideale \textit{bilatero} generato da $a$ in $A$, ossia: + + \[(a)=\{ba \mid b \in A\} \cup \{ab \mid b \in A\}.\] +\end{definition} + +\begin{definition} + Si dice che un ideale $I$ è \textit{principale} o \textbf{monogenerato}, quando $\exists a \in I \mid I = (a)$. +\end{definition} + +\begin{example} + In relazione all'\textit{Esempio \ref{exmpl:polinomi}}, l'ideale $I$ è + monogenerato\footnote{Non è un caso: $\RR[x]$, in quanto anello euclideo, si dimostra essere un PID (\textit{principal ideal domain}), ossia un dominio che ammette \textit{solo} ideali monogenerati.}. In particolare, $I=(x-2)$. +\end{example} + +\section{Anelli quoziente e teorema d'isomorfismo} + +Si definisce invece adesso il concetto di \textbf{anello quoziente}, in modo +completamente analogo a quello di \textit{gruppo quoziente}: + +\begin{definition} + Sia $A$ un anello e $I$ un suo ideale, si definisce $A/I$ l'anello ottenuto + quozientando $A$ per $I$. Gli elementi di tale anello sono le classi di equivalenza di $\sim$ (i.e. gli elementi di $A/{\sim}$), dove $\forall a$, $b \in A$, $a\sim b \iff a-b \in I$. Tali classi di equivalenza vengono indicate come + $a + I$, dove $a$ è un rappresentante della classe. L'anello è così dotato di due operazioni: + + \begin{itemize} + \item $\forall a$, $b \in A$, $(a+I)+(b+I)=(a+b)+I$, + \item $\forall a$, $b \in A$, $(a+I)(b+I)=ab+I$. + \end{itemize} +\end{definition} + +\begin{note} + L'addizione di $A/I$ è ben definita, dal momento che $I \nsg A$, in quanto sottogruppo di un gruppo abeliano. +\end{note} + +\begin{note} + Anche la moltiplicazione di $A/I$ è ben definita. Siano $a\sim a'$, $b \sim b'$ quattro elementi di $A$ tali che $a = a' + i_1$ e $b = b' + i_2$ con $i_1$, $i_2 \in I$. Allora $ab=(a'+i_1)(b'+i_2)=a'b' + \underbrace{i_1b' + i_2a' + i_1i_2}_{\in I} \implies ab \sim a'b'$. +\end{note} + +\begin{proposition} + \label{prop:quoziente_pieno} + $A/\{0\} \cong A$. +\end{proposition} + +\begin{proof} + Sia $\pi : A \to A/\{0\}$, $a \mapsto a + \{0\}$ l'omomorfismo di proiezione + al quoziente. Innanzitutto, $a \sim a' \iff a-a'=0 \iff a=a'$, per cui $\pi$ è + un monomorfismo (altrimenti si troverebbero due $a$, $b \mid a \neq b \,\land\, a \sim b$). Infine, $\pi$ è un epimorfismo, dal momento che $\forall a + \{0\} \in A/\{0\}, \, \pi(a) = a + \{0\}$. Pertanto $\pi$ è un isomorfismo. +\end{proof} + +Adesso è possibile enunciare il seguente fondamentale teorema: + +\begin{theorem}[\textit{Primo teorema d'isomorfismo}] + \label{th:primo_isomorfismo} + Sia $\phi : A \to B$ un omomorfismo. $A/\Ker \phi \cong \Imm \phi$. +\end{theorem} + +\begin{proof} + La dimostrazione procede in modo analogo a quanto visto per il teorema correlato + in teoria dei gruppi. \\ + + Sia $\zeta : A/\Ker \phi \to \Imm \phi$, $a + \Ker \phi \mapsto \phi(a)$. + Si verifica che $\zeta$ è un omomorfismo: essendolo già per i + gruppi, è sufficiente verificare che $\zeta((a+I)(b+I))=\zeta(ab+I)=\phi(ab)=\phi(a)\phi(b)=\zeta(a+I)\zeta(b+I)$. \\ + + $\zeta$ è chiaramente anche un epimorfismo, dal momento che $\forall \phi(a) \in \Imm \phi$, $\zeta(a + \Ker \phi) = \phi(a)$. Inoltre, dal momento che $\zeta(a + \Ker \phi) = 0 \iff \phi(a) = 0 \iff a + \Ker \phi = \Ker \phi$, ossia l'identità di $A/\Ker \phi$, si deduce anche che $\zeta$ è un monomorfismo. Pertanto $\zeta$ è un isomorfismo. +\end{proof} + +\begin{corollary} + Sia $\phi : A \to B$ un monomorfismo. $A \cong \Imm \phi$. +\end{corollary} + +\begin{proof} + Poiché $\phi$ è un monomorfismo, $\Ker \phi = \{0\}$. Allora, per il \textit{Primo teorema di isomorfismo}, $A/\{0\} \cong \Imm \phi$. Dalla + \textit{Proposizione \ref{prop:quoziente_pieno}}, si desume che $A \cong A/\{0\}$. Allora, per la proprietà transitiva degli isomorfismi, $A \cong \Imm \phi$. +\end{proof} + +\end{document} diff --git a/Algebra/Articoli/2. Anelli euclidei, PID e UFD/anelli_euclidei_pid_ufd.pdf b/Algebra/Articoli/2. 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zF~}zZuNw9&h-VPb;*mTKd{KnY4`2j%tOj5}W5LG&#hGyYfC~)Itz+@PeL>)dS!4{5 zfI*A_-suYGUBKdSJ&6V81L=o>O$Rtn;=#570vEt*2pAB3n7#te1{y(o07LqYz&)j* zi?^q>^C5drb~!mwX+!6u_T)`3X#-IB27UwWpm$SixH`CjPXG(@r(L4^0$} + { + $m \gets d$\; + $d \gets e \bmod d$\; + $e \gets m$\; + } +\end{algorithm} + +dove $e$ è l'MCD ricercato e l'operazione $\mathrm{mod}$ restituisce un resto della +divisione euclidea\footnote{Ossia $a \bmod b$ restituisce un $r$ tale che $\exists q + \mid a = bq+r$ con $r=0$ o $g(r)] (0, 0) -- (0.5, 0.5) node[align=center, below=3pt]{$b$}; + \draw[line width=0.7pt, ->] (0, 0) -- (-0.5, 0.5) node[align=center, below=2pt]{$ib$}; + + \draw[line width=0.5pt, ->] (0, 0) -- (0.5, 2.5) node[above=0.5pt]{$bq$}; + + \draw[line width=0.5pt, ->] (0, 0) -- (1, 2.5) node[below, right]{$a$}; + + \draw[densely dotted] (0.5, 2.5) -- (1, 2.5) node[below=4pt, left=2.5pt]{$r$}; + + \draw[line width=0.2pt, ->] (0, -1) -- (0, 3); + \draw[line width=0.2pt, ->] (-3, 0) -- (3, 0); + + \end{scope} + \end{tikzpicture} + \caption{Visualizzazione della divisione euclidea nel piano degli interi di Gauss.} + \label{fig:z_i} +\end{wrapfigure} + +La funzione grado coincide in particolare con il quadrato del modulo di un numero complesso, ossia: +\[g(z) : \ZZ[i] \setminus \{0\} \to \NN, \, a+bi \mapsto \left| a+bi \right|^2.\] + +Il vantaggio di quest'ultima definizione è l'enfasi sul collegamento tra la funzione grado +di $\ZZ$ e quella di $\ZZ[i].$ Infatti, se $a \in \ZZ$, il grado di $a$ in $\ZZ$ e in $\ZZ[i]$ +sono uno il quadrato dell'altro. In particolare, è possibile ridefinire il grado +di $\ZZ$ proprio in modo tale da farlo coincidere con quello di $\ZZ[i]$. \\ + +\newpage + +\begin{theorem} + $\ZZ[i]$ è un anello euclideo. +\end{theorem} + +\begin{proof} + Si verifica la prima proprietà della funzione grado. Siano $a$, $b \in \ZZ[i] \setminus \{0\}$, + allora $\left|a\right| \geq 1 \,\land\, \left|b\right| \geq 1$. Poiché + $\left|ab\right| = \left|a\right|\left|b\right|$\footnote{Questa interessante proprietà del modulo è alla base dell'identità di Brahmagupta-Fibonacci: $(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2.$}, si verifica facilmente che + $\left|ab\right| \geq \left|a\right|$, ossia che $g(ab) \geq g(a)$. \\ + + Si verifica infine che esiste una divisione euclidea, ossia che + $\forall a \in \ZZ[i]$, $\forall b \in \ZZ[i] \setminus \{0\}$, $\exists q$, $r \in \ZZ[i] \mid a = bq + r$ e $r=0 \,\lor\, g(r) < g(b)$. + Come si visualizza facilmente nella \textit{Figura \ref{fig:z_i}}, + tutti i multipli di $b$ formano un piano con basi $b$ e $ib$, dove + sicuramente esiste un certo $q$ tale che la distanza $\left|r\right| = \left|a-bq\right|$ sia minima. \\ + + Se $a$ è un multiplo di $b$, vale sicuramente che $a = bq$. Altrimenti dal momento che $r$ è sicuramente inquadrato in uno dei tasselli del piano, vale + sicuramente la seguente disuguaglianza, che lega il modulo di $r$ alla diagonale di + ogni quadrato: + + \[\left|r\right| \leq \frac{\left|b\right|}{\sqrt{2}}.\] + + Pertanto vale la seconda e ultima proprietà della funzione grado: + + \[\left|r\right| \leq \frac{\left|b\right|}{\sqrt{2}} < \left|b\right| \implies \left|r\right|^2 < \left|b\right|^2 \implies g(r) < g(b).\] +\end{proof} + +\subsection{Gli interi di Eisenstein: $\ZZ[\omega]$} + +Sulla scia di $\ZZ[i]$ è possibile definire anche l'anello degli +interi di Eisenstein, aggiungendo a $\ZZ$ la prima radice cubica +primitiva dell'unità in senso antiorario, ossia: + +\[\omega = e^{\frac{2\pi i}{3}} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i.\] + +In particolare, $\omega$ è una delle due radici dell'equazione +$z^2 + z + 1 = 0$, dove invece l'altra radice altro non è che +$\omega^2 = \overline{\omega}$. + +\begin{wrapfigure}{l}{0pt} + \begin{tikzpicture} + \begin{scope} + \clip (-2, -0.5) rectangle (3, 3); + \draw[step=0.25cm, gray!20!white, very thin] (-7, -3) grid (7, 3); + + \foreach \x in {-4,...,4} { + \draw[ultra thin, loosely dashdotted] (-3 + 0.87*\x, -3) -- (3 + 0.87*\x, 3); + } + + \foreach \y in {-4,...,5} { + \draw[ultra thin, loosely dashdotted] (-7, 1.8756443470179 + 0.65*\y) -- (7, -1.8756443470179 + 0.65*\y); + } + + \foreach \x in {-4,...,5} { + \draw[ultra thin, loosely dashed] (-7 + 0.6289*\x, 28.5025773880714) -- (7+ 0.65*\x, -28.5025773880714); + } + + \draw[line width=0.7pt, ->] (0, 0) -- (0.5, 0.5) node[align=center, below=3pt]{$b$}; + \draw[line width=0.7pt, ->] (0, 0) -- (-0.6830127018922, 0.1830127018922) node[align=center, below=2pt]{$\omega b$}; + + \draw[line width=0.5pt, ->] (0, 0) -- (0.71494, 2.41094) node[below=2pt, left=4pt]{$bq$}; + + \draw[line width=0.5pt, ->] (0, 0) -- (1.1, 2.7) node[below, right]{$a$}; + + \draw[densely dotted] (0.71494, 2.41094) -- (1.1, 2.7) node[above=3pt, left=2.5pt]{$r$}; + + \draw[line width=0.2pt, ->] (0, -1) -- (0, 3); + \draw[line width=0.2pt, ->] (-3, 0) -- (3, 0); + + \end{scope} + \end{tikzpicture} + \caption{Visualizzazione della divisione euclidea nel piano degli interi di Eisenstein.} + \label{fig:z_omega} +\end{wrapfigure} + +\vskip 0.1in + +La funzione grado in $\ZZ[\omega]$ deriva da quella di $\ZZ[i]$ e coincide ancora +con il quadrato del modulo del numero complesso. Si definisce quindi: + +\[g : \ZZ[\omega] \setminus \{0\}, \, a+b\omega \mapsto \left|a+b\omega\right|^2.\] + +Sviluppando il modulo è possibile ottenere una formula più concreta: + +\[ \left|a+b\omega\right|^2 = \left|\left(a-\frac{b}{2}\right) + \frac{b\sqrt{3}}{2}i\right|^2 =\] \\ + +\[= \left(a-\frac{b}{2}\right)^2 + \frac{3b^2}{4} = a^2 - ab + b^2.\] \\ + +\begin{theorem} + $\ZZ[\omega]$ è un anello euclideo. +\end{theorem} + +\begin{proof} + Sulla scia della dimostrazione presentata per $\ZZ[i]$, si verifica facilmente + la prima proprietà della funzione grado. Siano $a$, $b \in \ZZ[\omega]$, allora + $\left|a\right| \geq 1$ e $\left|b\right| \geq 1$. Poiché dalle proprietà + dei numeri complessi vale ancora $\left|a\right| \left|b\right| \geq \left|a\right|$, + la proprietà $g(ab) \geq g(a)$ è già verificata. \\ + + Si verifica infine la seconda e ultima proprietà della funzione grado. Come per + $\ZZ[i]$, i multipli di $b \in \ZZ[\omega]$ sono visualizzati su un piano che + ha per basi $b$ e $\omega b$ (come in $\textit{Figura \ref{fig:z_omega}}$), pertanto + esiste sicuramente un $q$ tale che la distanza $\left|a-bq\right|$ sia minima. \\ + + Se $a$ è multiplo di $b$, allora chiaramente $a = bq$. Altrimenti, $a$ è certamente + inquadrato in uno dei triangoli del piano, per cui vale la seguente disuguaglianza: + + \[\left|r\right| \leq \frac{\sqrt{3}}{2} \left|b\right|.\] + + Dunque la tesi è verificata: + + \[\left|r\right| \leq \frac{\sqrt{3}}{2} \left|b\right| < \left|b\right| \implies \left|r\right|^2 < \left|b\right|^2 \implies g(r) < g(b). \] +\end{proof} + +\section{Riferimenti bibliografici} + +\printbibliography[heading=none] + +\end{document} diff --git a/Algebra/Articoli/2. Anelli euclidei, PID e UFD/bibliography.bib b/Algebra/Articoli/2. Anelli euclidei, PID e UFD/bibliography.bib new file mode 100644 index 0000000..1bf709f --- /dev/null +++ b/Algebra/Articoli/2. Anelli euclidei, PID e UFD/bibliography.bib @@ -0,0 +1,20 @@ +@book{di2013algebra, + title={Algebra}, + author={Di Martino, P. and Dvornicich, R.}, + isbn={9788867410958}, + series={Didattica e Ricerca. Manuali}, + year={2013}, + publisher={Pisa University Press} +}, +@article{10.2307/2315810, + ISSN = {00029890, 19300972}, + URL = {http://www.jstor.org/stable/2315810}, + author = {M. A. Jodeit}, + journal = {The American Mathematical Monthly}, + number = {7}, + pages = {835--836}, + publisher = {Mathematical Association of America}, + title = {Uniqueness in the Division Algorithm}, + volume = {74}, + year = {1967} +} diff --git a/Algebra/Articoli/3. Proprietà fondamentali di Z[i], di Z_p[x], di Z[x] e di Q[x]/proprietà_fondamentali_zi_zx_zpx_qx.pdf b/Algebra/Articoli/3. 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+ +
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+

Algoritmo di rappresentazione dei polinomi simmetrici negli $e_i$

Lo scopo di questo notebook è implementare l'algoritmo impiegato nella dimostrazione del Teorema fondamentale dei polinomi simmetrici, che, dato un campo $F$, asserisce il seguente isomorfismo:

+$$\operatorname{Sym}[X_n] \cong F[e_1, \ldots, e_n],$$

dove $X_n$ è l'insieme $\{x_1, \ldots, x_n\}$.

+

Innanzitutto, si sceglie un $n \in \mathbb{N}^+$, che rappresenta il numero di variabili di cui è composto il polinomio simmetrico, +e si crea l'anello di polinomi $\mathbb{Q}[x_1, \ldots, x_n]$, nel quale vale il degree lexicographic order (deglex). Vengono +infine create delle variabili simboliche a rappresentare i polinomi simmetrici elementari.

+ +
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+
In [1]:
+
+
+
%display latex
+n = 5
+
+P = PolynomialRing(QQ, ",".join(f"x_{i}" for i in range(1, n+1)), order='deglex')
+P.inject_variables()
+
+var(",".join(f"e_{i}" for i in range(n+1)))
+
+ +
+
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+ +
+
+ + +
+ +
+ + +
+
Defining x_1, x_2, x_3, x_4, x_5
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Out[1]:
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Si definisce il valore che deve assumere la funzione $e(d)$, che rappresenta il polinomio simmetrico $e_d$, nel +seguente modo:

+$$e(d) = \begin{cases}1 & \text{se }d=0, \\ \displaystyle \sum_{1\leq i_1 < i_2 < \cdots < i_d \leq n} \underbrace{x_{i_1} \cdots x_{i_d}}_{d\text{ volte}} & \text{altrimenti.} \end{cases}$$ +
+
+
+
+
+
In [2]:
+
+
+
from functools import reduce
+from itertools import combinations
+from operator import mul
+
+def e(d: Integer):
+    if d == 0:
+        return 1
+    
+    return sum(reduce(mul, [P(f"x_{j}") for j in i], 1) for i in combinations(range(1, n+1), d))
+
+ +
+
+
+ +
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+

Si definisce una funzione $\beta(\operatorname{exp\_lt})$ che prende in ingresso una tupla ordinata $\operatorname{exp\_lt} \in \mathbb{N}^n$ e restituisce una tupla dello stesso tipo definita nel seguente modo:

+$$\beta(\operatorname{exp\_lt})_i = \begin{cases} \operatorname{exp\_lt}_i - \operatorname{exp\_lt}_{i+1} & \text{se }1 \leq i < n, \\ \operatorname{exp\_lt}_i & \text{altrimenti.} \end{cases}$$ +
+
+
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+
In [3]:
+
+
+
def beta(exp_lt):
+    return [e - exp_lt[i+1] if i != n-1 else e for i, e in enumerate(exp_lt)]
+
+ +
+
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+ +
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+
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+

Si definiscono due funzioni analoghe, dette $\operatorname{e\_prod}$ e $\operatorname{e\_prod\_value}$. La seconda restituisce lo stesso polinomio di $\operatorname{e\_prod}$ sostituendo ai simboli $e_i$ i corrispettivi valori $e(i)$. Pertanto si definisce solo il valore della prima funzione.

+

Tale funzione $\operatorname{e\_prod}$ prende in ingresso una tupla $b \in \mathbb{N}^n$ e restituisce $e^b = {e_1}^{b_1} {e_2}^{b_2} \cdots {e_n}^{b_n}$.

+ +
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In [4]:
+
+
+
def e_prod(b):
+    return reduce(mul, [eval(f"e_{i+1}")^k for i, k in enumerate(b)], 1)
+
+
+def e_prod_value(b):
+    return reduce(mul, [e(i+1)^k for i, k in enumerate(b)], 1)
+
+ +
+
+
+ +
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+

Si definisce infine la funzione $\operatorname{combination}(\operatorname{poly}(x))$, che prende in ingresso un polinomio simmetrico $\operatorname{poly}(x)$ e ne restituisce la rappresentazione in $F[e_1, \ldots, e_n]$, secondo il seguente algoritmo.

+
    +
  • se $\operatorname{poly}(x)$ è $0$ o ha grado nullo, la sua rappresentazione in $F[e_1, \ldots, e_n]$ è già $\operatorname{poly}(x)$, e quindi la funzione restituisce il polinomio in ingresso senza modificarlo,
  • +
  • altrimenti, si considera, secondo il deglex, il leading term di $\operatorname{poly}(x)$, detto $\operatorname{lt}$:
      +
    • detta $\alpha$ la tupla ordinata degli esponenti di $\operatorname{lt}$, si calcola $\beta(\alpha)$, detto $b$,
    • +
    • detto $c$ il coefficiente razionale di $\operatorname{lt}$, si restituisce la rappresentazione simbolica $c \cdot \operatorname{e\_prod(b)}$, a cui si aggiunge ricorsivamente la rappresentazione del polinomio $\operatorname{poly(x)} -\, c \cdot \operatorname{e\_prod\_value(b)}$, ottenuta reiterando l'algoritmo su di esso.
    • +
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  • +
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In [5]:
+
+
+
def combination(poly):
+    if isinstance(poly, Integer) or isinstance(poly, Rational) or poly == 0 or poly.degree() == 0:
+        return poly
+    
+    lt = sorted(list(poly), reverse=True)[0]
+    
+    try:
+        b = beta(lt[1].exponents()[0])
+        return lt[0] * e_prod(b) + combination(poly - lt[0] * e_prod_value(b))
+    except (AttributeError, TypeError):
+        raise TypeError("The given polynomial is not of symmetric kind")
+
+ +
+
+
+ +
+
+
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+

Si sceglie inoltre un polinomio $\operatorname{poly}(x)$ che deve essere simmetrico -- qualora non fosse tale, l'algoritmo non terminerà con successo (se terminasse con successo, il polinomio sarebbe combinazione di polinomi simmetrici, e sarebbe dunque anch'esso simmetrico, ↯).

+ +
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In [6]:
+
+
+
# Il polinomio deve essere simmetrico...
+poly = x_1 + x_2 + x_3 + x_4 + x_5 + (x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 - x_1*x_2*x_3*x_4*x_5)^2
+poly
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Out[6]:
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+

Si calcola adesso la rappresentazione di $\operatorname{poly}(x)$ in funzione dei polinomi simmetrici elementari secondo l'implementazione di $\operatorname{combination}$:

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In [7]:
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combination(poly)
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Data una somma di potenze simmetrica, è possibile implementare un algoritmo di rappresentazione negli $e_i$ secondo le identità di Newton-Girard. Si definisce allora la funzione $\operatorname{newton\_girard}(k)$, che prende in ingresso un $k \in \mathbb{N}$ e restituisce la rappresentazione di $\sum_{i=1}^n {x_i}^k$ nel seguente modo:

+$$\operatorname{newton\_girard}(k) = \begin{cases} n & \text{se } k = 0, \\ (-1)^{k+1} \cdot (k e_{k} + \sum_{i=1}^{k-1} (-1)^i \operatorname{newton\_girard}(i) \, e_{k-1}) & \text{altrimenti}, \end{cases}$$

dove si pone $e_i = 0$ per ogni $i > n$.

+

La funzione è implementata con l'ausilio di un sistema di caching, affinché non vengano ricalcolati i valori già calcolati.

+ +
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In [8]:
+
+
+
from functools import lru_cache
+
+@lru_cache(maxsize=None)
+def newton_girard(k):    
+    if k == 0:
+        return n
+    
+    return ((-1)**(k+1) * ((k * eval(f"e_{k}") if k <= n else 0) +
+                           sum((-1)**i * newton_girard(i) * eval(f"e_{k - i}") for i in range(max(1, k-n), k)))).expand()
+
+ +
+
+
+ +
+
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+
+

Si definisce infine la funzione $\operatorname{sum\_of\_powers}(k)$, che, dato in ingresso un $k \in \mathbb{N}$, restituisce la rappresentazione di $\sum_{i=1}^n (x_i)^k$ secondo la funzione $\operatorname{combination}$ già definita.

+ +
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In [9]:
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+
def sum_of_powers(k):
+    return combination(sum(eval(f"x_{i}")^k for i in range(1, n+1)))
+
+ +
+
+
+ +
+
+
+
+

Si sceglie infine un $k \in \mathbb{N}$ al quale elevare tutti le variabili della somma.

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In [10]:
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k = 8
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Si computa la rappresentazione di $\sum_{i=1}^n (x_i)^k$ prima secondo $\operatorname{newton\_girard}$, e poi secondo $\operatorname{sum\_of\_powers}$, e si verifica che siano uguali.

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In [11]:
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a = newton_girard(k)
+a
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Out[11]:
+ + + +
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In [12]:
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b = sum_of_powers(k)
+b
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+ +
Out[12]:
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In [13]:
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a - b
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+

Sia ora:

+$$f(x) = x^n + a_{n-1} x^{n-1} \ldots + a_0,$$

con radici $x_1$, $x_2$, ..., $x_n$. Allora si definisce discriminante di $f$ la seguente produttoria:

+$$\Delta(f) = \displaystyle \prod_{1 \leq i < j \leq n} (x_i - x_j)^2.$$

In particolare, vale la seguente proprietà:

+$$\exists\, i \neq j \mid x_i = x_j \iff \Delta(f) = 0.$$ +
+
+
+
+
+
+

Si verifica facilmente che $\Delta(f)$ è un polinomio simmetrico. Poiché ogni permutazione è una composizione di trasposizioni, è sufficiente verificare che invertendo due radici $x_i$ e $x_j$ il discriminante rimanga invariato:

+
    +
  • i fattori che contengono solo uno tra $x_i$ e $x_j$ mantengono la somma della base invariata o al più cambiano di segno, e dunque, elevando al quadrato, rimangono invariati,
  • +
  • il fattore che contiene sia $x_i$ che $x_j$ cambia la propria base di segno, ma rimane invariato elevando al quadrato.
  • +
+

Poiché $\Delta(f)$ è un polinomio simmetrico nelle variabili $x_1$, ..., $x_n$, per il Teorema fondamentale dei polinomi simmetrici, si scrive in modo unico in funzione dei polinomi simmetrici elementari $e_i(x_1, \ldots, x_n)$. Tuttavia tali polinomi simmetrici elementari altro non sono che i coefficienti di $f(x)$, a meno del segno. In particolare vale che:

+$$e_i(x_1, \ldots, x_n) = (-1)^{i} a_{n-i}, \quad \text{per } 0 \leq i \leq n.$$ +
+
+
+
+
+
+

Si definiscono quindi i simboli $a_0$, ..., $a_n$ e si costruisce una funzione $\Delta(n)$ che restituisce il discriminante di un generico polinomio di $n$-esimo grado dato in ingresso in funzione degli $a_i$ mediante $\operatorname{combination}$.

+ +
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+
In [14]:
+
+
+
var(",".join(f"a_{i}" for i in range(n+1)))
+
+def delta(n):
+    
+    f = reduce(mul, (eval(f"(x_{i}-x_{j})")^2 for i, j in combinations(range(1, n+1), 2)), 1)
+    c = combination(f)
+
+    for i in range(1, n+1):
+        if i % 2:
+            c = eval(f"c.substitute(e_{i}=-a_{n-i})")
+        else:
+            c = eval(f"c.substitute(e_{i}=a_{n-i})")
+
+    return c
+
+ +
+
+
+ +
+
+
+
+

Dunque, per l'$n$ scelto in partenza, il discriminante del generico polinomio di $n$-esimo grado è il seguente:

+ +
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In [15]:
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delta(n)
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Out[15]:
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+ +
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+ +
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+ +
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+

Infine si costruisce la funzione $\operatorname{evaluate\_delta}$ che, dato in ingresso un polinomio $f(x)$ di $n$-esimo grado, restituisce il valore di $\Delta(f)$, sostituendo agli $a_i$ generici di $\operatorname{delta}(n)$ i valori dei coefficienti di $f$.

+ +
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+
In [16]:
+
+
+
def evaluate_delta(f):
+    d = delta(n)
+    
+    for i, a in enumerate(f.list()):
+        d = eval(f"d.substitute(a_{i}=a)")
+        
+    return d
+
+ +
+
+
+ +
+
+
+
+

Si verifica adesso che per un polinomio con radici multiple il valore di $\operatorname{evaluate\_delta}$ è precisamente zero:

+ +
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+
In [17]:
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f = (x-2)^2*(x-3)*(x-4)*(x-5)
+f
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+ +
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+ +
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+ + +
+ +
Out[17]:
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In [18]:
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f.expand()
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+ +
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+ +
Out[18]:
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In [19]:
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evaluate_delta(f)
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+ +
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+ +
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+ + +
+ +
Out[19]:
+ + + +
+ +
+ +
+ +
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+ +
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+

Infine, si fa lo stesso con un polinomio con radici distinte, per verificare che $\operatorname{evaluate\_delta}$ è strettamente diverso da zero.

+ +
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+
+
+
+
In [20]:
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g = (x+2)*(x+I)*(x-I)*(x-1)*(x-2)
+g
+
+ +
+
+
+ +
+
+ + +
+ +
Out[20]:
+ + + +
+ +
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+ +
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+ +
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In [21]:
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g.expand()
+
+ +
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+ +
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+ + +
+ +
Out[21]:
+ + + +
+ +
+ +
+ +
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+ +
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+
In [22]:
+
+
+
evaluate_delta(g)
+
+ +
+
+
+ +
+
+ + +
+ +
Out[22]:
+ + + +
+ +
+ +
+ +
+
+ +
+
+
+
+
+

(c) 2022, ~videtta

+ +
+
+
+
+
+ + + + + + diff --git a/Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.ipynb b/Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.ipynb new file mode 100644 index 0000000..57c0822 --- /dev/null +++ b/Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.ipynb @@ -0,0 +1,735 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Algoritmo di rappresentazione dei polinomi simmetrici negli $e_i$\n", + "\n", + "Lo scopo di questo notebook è implementare l'algoritmo impiegato nella dimostrazione del _Teorema fondamentale dei polinomi simmetrici_, che, dato un campo $F$, asserisce il seguente isomorfismo:\n", + "\n", + "$$\\operatorname{Sym}[X_n] \\cong F[e_1, \\ldots, e_n],$$\n", + "\n", + "dove $X_n$ è l'insieme $\\{x_1, \\ldots, x_n\\}$.\n", + "\n", + "Innanzitutto, si sceglie un $n \\in \\mathbb{N}^+$, che rappresenta il numero di variabili di cui è composto il polinomio simmetrico,\n", + "e si crea l'anello di polinomi $\\mathbb{Q}[x_1, \\ldots, x_n]$, nel quale vale il _degree lexicographic order_ (**deglex**). Vengono\n", + "infine create delle variabili simboliche a rappresentare i polinomi simmetrici elementari." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Defining x_1, x_2, x_3, x_4, x_5\n" + ] + }, + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(e_{0}, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}\\right)\n", + "\\end{math}" + ], + "text/plain": [ + "(e_0, e_1, e_2, e_3, e_4, e_5)" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "%display latex\n", + "n = 5\n", + "\n", + "P = PolynomialRing(QQ, \",\".join(f\"x_{i}\" for i in range(1, n+1)), order='deglex')\n", + "P.inject_variables()\n", + "\n", + "var(\",\".join(f\"e_{i}\" for i in range(n+1)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si definisce il valore che deve assumere la funzione $e(d)$, che rappresenta il polinomio simmetrico $e_d$, nel\n", + "seguente modo:\n", + "\n", + "$$e(d) = \\begin{cases}1 & \\text{se }d=0, \\\\ \\displaystyle \\sum_{1\\leq i_1 < i_2 < \\cdots < i_d \\leq n} \\underbrace{x_{i_1} \\cdots x_{i_d}}_{d\\text{ volte}} & \\text{altrimenti.} \\end{cases}$$" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "from functools import reduce\n", + "from itertools import combinations\n", + "from operator import mul\n", + "\n", + "def e(d: Integer):\n", + " if d == 0:\n", + " return 1\n", + " \n", + " return sum(reduce(mul, [P(f\"x_{j}\") for j in i], 1) for i in combinations(range(1, n+1), d))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si definisce una funzione $\\beta(\\operatorname{exp\\_lt})$ che prende in ingresso una tupla ordinata $\\operatorname{exp\\_lt} \\in \\mathbb{N}^n$ e restituisce una tupla dello stesso tipo definita nel seguente modo:\n", + "\n", + "$$\\beta(\\operatorname{exp\\_lt})_i = \\begin{cases} \\operatorname{exp\\_lt}_i - \\operatorname{exp\\_lt}_{i+1} & \\text{se }1 \\leq i < n, \\\\ \\operatorname{exp\\_lt}_i & \\text{altrimenti.} \\end{cases}$$" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "def beta(exp_lt):\n", + " return [e - exp_lt[i+1] if i != n-1 else e for i, e in enumerate(exp_lt)]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si definiscono due funzioni analoghe, dette $\\operatorname{e\\_prod}$ e $\\operatorname{e\\_prod\\_value}$. La seconda restituisce lo stesso polinomio di $\\operatorname{e\\_prod}$ sostituendo ai simboli $e_i$ i corrispettivi valori $e(i)$. Pertanto si definisce solo il valore della prima funzione.\n", + "\n", + "Tale funzione $\\operatorname{e\\_prod}$ prende in ingresso una tupla $b \\in \\mathbb{N}^n$ e restituisce $e^b = {e_1}^{b_1} {e_2}^{b_2} \\cdots {e_n}^{b_n}$." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "def e_prod(b):\n", + " return reduce(mul, [eval(f\"e_{i+1}\")^k for i, k in enumerate(b)], 1)\n", + "\n", + "\n", + "def e_prod_value(b):\n", + " return reduce(mul, [e(i+1)^k for i, k in enumerate(b)], 1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si definisce infine la funzione $\\operatorname{combination}(\\operatorname{poly}(x))$, che prende in ingresso un polinomio simmetrico $\\operatorname{poly}(x)$ e ne restituisce la rappresentazione in $F[e_1, \\ldots, e_n]$, secondo il seguente algoritmo.\n", + "\n", + " - se $\\operatorname{poly}(x)$ è $0$ o ha grado nullo, la sua rappresentazione in $F[e_1, \\ldots, e_n]$ è già $\\operatorname{poly}(x)$, e quindi la funzione restituisce il polinomio in ingresso senza modificarlo,\n", + " - altrimenti, si considera, secondo il _deglex_, il _leading term_ di $\\operatorname{poly}(x)$, detto $\\operatorname{lt}$:\n", + " - detta $\\alpha$ la tupla ordinata degli esponenti di $\\operatorname{lt}$, si calcola $\\beta(\\alpha)$, detto $b$,\n", + " - detto $c$ il coefficiente razionale di $\\operatorname{lt}$, si restituisce la rappresentazione simbolica $c \\cdot \\operatorname{e\\_prod(b)}$, a cui si aggiunge ricorsivamente la rappresentazione del polinomio $\\operatorname{poly(x)} -\\, c \\cdot \\operatorname{e\\_prod\\_value(b)}$, ottenuta reiterando l'algoritmo su di esso." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "def combination(poly):\n", + " if isinstance(poly, Integer) or isinstance(poly, Rational) or poly == 0 or poly.degree() == 0:\n", + " return poly\n", + " \n", + " lt = sorted(list(poly), reverse=True)[0]\n", + " \n", + " try:\n", + " b = beta(lt[1].exponents()[0])\n", + " return lt[0] * e_prod(b) + combination(poly - lt[0] * e_prod_value(b))\n", + " except (AttributeError, TypeError):\n", + " raise TypeError(\"The given polynomial is not of symmetric kind\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si sceglie inoltre un polinomio $\\operatorname{poly}(x)$ che _deve_ essere simmetrico -- qualora non fosse tale, l'algoritmo non terminerà con successo (se terminasse con successo, il polinomio sarebbe combinazione di polinomi simmetrici, e sarebbe dunque anch'esso simmetrico, ↯)." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x_{1}^{2} x_{2}^{2} x_{3}^{2} x_{4}^{2} x_{5}^{2} - 2 x_{1}^{3} x_{2} x_{3} x_{4} x_{5} - 2 x_{1} x_{2}^{3} x_{3} x_{4} x_{5} - 2 x_{1} x_{2} x_{3}^{3} x_{4} x_{5} - 2 x_{1} x_{2} x_{3} x_{4}^{3} x_{5} - 2 x_{1} x_{2} x_{3} x_{4} x_{5}^{3} + x_{1}^{4} + 2 x_{1}^{2} x_{2}^{2} + 2 x_{1}^{2} x_{3}^{2} + 2 x_{1}^{2} x_{4}^{2} + 2 x_{1}^{2} x_{5}^{2} + x_{2}^{4} + 2 x_{2}^{2} x_{3}^{2} + 2 x_{2}^{2} x_{4}^{2} + 2 x_{2}^{2} x_{5}^{2} + x_{3}^{4} + 2 x_{3}^{2} x_{4}^{2} + 2 x_{3}^{2} x_{5}^{2} + x_{4}^{4} + 2 x_{4}^{2} x_{5}^{2} + x_{5}^{4} + x_{1} + x_{2} + x_{3} + x_{4} + x_{5}\n", + "\\end{math}" + ], + "text/plain": [ + "x_1^2*x_2^2*x_3^2*x_4^2*x_5^2 - 2*x_1^3*x_2*x_3*x_4*x_5 - 2*x_1*x_2^3*x_3*x_4*x_5 - 2*x_1*x_2*x_3^3*x_4*x_5 - 2*x_1*x_2*x_3*x_4^3*x_5 - 2*x_1*x_2*x_3*x_4*x_5^3 + x_1^4 + 2*x_1^2*x_2^2 + 2*x_1^2*x_3^2 + 2*x_1^2*x_4^2 + 2*x_1^2*x_5^2 + x_2^4 + 2*x_2^2*x_3^2 + 2*x_2^2*x_4^2 + 2*x_2^2*x_5^2 + x_3^4 + 2*x_3^2*x_4^2 + 2*x_3^2*x_5^2 + x_4^4 + 2*x_4^2*x_5^2 + x_5^4 + x_1 + x_2 + x_3 + x_4 + x_5" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Il polinomio deve essere simmetrico...\n", + "poly = x_1 + x_2 + x_3 + x_4 + x_5 + (x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 - x_1*x_2*x_3*x_4*x_5)^2\n", + "poly" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si calcola adesso la rappresentazione di $\\operatorname{poly}(x)$ in funzione dei polinomi simmetrici elementari secondo l'implementazione di $\\operatorname{combination}$:" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1}^{4} - 4 \\, e_{1}^{2} e_{2} - 2 \\, e_{1}^{2} e_{5} + 4 \\, e_{2}^{2} + 4 \\, e_{2} e_{5} + e_{5}^{2} + e_{1}\n", + "\\end{math}" + ], + "text/plain": [ + "e_1^4 - 4*e_1^2*e_2 - 2*e_1^2*e_5 + 4*e_2^2 + 4*e_2*e_5 + e_5^2 + e_1" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "combination(poly)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Data una somma di potenze simmetrica, è possibile implementare un algoritmo di rappresentazione negli $e_i$ secondo le [identità di Newton-Girard](https://it.wikipedia.org/w/index.php?title=Identit%C3%A0_di_Newton&oldid=127634236#Enunciato_in_forma_relativa_ai_polinomi_simmetrici_elementari). Si definisce allora la funzione $\\operatorname{newton\\_girard}(k)$, che prende in ingresso un $k \\in \\mathbb{N}$ e restituisce la rappresentazione di $\\sum_{i=1}^n {x_i}^k$ nel seguente modo:\n", + "\n", + "$$\\operatorname{newton\\_girard}(k) = \\begin{cases} n & \\text{se } k = 0, \\\\ (-1)^{k+1} \\cdot (k e_{k} + \\sum_{i=1}^{k-1} (-1)^i \\operatorname{newton\\_girard}(i) \\, e_{k-1}) & \\text{altrimenti}, \\end{cases}$$\n", + "\n", + "dove si pone $e_i = 0$ per ogni $i > n$.\n", + "\n", + "La funzione è implementata con l'ausilio di un sistema di _caching_, affinché non vengano ricalcolati i valori già calcolati." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "from functools import lru_cache\n", + "\n", + "@lru_cache(maxsize=None)\n", + "def newton_girard(k): \n", + " if k == 0:\n", + " return n\n", + " \n", + " return ((-1)**(k+1) * ((k * eval(f\"e_{k}\") if k <= n else 0) +\n", + " sum((-1)**i * newton_girard(i) * eval(f\"e_{k - i}\") for i in range(max(1, k-n), k)))).expand()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si definisce infine la funzione $\\operatorname{sum\\_of\\_powers}(k)$, che, dato in ingresso un $k \\in \\mathbb{N}$, restituisce la rappresentazione di $\\sum_{i=1}^n (x_i)^k$ secondo la funzione $\\operatorname{combination}$ già definita." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "def sum_of_powers(k):\n", + " return combination(sum(eval(f\"x_{i}\")^k for i in range(1, n+1)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si sceglie infine un $k \\in \\mathbb{N}$ al quale elevare tutti le variabili della somma." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "k = 8" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si computa la rappresentazione di $\\sum_{i=1}^n (x_i)^k$ prima secondo $\\operatorname{newton\\_girard}$, e poi secondo $\\operatorname{sum\\_of\\_powers}$, e si verifica che siano uguali." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1}^{8} - 8 \\, e_{1}^{6} e_{2} + 20 \\, e_{1}^{4} e_{2}^{2} + 8 \\, e_{1}^{5} e_{3} - 16 \\, e_{1}^{2} e_{2}^{3} - 32 \\, e_{1}^{3} e_{2} e_{3} - 8 \\, e_{1}^{4} e_{4} + 2 \\, e_{2}^{4} + 24 \\, e_{1} e_{2}^{2} e_{3} + 12 \\, e_{1}^{2} e_{3}^{2} + 24 \\, e_{1}^{2} e_{2} e_{4} + 8 \\, e_{1}^{3} e_{5} - 8 \\, e_{2} e_{3}^{2} - 8 \\, e_{2}^{2} e_{4} - 16 \\, e_{1} e_{3} e_{4} - 16 \\, e_{1} e_{2} e_{5} + 4 \\, e_{4}^{2} + 8 \\, e_{3} e_{5}\n", + "\\end{math}" + ], + "text/plain": [ + "e_1^8 - 8*e_1^6*e_2 + 20*e_1^4*e_2^2 + 8*e_1^5*e_3 - 16*e_1^2*e_2^3 - 32*e_1^3*e_2*e_3 - 8*e_1^4*e_4 + 2*e_2^4 + 24*e_1*e_2^2*e_3 + 12*e_1^2*e_3^2 + 24*e_1^2*e_2*e_4 + 8*e_1^3*e_5 - 8*e_2*e_3^2 - 8*e_2^2*e_4 - 16*e_1*e_3*e_4 - 16*e_1*e_2*e_5 + 4*e_4^2 + 8*e_3*e_5" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "a = newton_girard(k)\n", + "a" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{1}^{8} - 8 \\, e_{1}^{6} e_{2} + 20 \\, e_{1}^{4} e_{2}^{2} + 8 \\, e_{1}^{5} e_{3} - 16 \\, e_{1}^{2} e_{2}^{3} - 32 \\, e_{1}^{3} e_{2} e_{3} - 8 \\, e_{1}^{4} e_{4} + 2 \\, e_{2}^{4} + 24 \\, e_{1} e_{2}^{2} e_{3} + 12 \\, e_{1}^{2} e_{3}^{2} + 24 \\, e_{1}^{2} e_{2} e_{4} + 8 \\, e_{1}^{3} e_{5} - 8 \\, e_{2} e_{3}^{2} - 8 \\, e_{2}^{2} e_{4} - 16 \\, e_{1} e_{3} e_{4} - 16 \\, e_{1} e_{2} e_{5} + 4 \\, e_{4}^{2} + 8 \\, e_{3} e_{5}\n", + "\\end{math}" + ], + "text/plain": [ + "e_1^8 - 8*e_1^6*e_2 + 20*e_1^4*e_2^2 + 8*e_1^5*e_3 - 16*e_1^2*e_2^3 - 32*e_1^3*e_2*e_3 - 8*e_1^4*e_4 + 2*e_2^4 + 24*e_1*e_2^2*e_3 + 12*e_1^2*e_3^2 + 24*e_1^2*e_2*e_4 + 8*e_1^3*e_5 - 8*e_2*e_3^2 - 8*e_2^2*e_4 - 16*e_1*e_3*e_4 - 16*e_1*e_2*e_5 + 4*e_4^2 + 8*e_3*e_5" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "b = sum_of_powers(k)\n", + "b" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}0\n", + "\\end{math}" + ], + "text/plain": [ + "0" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "a - b" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Sia ora:\n", + "\n", + "$$f(x) = x^n + a_{n-1} x^{n-1} \\ldots + a_0,$$\n", + "\n", + "con radici $x_1$, $x_2$, ..., $x_n$. Allora si definisce **discriminante** di $f$ la seguente produttoria:\n", + "\n", + "$$\\Delta(f) = \\displaystyle \\prod_{1 \\leq i < j \\leq n} (x_i - x_j)^2.$$\n", + "\n", + "In particolare, vale la seguente proprietà:\n", + "\n", + "$$\\exists\\, i \\neq j \\mid x_i = x_j \\iff \\Delta(f) = 0.$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si verifica facilmente che $\\Delta(f)$ è un polinomio simmetrico. Poiché ogni permutazione è una composizione di trasposizioni, è sufficiente verificare che invertendo due radici $x_i$ e $x_j$ il discriminante rimanga invariato:\n", + "\n", + " - i fattori che contengono solo uno tra $x_i$ e $x_j$ mantengono la somma della base invariata o al più cambiano di segno, e dunque, elevando al quadrato, rimangono invariati,\n", + " - il fattore che contiene sia $x_i$ che $x_j$ cambia la propria base di segno, ma rimane invariato elevando al quadrato.\n", + " \n", + "Poiché $\\Delta(f)$ è un polinomio simmetrico nelle variabili $x_1$, ..., $x_n$, per il _Teorema fondamentale dei polinomi simmetrici_, si scrive in modo unico in funzione dei polinomi simmetrici elementari $e_i(x_1, \\ldots, x_n)$. Tuttavia tali polinomi simmetrici elementari altro non sono che i coefficienti di $f(x)$, a meno del segno. In particolare vale che:\n", + "\n", + "$$e_i(x_1, \\ldots, x_n) = (-1)^{i} a_{n-i}, \\quad \\text{per } 0 \\leq i \\leq n.$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si definiscono quindi i simboli $a_0$, ..., $a_n$ e si costruisce una funzione $\\Delta(n)$ che restituisce il discriminante di un generico polinomio di $n$-esimo grado dato in ingresso in funzione degli $a_i$ mediante $\\operatorname{combination}$." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [], + "source": [ + "var(\",\".join(f\"a_{i}\" for i in range(n+1)))\n", + "\n", + "def delta(n):\n", + " \n", + " f = reduce(mul, (eval(f\"(x_{i}-x_{j})\")^2 for i, j in combinations(range(1, n+1), 2)), 1)\n", + " c = combination(f)\n", + "\n", + " for i in range(1, n+1):\n", + " if i % 2:\n", + " c = eval(f\"c.substitute(e_{i}=-a_{n-i})\")\n", + " else:\n", + " c = eval(f\"c.substitute(e_{i}=a_{n-i})\")\n", + "\n", + " return c" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Dunque, per l'$n$ scelto in partenza, il discriminante del generico polinomio di $n$-esimo grado è il seguente:" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}a_{1}^{2} a_{2}^{2} a_{3}^{2} a_{4}^{2} - 4 \\, a_{0} a_{2}^{3} a_{3}^{2} a_{4}^{2} - 4 \\, a_{1}^{3} a_{3}^{3} a_{4}^{2} + 18 \\, a_{0} a_{1} a_{2} a_{3}^{3} a_{4}^{2} - 27 \\, a_{0}^{2} a_{3}^{4} a_{4}^{2} - 4 \\, a_{1}^{2} a_{2}^{3} a_{4}^{3} + 16 \\, a_{0} a_{2}^{4} a_{4}^{3} + 18 \\, a_{1}^{3} a_{2} a_{3} a_{4}^{3} - 80 \\, a_{0} a_{1} a_{2}^{2} a_{3} a_{4}^{3} - 6 \\, a_{0} a_{1}^{2} a_{3}^{2} a_{4}^{3} + 144 \\, a_{0}^{2} a_{2} a_{3}^{2} a_{4}^{3} - 27 \\, a_{1}^{4} a_{4}^{4} + 144 \\, a_{0} a_{1}^{2} a_{2} a_{4}^{4} - 128 \\, a_{0}^{2} a_{2}^{2} a_{4}^{4} - 192 \\, a_{0}^{2} a_{1} a_{3} a_{4}^{4} + 256 \\, a_{0}^{3} a_{4}^{5} - 4 \\, a_{1}^{2} a_{2}^{2} a_{3}^{3} + 16 \\, a_{0} a_{2}^{3} a_{3}^{3} + 16 \\, a_{1}^{3} a_{3}^{4} - 72 \\, a_{0} a_{1} a_{2} a_{3}^{4} + 108 \\, a_{0}^{2} a_{3}^{5} + 18 \\, a_{1}^{2} a_{2}^{3} a_{3} a_{4} - 72 \\, a_{0} a_{2}^{4} a_{3} a_{4} - 80 \\, a_{1}^{3} a_{2} a_{3}^{2} a_{4} + 356 \\, a_{0} a_{1} a_{2}^{2} a_{3}^{2} a_{4} + 24 \\, a_{0} a_{1}^{2} a_{3}^{3} a_{4} - 630 \\, a_{0}^{2} a_{2} a_{3}^{3} a_{4} - 6 \\, a_{1}^{3} a_{2}^{2} a_{4}^{2} + 24 \\, a_{0} a_{1} a_{2}^{3} a_{4}^{2} + 144 \\, a_{1}^{4} a_{3} a_{4}^{2} - 746 \\, a_{0} a_{1}^{2} a_{2} a_{3} a_{4}^{2} + 560 \\, a_{0}^{2} a_{2}^{2} a_{3} a_{4}^{2} + 1020 \\, a_{0}^{2} a_{1} a_{3}^{2} a_{4}^{2} - 36 \\, a_{0} a_{1}^{3} a_{4}^{3} + 160 \\, a_{0}^{2} a_{1} a_{2} a_{4}^{3} - 1600 \\, a_{0}^{3} a_{3} a_{4}^{3} - 27 \\, a_{1}^{2} a_{2}^{4} + 108 \\, a_{0} a_{2}^{5} + 144 \\, a_{1}^{3} a_{2}^{2} a_{3} - 630 \\, a_{0} a_{1} a_{2}^{3} a_{3} - 128 \\, a_{1}^{4} a_{3}^{2} + 560 \\, a_{0} a_{1}^{2} a_{2} a_{3}^{2} + 825 \\, a_{0}^{2} a_{2}^{2} a_{3}^{2} - 900 \\, a_{0}^{2} a_{1} a_{3}^{3} - 192 \\, a_{1}^{4} a_{2} a_{4} + 1020 \\, a_{0} a_{1}^{2} a_{2}^{2} a_{4} - 900 \\, a_{0}^{2} a_{2}^{3} a_{4} + 160 \\, a_{0} a_{1}^{3} a_{3} a_{4} - 2050 \\, a_{0}^{2} a_{1} a_{2} a_{3} a_{4} + 2250 \\, a_{0}^{3} a_{3}^{2} a_{4} - 50 \\, a_{0}^{2} a_{1}^{2} a_{4}^{2} + 2000 \\, a_{0}^{3} a_{2} a_{4}^{2} + 256 \\, a_{1}^{5} - 1600 \\, a_{0} a_{1}^{3} a_{2} + 2250 \\, a_{0}^{2} a_{1} a_{2}^{2} + 2000 \\, a_{0}^{2} a_{1}^{2} a_{3} - 3750 \\, a_{0}^{3} a_{2} a_{3} - 2500 \\, a_{0}^{3} a_{1} a_{4} + 3125 \\, a_{0}^{4}\n", + "\\end{math}" + ], + "text/plain": [ + "a_1^2*a_2^2*a_3^2*a_4^2 - 4*a_0*a_2^3*a_3^2*a_4^2 - 4*a_1^3*a_3^3*a_4^2 + 18*a_0*a_1*a_2*a_3^3*a_4^2 - 27*a_0^2*a_3^4*a_4^2 - 4*a_1^2*a_2^3*a_4^3 + 16*a_0*a_2^4*a_4^3 + 18*a_1^3*a_2*a_3*a_4^3 - 80*a_0*a_1*a_2^2*a_3*a_4^3 - 6*a_0*a_1^2*a_3^2*a_4^3 + 144*a_0^2*a_2*a_3^2*a_4^3 - 27*a_1^4*a_4^4 + 144*a_0*a_1^2*a_2*a_4^4 - 128*a_0^2*a_2^2*a_4^4 - 192*a_0^2*a_1*a_3*a_4^4 + 256*a_0^3*a_4^5 - 4*a_1^2*a_2^2*a_3^3 + 16*a_0*a_2^3*a_3^3 + 16*a_1^3*a_3^4 - 72*a_0*a_1*a_2*a_3^4 + 108*a_0^2*a_3^5 + 18*a_1^2*a_2^3*a_3*a_4 - 72*a_0*a_2^4*a_3*a_4 - 80*a_1^3*a_2*a_3^2*a_4 + 356*a_0*a_1*a_2^2*a_3^2*a_4 + 24*a_0*a_1^2*a_3^3*a_4 - 630*a_0^2*a_2*a_3^3*a_4 - 6*a_1^3*a_2^2*a_4^2 + 24*a_0*a_1*a_2^3*a_4^2 + 144*a_1^4*a_3*a_4^2 - 746*a_0*a_1^2*a_2*a_3*a_4^2 + 560*a_0^2*a_2^2*a_3*a_4^2 + 1020*a_0^2*a_1*a_3^2*a_4^2 - 36*a_0*a_1^3*a_4^3 + 160*a_0^2*a_1*a_2*a_4^3 - 1600*a_0^3*a_3*a_4^3 - 27*a_1^2*a_2^4 + 108*a_0*a_2^5 + 144*a_1^3*a_2^2*a_3 - 630*a_0*a_1*a_2^3*a_3 - 128*a_1^4*a_3^2 + 560*a_0*a_1^2*a_2*a_3^2 + 825*a_0^2*a_2^2*a_3^2 - 900*a_0^2*a_1*a_3^3 - 192*a_1^4*a_2*a_4 + 1020*a_0*a_1^2*a_2^2*a_4 - 900*a_0^2*a_2^3*a_4 + 160*a_0*a_1^3*a_3*a_4 - 2050*a_0^2*a_1*a_2*a_3*a_4 + 2250*a_0^3*a_3^2*a_4 - 50*a_0^2*a_1^2*a_4^2 + 2000*a_0^3*a_2*a_4^2 + 256*a_1^5 - 1600*a_0*a_1^3*a_2 + 2250*a_0^2*a_1*a_2^2 + 2000*a_0^2*a_1^2*a_3 - 3750*a_0^3*a_2*a_3 - 2500*a_0^3*a_1*a_4 + 3125*a_0^4" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "delta(n)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Infine si costruisce la funzione $\\operatorname{evaluate\\_delta}$ che, dato in ingresso un polinomio $f(x)$ di $n$-esimo grado, restituisce il valore di $\\Delta(f)$, sostituendo agli $a_i$ generici di $\\operatorname{delta}(n)$ i valori dei coefficienti di $f$." + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "def evaluate_delta(f):\n", + " d = delta(n)\n", + " \n", + " for i, a in enumerate(f.list()):\n", + " d = eval(f\"d.substitute(a_{i}=a)\")\n", + " \n", + " return d" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si verifica adesso che per un polinomio con radici multiple il valore di $\\operatorname{evaluate\\_delta}$ è precisamente zero:" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x - 2\\right)}^{2} {\\left(x - 3\\right)} {\\left(x - 4\\right)} {\\left(x - 5\\right)}\n", + "\\end{math}" + ], + "text/plain": [ + "(x - 2)^2*(x - 3)*(x - 4)*(x - 5)" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "f = (x-2)^2*(x-3)*(x-4)*(x-5)\n", + "f" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{5} - 16 \\, x^{4} + 99 \\, x^{3} - 296 \\, x^{2} + 428 \\, x - 240\n", + "\\end{math}" + ], + "text/plain": [ + "x^5 - 16*x^4 + 99*x^3 - 296*x^2 + 428*x - 240" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "f.expand()" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}0\n", + "\\end{math}" + ], + "text/plain": [ + "0" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evaluate_delta(f)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Infine, si fa lo stesso con un polinomio con radici distinte, per verificare che $\\operatorname{evaluate\\_delta}$ è strettamente diverso da zero." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 2\\right)} {\\left(x + i\\right)} {\\left(x - i\\right)} {\\left(x - 1\\right)} {\\left(x - 2\\right)}\n", + "\\end{math}" + ], + "text/plain": [ + "(x + 2)*(x + I)*(x - I)*(x - 1)*(x - 2)" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g = (x+2)*(x+I)*(x-I)*(x-1)*(x-2)\n", + "g" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{5} - x^{4} - 3 \\, x^{3} + 3 \\, x^{2} - 4 \\, x + 4\n", + "\\end{math}" + ], + "text/plain": [ + "x^5 - x^4 - 3*x^3 + 3*x^2 - 4*x + 4" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g.expand()" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "\\begin{math}\n", + "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-1440000\n", + "\\end{math}" + ], + "text/plain": [ + "-1440000" + ] + }, + "execution_count": 22, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evaluate_delta(g)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***\n", + "(c) 2022, [~videtta](https://poisson.phc.dm.unipi.it/~videtta/)" + ] + } + ], + "metadata": { + "celltoolbar": "Edit Metadata", + "kernelspec": { + "display_name": "SageMath 9.2", + "language": "sage", + "name": "sagemath" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.7" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Geometria/Articoli/1. 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zf+C$4+mn&k$JopjQf+^G2^*vTRc!DL>=r>B;ZOd~f0q9L&F!bDY-SG6ByDSI=0>W+ zObTRU)`MqKvGOqcy$H{wL8=2JWhDiIHY(fMIg?>LPCk4!~2I$ zh@fK9qQ1w;jZkNtQuil?We`JDI`-@viFs?aXN0K;?f1ig^M!@|9xWzKU`5u@FE=TR zFU{E`clwJ-{NpXyMYphW@X4t_lgy$tQj@5*G>#QH5@i^n-}ddoNB#q{`usTOSkrGJ z$^%M{k@;V@C^JlB0Rxop`lY{#*GMCCi#tlkrc+x;5p&Coe)W`cNG-?q?@la(Ck>NS z-y_?~QxA$Ic`FMiO&d}-BE4x<1ENN2B>?1 |V|$.}. +\end{theorem} + +\begin{proof} +Se $\dual V$ è di dimensione infinita, anche $V$ deve esserlo +necessariamente, altrimenti, per il \textit{Teorema \ref{isom}} +dovrebbe esserlo anche $\dual V$. \\ + +Sia allora $V$ di dimensione infinita e sia $A_i$ una famiglia +di indici che enumeri $i$ elementi della base di $V$. + +Si consideri l'insieme linearmente indipendente $I_n = \{\vec v_{\alpha}^* \}_{\alpha \in A_n}$ con\footnote{Ancora una volta +questa definizione ricalca il delta di Dirac.}: + +\[\vec v_{\alpha}^*\left(\vec v_{\beta}\right) = \begin{cases}1 & \text{se } \alpha = \beta \\ 0 & \text{altrimenti} \end{cases}\] + +Si assuma l'esistenza di una base $\BB$ di $\dual V$ di +cardinalità finita, e sia $| \BB | = n \in \NN$. Ogni insieme $P \subset V$ linearmente indipendente è t.c. $|P| \leq n$. +Tuttavia $|I_{n+1}|=n+1>n$, \Lightning. + +\end{proof} + +\section{Esercizi} + +\begin{exercise} +Si dimostri che l'insieme $I_n$ è linearmente indipendente in $\dual V$, dato $V$ spazio vettoriale di dimensione infinita. +\end{exercise} + +\begin{exercise} +Dato $V$ uno spazio vettoriale di dimensione finita, si esibisca +una base per $\dual {\left( \dual V \right)} = \LL{\LL{V}{ \KK}}{\KK}$, il cosiddetto \textbf{spazio biduale}. +\end{exercise} + +\end{document} diff --git a/Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.html b/Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.html new file mode 100644 index 0000000..0157601 --- /dev/null +++ b/Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.html @@ -0,0 +1,13943 @@ + + + + +Algoritmo di estrazione della forma canonica di Jordan + + + + + + + + + + + + + + + + + + + + + + +
+
+ + +
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+
+

Lo scopo di questo notebook è illustrare un breve e semplice algoritmo di estrazione della forma canonica di Jordan di una matrice $A \in M(n, \mathbb{C})$ qualsiasi. Si ricorda che la forma canonica di Jordan è un invariante completo per similitudine +di una matrice e che è una matrice a blocchi diagonale composta da più blocchi, detti per l'appunto di Jordan, della seguente forma:

+$$ J_{\lambda, m} = \begin{bmatrix} +\lambda & 1 & \; & \; \\ +\; & \lambda & \ddots & \; \\ +\; & \; & \ddots & 1 \\ +\; & \; & \; & \lambda +\end{bmatrix} \in M(m, \mathbb{C}). $$

Innanzitutto si sceglie un $n \in \mathbb{N}^+$ che rappresenta la taglia della matrice quadrata data in esame all'algoritmo e si crea la matrice $I = I_n$, ossia la matrice identità di taglia $n$.

+ +
+
+
+
+
+
In [35]:
+
+
+
%display latex
+from IPython.display import display, Markdown, Latex
+
+n = 10
+I = matrix.identity(n)
+
+ +
+
+
+ +
+
+
+
+

Si definisce adesso una funzione $\operatorname{jordan}(A)$ che prende in ingresso una matrice $A \in M(n, \mathbb{C})$ senza restituire alcun risultato. Tale funzione illustra passo passo in output l'algoritmo di estrazione della forma canonica di Jordan della matrice $A$ nei seguenti passi:

+
    +
  • Calcola il polinomio caratteristico $p_A(x)$ di $A$ e lo fattorizza;
  • +
  • Ne deduce lo spettro $\operatorname{sp}(A)$ e si riduce a studiare i blocchi relativi a ciascun autovalore;
  • +
  • Per ogni autovalore $\lambda$, calcola le dimensioni dei kernel delle potenze di $B = A - \lambda I$ fino a che non viene raggiunta la molteplicità algebrica di $\lambda$;
  • +
  • Infine, per l'autovalore $\lambda$, calcola il numero di blocchi $b_i$ di taglia $i$ secondo la formula $b_i = 2 \dim \ker B^i - \dim \ker B^{i-1} - \dim \ker B^{i+1}$;
  • +
  • Compone tutti i blocchi in una matrice a blocchi diagonale, ossia la forma canonica di Jordan.
  • +
+ +
+
+
+
+
+
In [58]:
+
+
+
def jordan(A):
+    display(Latex(r"Innanzitutto, si calcola il polinomio caratteristico di $A$, che risulta essere $p_A(x) = " + latex(factor(charpoly(A)(SR('x')))) + "$."))
+    eig = set(A.eigenvalues())
+    sp = ", ".join(map(latex, eig))
+
+    display(Latex(r"Allora, calcolandone le radici, si ricava $\operatorname{sp}(A) = \{" + sp + r"\}$."))
+    display(Markdown("---"))
+
+    Js = []
+
+    for i, t in enumerate(eig, start=1):
+        B = A - t*I
+        r = rank(B)
+        k = n - r
+
+        display(Latex(f"({i}) " + r"Consideriamo $\lambda = " + latex(t) + r"$ e $B=A-\lambda I=" + latex(B) + r".$"))
+        display(Latex(r"$B$ ha rango $" + str(r) + r"$ e quindi $\mu_g(\lambda) = " + str(k) + r"$."))
+
+        if k == 1:
+            display(Latex(r"Allora a $\lambda$ sarà dedicato esattamente un blocco nella forma canonica di Jordan."))
+        else:
+            display(Latex(r"Allora a $\lambda$ saranno dedicati esattamente $" + str(k) + "$ blocchi nella forma canonica di Jordan."))
+
+        K = [k]
+        k_1 = k
+        k_2 = n-rank(B^2)
+
+        i = 3
+        while k_1 != k_2:
+            K.append(k_2)
+            k_1 = k_2
+            k_2 = n-rank(B^i)
+            i += 1
+
+        display(Latex(r"Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\mu_a(\lambda) = " + str(K[-1]) + "$:"))
+
+        for i, k in enumerate(K, start=1):
+            if i == 1:
+                if i == len(K):
+                    display(Latex(r"‎ ‎ ‎ ‎ • $\dim \ker B = " + str(k) + "$."))
+                else:
+                    display(Latex(r"‎ ‎ ‎ ‎ • $\dim \ker B = " + str(k) + "$;"))
+            else:
+                if i == len(K):
+                    display(Latex(r"‎ ‎ ‎ ‎ • $\dim \ker B^" + str(i) + r" = " + str(k) + "$."))
+                else:
+                    display(Latex(r"‎ ‎ ‎ ‎ • $\dim \ker B^" + str(i) + r" = " + str(k) + "$;"))
+
+        display(Latex(r"Pertanto a $\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$):"))
+
+        b = {}
+
+        for i, k in enumerate(K, start=1):
+            if i == len(K):
+                if i == 2:
+                    b[i] = k - K[0]
+                    display(Latex(r"‎ ‎ ‎ ‎ • $b_2 = \dim \ker B^2 - \dim \ker B = " + str(b[i]) + "$."))
+                elif i == 1:
+                    b[i] = k
+                    display(Latex(r"‎ ‎ ‎ ‎ • $b_1 = \dim \ker B = " + str(b[i]) + "$."))
+                else:
+                    b[i] = k - K[i-2]
+                    display(Latex(r"‎ ‎ ‎ ‎ • $b_" + str(i) + r" = \dim \ker B^" + str(i) + r" - \dim \ker B^" + str(i-1) + " = " + str(b[i]) + "$."))
+            elif i == 1:
+                b[i] = 2*k - K[1]
+                display(Latex(r"‎ ‎ ‎ ‎ • $b_1 = 2 \dim \ker B - \dim \ker B^2 = " + str(b[i]) + "$;"))
+            else:
+                if i != 2:
+                    b[i] = 2*k - K[i-2] - K[i]
+                    display(Latex(r"‎ ‎ ‎ ‎ • $b_" + str(i) + r" = 2 \dim \ker B^" + str(i) + r" - \dim \ker B^" + str(i-1) + r" - \dim \ker B^" + str(i+1) + " = " + str(b[i]) + "$;"))
+                else:
+                    b[i] = 2*k - K[0] - K[2]
+                    display(Latex(r"‎ ‎ ‎ ‎ • $b_" + str(i) + r" = 2 \dim \ker B^2 - \dim \ker B - \dim \ker B^3 = " + str(b[i]) + "$;"))
+        
+        JBs = [jordan_block(t, N) for N, j in b.items() for _ in range(j)]
+        Js.extend(JBs)
+        
+        display(Latex(r"Allora l'insieme di tutti i blocchi relativi a $\lambda$ sarà rappresentato dalla seguente matrice: $J_{\lambda} = " + latex(block_diagonal_matrix(JBs)) + "$."))
+        display(Markdown("---"))
+            
+    JNF = block_diagonal_matrix(Js)
+    
+    display(Latex(r"La forma canonica di Jordan di $A$ sarà allora $J = " + latex(JNF) + "$."))
+
+ +
+
+
+ +
+
+
+
+

Si sceglie adesso una matrice $A \in M(n, \mathbb{C})$ di cui si vuole calcolare la forma canonica di Jordan.

+ +
+
+
+
+
+
In [59]:
+
+
+
# A deve avere esattamente n^2 elementi, essendo n x n...
+A = matrix(SR, n, [1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3])
+A
+
+ +
+
+
+ +
+
+ + +
+ +
Out[59]:
+ + + +
+ +
+ +
+ +
+
+ +
+
+
+
+

Si calcola infine la forma canonica di Jordan della matrice $A$ mediante la funzione $\operatorname{jordan}$.

+ +
+
+
+
+
+
In [60]:
+
+
+
jordan(A)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Innanzitutto, si calcola il polinomio caratteristico di $A$, che risulta essere $p_A(x) = {\left(x - 1\right)}^{4} {\left(x - 2\right)}^{3} {\left(x - 3\right)}^{3} $. +
+ +
+ +
+ +
+ + + + +
+Allora, calcolandone le radici, si ricava $\operatorname{sp}(A) = \{1, 2, 3\}$. +
+ +
+ +
+ +
+ + + +
+
+ +
+ +
+ +
+ +
+ + + + +
+(1) Consideriamo $\lambda = 1 $ e $B=A-\lambda I= \left(\begin{array}{rrrrrrrrrr} +0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 +\end{array}\right) .$ +
+ +
+ +
+ +
+ + + + +
+$B$ ha rango $9$ e quindi $\mu_g(\lambda) = 1$. +
+ +
+ +
+ +
+ + + + +
+Allora a $\lambda$ sarà dedicato esattamente un blocco nella forma canonica di Jordan. +
+ +
+ +
+ +
+ + + + +
+Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\mu_a(\lambda) = 4$: +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B = 1$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B^2 = 2$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B^3 = 3$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B^4 = 4$. +
+ +
+ +
+ +
+ + + + +
+Pertanto a $\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$): +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_1 = 2 \dim \ker B - \dim \ker B^2 = 0$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_2 = 2 \dim \ker B^2 - \dim \ker B - \dim \ker B^3 = 0$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_3 = 2 \dim \ker B^3 - \dim \ker B^2 - \dim \ker B^4 = 0$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_4 = \dim \ker B^4 - \dim \ker B^3 = 1$. +
+ +
+ +
+ +
+ + + + +
+Allora l'insieme di tutti i blocchi relativi a $\lambda$ sarà rappresentato dalla seguente matrice: $J_{\lambda} = \left(\begin{array}{rrrr} +1 & 1 & 0 & 0 \\ +0 & 1 & 1 & 0 \\ +0 & 0 & 1 & 1 \\ +0 & 0 & 0 & 1 +\end{array}\right) $. +
+ +
+ +
+ +
+ + + +
+
+ +
+ +
+ +
+ +
+ + + + +
+(2) Consideriamo $\lambda = 2 $ e $B=A-\lambda I= \left(\begin{array}{rrrrrrrrrr} +-1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & -1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & -1 & 1 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 +\end{array}\right) .$ +
+ +
+ +
+ +
+ + + + +
+$B$ ha rango $8$ e quindi $\mu_g(\lambda) = 2$. +
+ +
+ +
+ +
+ + + + +
+Allora a $\lambda$ saranno dedicati esattamente $2$ blocchi nella forma canonica di Jordan. +
+ +
+ +
+ +
+ + + + +
+Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\mu_a(\lambda) = 3$: +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B = 2$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B^2 = 3$. +
+ +
+ +
+ +
+ + + + +
+Pertanto a $\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$): +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_1 = 2 \dim \ker B - \dim \ker B^2 = 1$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_2 = \dim \ker B^2 - \dim \ker B = 1$. +
+ +
+ +
+ +
+ + + + +
+Allora l'insieme di tutti i blocchi relativi a $\lambda$ sarà rappresentato dalla seguente matrice: $J_{\lambda} = \left(\begin{array}{r|rr} +2 & 0 & 0 \\ +\hline + 0 & 2 & 1 \\ +0 & 0 & 2 +\end{array}\right) $. +
+ +
+ +
+ +
+ + + +
+
+ +
+ +
+ +
+ +
+ + + + +
+(3) Consideriamo $\lambda = 3 $ e $B=A-\lambda I= \left(\begin{array}{rrrrrrrrrr} +-2 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & -2 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & -2 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & -2 & 1 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 +\end{array}\right) .$ +
+ +
+ +
+ +
+ + + + +
+$B$ ha rango $9$ e quindi $\mu_g(\lambda) = 1$. +
+ +
+ +
+ +
+ + + + +
+Allora a $\lambda$ sarà dedicato esattamente un blocco nella forma canonica di Jordan. +
+ +
+ +
+ +
+ + + + +
+Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\mu_a(\lambda) = 3$: +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B = 1$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B^2 = 2$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $\dim \ker B^3 = 3$. +
+ +
+ +
+ +
+ + + + +
+Pertanto a $\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$): +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_1 = 2 \dim \ker B - \dim \ker B^2 = 0$; +
+ +
+ +
+ +
+ + + + +
+‎ ‎ ‎ ‎ • $b_2 = 2 \dim \ker B^2 - \dim \ker B - \dim \ker B^3 = 0$; +
+ +
+ +
+ +
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+‎ ‎ ‎ ‎ • $b_3 = \dim \ker B^3 - \dim \ker B^2 = 1$. +
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+Allora l'insieme di tutti i blocchi relativi a $\lambda$ sarà rappresentato dalla seguente matrice: $J_{\lambda} = \left(\begin{array}{rrr} +3 & 1 & 0 \\ +0 & 3 & 1 \\ +0 & 0 & 3 +\end{array}\right) $. +
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+La forma canonica di Jordan di $A$ sarà allora $J = \left(\begin{array}{rrrr|r|rr|rrr} +1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ +0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ +\hline + 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ +\hline + 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ +\hline + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 +\end{array}\right) $. +
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+

(c) 2023, ~videtta

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+ + + + + + diff --git a/Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.ipynb b/Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.ipynb new file mode 100644 index 0000000..6b07c9a --- /dev/null +++ b/Geometria/Notebook/1. Forma canonica di Jordan/Algoritmo di estrazione della forma canonica di Jordan.ipynb @@ -0,0 +1,840 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Algoritmo di estrazione della forma canonica di Jordan" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Lo scopo di questo notebook è illustrare un breve e semplice algoritmo di estrazione della forma canonica di Jordan di una matrice $A \\in M(n, \\mathbb{C})$ qualsiasi. Si ricorda che la forma canonica di Jordan è un invariante completo per similitudine\n", + "di una matrice e che è una matrice a blocchi diagonale composta da più blocchi, detti per l'appunto di Jordan, della seguente forma:\n", + "\n", + "$$ J_{\\lambda, m} = \\begin{bmatrix}\n", + "\\lambda & 1 & \\; & \\; \\\\\n", + "\\; & \\lambda & \\ddots & \\; \\\\\n", + "\\; & \\; & \\ddots & 1 \\\\\n", + "\\; & \\; & \\; & \\lambda \n", + "\\end{bmatrix} \\in M(m, \\mathbb{C}). $$\n", + "\n", + "Innanzitutto si sceglie un $n \\in \\mathbb{N}^+$ che rappresenta la taglia della matrice quadrata data in esame all'algoritmo e si crea la matrice $I = I_n$, ossia la matrice identità di taglia $n$." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "%display latex\n", + "from IPython.display import display, Markdown, Latex\n", + "\n", + "n = 10\n", + "I = matrix.identity(n)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si definisce adesso una funzione $\\operatorname{jordan}(A)$ che prende in ingresso una matrice $A \\in M(n, \\mathbb{C})$ senza restituire alcun risultato. Tale funzione illustra passo passo in output l'algoritmo di estrazione della forma canonica di Jordan della matrice $A$ nei seguenti passi:\n", + "\n", + " - Calcola il polinomio caratteristico $p_A(x)$ di $A$ e lo fattorizza;\n", + " - Ne deduce lo spettro $\\operatorname{sp}(A)$ e si riduce a studiare i blocchi relativi a ciascun autovalore;\n", + " - Per ogni autovalore $\\lambda$, calcola le dimensioni dei kernel delle potenze di $B = A - \\lambda I$ fino a che non viene raggiunta la molteplicità algebrica di $\\lambda$;\n", + " - Infine, per l'autovalore $\\lambda$, calcola il numero di blocchi $b_i$ di taglia $i$ secondo la formula $b_i = 2 \\dim \\ker B^i - \\dim \\ker B^{i-1} - \\dim \\ker B^{i+1}$;\n", + " - Compone tutti i blocchi in una matrice a blocchi diagonale, ossia la forma canonica di Jordan." + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "metadata": {}, + "outputs": [], + "source": [ + "def jordan(A):\n", + " display(Latex(r\"Innanzitutto, si calcola il polinomio caratteristico di $A$, che risulta essere $p_A(x) = \" + latex(factor(charpoly(A)(SR('x')))) + \"$.\"))\n", + " eig = set(A.eigenvalues())\n", + " sp = \", \".join(map(latex, eig))\n", + "\n", + " display(Latex(r\"Allora, calcolandone le radici, si ricava $\\operatorname{sp}(A) = \\{\" + sp + r\"\\}$.\"))\n", + " display(Markdown(\"---\"))\n", + "\n", + " Js = []\n", + "\n", + " for i, t in enumerate(eig, start=1):\n", + " B = A - t*I\n", + " r = rank(B)\n", + " k = n - r\n", + "\n", + " display(Latex(f\"({i}) \" + r\"Consideriamo $\\lambda = \" + latex(t) + r\"$ e $B=A-\\lambda I=\" + latex(B) + r\".$\"))\n", + " display(Latex(r\"$B$ ha rango $\" + str(r) + r\"$ e quindi $\\mu_g(\\lambda) = \" + str(k) + r\"$.\"))\n", + "\n", + " if k == 1:\n", + " display(Latex(r\"Allora a $\\lambda$ sarà dedicato esattamente un blocco nella forma canonica di Jordan.\"))\n", + " else:\n", + " display(Latex(r\"Allora a $\\lambda$ saranno dedicati esattamente $\" + str(k) + \"$ blocchi nella forma canonica di Jordan.\"))\n", + "\n", + " K = [k]\n", + " k_1 = k\n", + " k_2 = n-rank(B^2)\n", + "\n", + " i = 3\n", + " while k_1 != k_2:\n", + " K.append(k_2)\n", + " k_1 = k_2\n", + " k_2 = n-rank(B^i)\n", + " i += 1\n", + "\n", + " display(Latex(r\"Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\\mu_a(\\lambda) = \" + str(K[-1]) + \"$:\"))\n", + "\n", + " for i, k in enumerate(K, start=1):\n", + " if i == 1:\n", + " if i == len(K):\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $\\dim \\ker B = \" + str(k) + \"$.\"))\n", + " else:\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $\\dim \\ker B = \" + str(k) + \"$;\"))\n", + " else:\n", + " if i == len(K):\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $\\dim \\ker B^\" + str(i) + r\" = \" + str(k) + \"$.\"))\n", + " else:\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $\\dim \\ker B^\" + str(i) + r\" = \" + str(k) + \"$;\"))\n", + "\n", + " display(Latex(r\"Pertanto a $\\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$):\"))\n", + "\n", + " b = {}\n", + "\n", + " for i, k in enumerate(K, start=1):\n", + " if i == len(K):\n", + " if i == 2:\n", + " b[i] = k - K[0]\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $b_2 = \\dim \\ker B^2 - \\dim \\ker B = \" + str(b[i]) + \"$.\"))\n", + " elif i == 1:\n", + " b[i] = k\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $b_1 = \\dim \\ker B = \" + str(b[i]) + \"$.\"))\n", + " else:\n", + " b[i] = k - K[i-2]\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $b_\" + str(i) + r\" = \\dim \\ker B^\" + str(i) + r\" - \\dim \\ker B^\" + str(i-1) + \" = \" + str(b[i]) + \"$.\"))\n", + " elif i == 1:\n", + " b[i] = 2*k - K[1]\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $b_1 = 2 \\dim \\ker B - \\dim \\ker B^2 = \" + str(b[i]) + \"$;\"))\n", + " else:\n", + " if i != 2:\n", + " b[i] = 2*k - K[i-2] - K[i]\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $b_\" + str(i) + r\" = 2 \\dim \\ker B^\" + str(i) + r\" - \\dim \\ker B^\" + str(i-1) + r\" - \\dim \\ker B^\" + str(i+1) + \" = \" + str(b[i]) + \"$;\"))\n", + " else:\n", + " b[i] = 2*k - K[0] - K[2]\n", + " display(Latex(r\"‎ ‎ ‎ ‎ • $b_\" + str(i) + r\" = 2 \\dim \\ker B^2 - \\dim \\ker B - \\dim \\ker B^3 = \" + str(b[i]) + \"$;\"))\n", + " \n", + " JBs = [jordan_block(t, N) for N, j in b.items() for _ in range(j)]\n", + " Js.extend(JBs)\n", + " \n", + " display(Latex(r\"Allora l'insieme di tutti i blocchi relativi a $\\lambda$ sarà rappresentato dalla seguente matrice: $J_{\\lambda} = \" + latex(block_diagonal_matrix(JBs)) + \"$.\"))\n", + " display(Markdown(\"---\"))\n", + " \n", + " JNF = block_diagonal_matrix(Js)\n", + " \n", + " display(Latex(r\"La forma canonica di Jordan di $A$ sarà allora $J = \" + latex(JNF) + \"$.\"))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si sceglie adesso una matrice $A \\in M(n, \\mathbb{C})$ di cui si vuole calcolare la forma canonica di Jordan." + ] + }, + { + "cell_type": "code", + "execution_count": 59, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/html": [ + "" + ], + "text/latex": [ + "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrrr}\n", + "1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 2 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3\n", + "\\end{array}\\right)$$" + ], + "text/plain": [ + "[1 1 0 0 0 0 1 0 1 0]\n", + "[0 1 1 0 0 0 1 0 1 0]\n", + "[0 0 1 1 0 0 1 0 1 0]\n", + "[0 0 0 1 1 0 1 0 1 0]\n", + "[0 0 0 0 2 0 1 0 1 0]\n", + "[0 0 0 0 0 2 1 0 1 0]\n", + "[0 0 0 0 0 0 2 1 1 0]\n", + "[0 0 0 0 0 0 0 3 1 0]\n", + "[0 0 0 0 0 0 0 0 3 1]\n", + "[0 0 0 0 0 0 0 0 0 3]" + ] + }, + "execution_count": 59, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# A deve avere esattamente n^2 elementi, essendo n x n...\n", + "A = matrix(SR, n, [1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3])\n", + "A" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si calcola infine la forma canonica di Jordan della matrice $A$ mediante la funzione $\\operatorname{jordan}$." + ] + }, + { + "cell_type": "code", + "execution_count": 60, + "metadata": { + "scrolled": false + }, + "outputs": [ + { + "data": { + "text/latex": [ + "Innanzitutto, si calcola il polinomio caratteristico di $A$, che risulta essere $p_A(x) = {\\left(x - 1\\right)}^{4} {\\left(x - 2\\right)}^{3} {\\left(x - 3\\right)}^{3} $." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Allora, calcolandone le radici, si ricava $\\operatorname{sp}(A) = \\{1, 2, 3\\}$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/markdown": [ + "---" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "(1) Consideriamo $\\lambda = 1 $ e $B=A-\\lambda I= \\left(\\begin{array}{rrrrrrrrrr}\n", + "0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\n", + "\\end{array}\\right) .$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$B$ ha rango $9$ e quindi $\\mu_g(\\lambda) = 1$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Allora a $\\lambda$ sarà dedicato esattamente un blocco nella forma canonica di Jordan." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\\mu_a(\\lambda) = 4$:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B = 1$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B^2 = 2$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B^3 = 3$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B^4 = 4$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Pertanto a $\\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$):" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_1 = 2 \\dim \\ker B - \\dim \\ker B^2 = 0$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_2 = 2 \\dim \\ker B^2 - \\dim \\ker B - \\dim \\ker B^3 = 0$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_3 = 2 \\dim \\ker B^3 - \\dim \\ker B^2 - \\dim \\ker B^4 = 0$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_4 = \\dim \\ker B^4 - \\dim \\ker B^3 = 1$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Allora l'insieme di tutti i blocchi relativi a $\\lambda$ sarà rappresentato dalla seguente matrice: $J_{\\lambda} = \\left(\\begin{array}{rrrr}\n", + "1 & 1 & 0 & 0 \\\\\n", + "0 & 1 & 1 & 0 \\\\\n", + "0 & 0 & 1 & 1 \\\\\n", + "0 & 0 & 0 & 1\n", + "\\end{array}\\right) $." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/markdown": [ + "---" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "(2) Consideriamo $\\lambda = 2 $ e $B=A-\\lambda I= \\left(\\begin{array}{rrrrrrrrrr}\n", + "-1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & -1 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & -1 & 1 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\n", + "\\end{array}\\right) .$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$B$ ha rango $8$ e quindi $\\mu_g(\\lambda) = 2$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Allora a $\\lambda$ saranno dedicati esattamente $2$ blocchi nella forma canonica di Jordan." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\\mu_a(\\lambda) = 3$:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B = 2$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B^2 = 3$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Pertanto a $\\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$):" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_1 = 2 \\dim \\ker B - \\dim \\ker B^2 = 1$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_2 = \\dim \\ker B^2 - \\dim \\ker B = 1$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Allora l'insieme di tutti i blocchi relativi a $\\lambda$ sarà rappresentato dalla seguente matrice: $J_{\\lambda} = \\left(\\begin{array}{r|rr}\n", + "2 & 0 & 0 \\\\\n", + "\\hline\n", + " 0 & 2 & 1 \\\\\n", + "0 & 0 & 2\n", + "\\end{array}\\right) $." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/markdown": [ + "---" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "(3) Consideriamo $\\lambda = 3 $ e $B=A-\\lambda I= \\left(\\begin{array}{rrrrrrrrrr}\n", + "-2 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & -2 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & -2 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & -2 & 1 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n", + "\\end{array}\\right) .$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$B$ ha rango $9$ e quindi $\\mu_g(\\lambda) = 1$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Allora a $\\lambda$ sarà dedicato esattamente un blocco nella forma canonica di Jordan." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Si calcolano adesso le dimensioni dei kernel fino a quando non viene raggiunta la molteplicità algebrica $\\mu_a(\\lambda) = 3$:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B = 1$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B^2 = 2$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $\\dim \\ker B^3 = 3$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Pertanto a $\\lambda$ sono assegnati i seguenti blocchi (dove $b_n$ indica il numero di blocchi di taglia $n$):" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_1 = 2 \\dim \\ker B - \\dim \\ker B^2 = 0$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_2 = 2 \\dim \\ker B^2 - \\dim \\ker B - \\dim \\ker B^3 = 0$;" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "‎ ‎ ‎ ‎ • $b_3 = \\dim \\ker B^3 - \\dim \\ker B^2 = 1$." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Allora l'insieme di tutti i blocchi relativi a $\\lambda$ sarà rappresentato dalla seguente matrice: $J_{\\lambda} = \\left(\\begin{array}{rrr}\n", + "3 & 1 & 0 \\\\\n", + "0 & 3 & 1 \\\\\n", + "0 & 0 & 3\n", + "\\end{array}\\right) $." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/markdown": [ + "---" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "La forma canonica di Jordan di $A$ sarà allora $J = \\left(\\begin{array}{rrrr|r|rr|rrr}\n", + "1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\hline\n", + " 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\hline\n", + " 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\\\\n", + "\\hline\n", + " 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 1 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3\n", + "\\end{array}\\right) $." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "jordan(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "***\n", + "(c) 2023, [~videtta](https://poisson.phc.dm.unipi.it/~videtta/)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 9.3", + "language": "sage", + "name": "sagemath" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.10" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..13ca539 --- /dev/null +++ b/LICENSE @@ -0,0 +1,156 @@ +Creative Commons Attribution 4.0 International + + Creative Commons Corporation (“Creative Commons”) is not a law firm and does not provide legal services or legal advice. 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For the avoidance of doubt, this paragraph does not form part of the public licenses. + +Creative Commons may be contacted at creativecommons.org. diff --git a/README.md b/README.md new file mode 100644 index 0000000..cb98037 --- /dev/null +++ b/README.md @@ -0,0 +1,12 @@ +# Raccolta di articoli e scritti 📝 + +Questo repository raccoglie tutti gli articoletti e tutti i notebook che ho scritto. Ogni cartella racchiude una materia differente, e a sua volta ogni materia è divisa in due sezioni, una dedicata agli scritti in [LaTeX](https://www.latex-project.org/) e una dedicata ai notebook creati su [SageMath](https://www.sagemath.org/). + +Sicuramente questi scritti non sono perfetti, e potrebbero dunque contenere alcuni errori. Nell'eventualità in cui vogliate segnalarmeli, potete farlo scrivendomi a g.videtta1@studenti.unipi.it, o aprendo un issue su questo repository. + +Riguardo la licenza con cui pubblico questi scritti, vale la stessa premessa che ho fatto per [i miei appunti](https://notes.hearot.it). + +Di seguito elenco gli strumenti che più utilizzo per scrivere i miei articoletti: + +- [TeXstudio](https://www.texstudio.org/) (assieme al foglio di stile che trovate sotto la cartella `tex` nel repository dei [miei appunti](https://notes.hearot.it)), +- [Overleaf](https://overleaf.com/), anche hostato dal [Dipartimento di Matematica](https://tex.dm.unipi.it) (spesso e volentieri solo per articoli coscritti).

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Proprietà fondamentali di Z[i], di Z_p[x], di Z[x] e di Q[x]/proprietà_fondamentali_zi_zx_zpx_qx.tex b/Algebra/Articoli/3. Proprietà fondamentali di Z[i], di Z_p[x], di Z[x] e di Q[x]/proprietà_fondamentali_zi_zx_zpx_qx.tex new file mode 100644 index 0000000..8d1de44 --- /dev/null +++ b/Algebra/Articoli/3. Proprietà fondamentali di Z[i], di Z_p[x], di Z[x] e di Q[x]/proprietà_fondamentali_zi_zx_zpx_qx.tex @@ -0,0 +1,709 @@ +\documentclass[a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage[italian]{babel} +\usepackage{algorithm2e} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsopn} +\usepackage{bm} +\usepackage{csquotes} +\usepackage{enumerate} +\usepackage{mathtools} +\usepackage{marvosym} +\usepackage{tikz} +\usepackage{wrapfig} +\usepackage{xpatch} + +\usepackage[colorlinks=false,bookmarksopen=true]{hyperref} +\usepackage{bookmark} + +\newcommand{\norm}[1]{\left|#1\right|} + +\newcommand{\zeroset}{\{0\}} +\newcommand{\setminuszero}{\setminus \{0\}} + +\newcommand{\BB}{\mathcal{B}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\FF}{\mathbb{F}_2} +\newcommand{\HH}{\mathbb{H}} +\newcommand{\KK}{\mathbb{K}} +\newcommand{\NN}{\mathbb{N}} +\newcommand{\QQ}{\mathbb{Q}} +\newcommand{\RR}{\mathbb{R}} +\newcommand{\ZZ}{\mathbb{Z}} +\newcommand{\ZZp}{\mathbb{Z}_p} + +\newcommand{\CCx}{\mathbb{C}[x]} +\newcommand{\QQx}{\mathbb{Q}[x]} +\newcommand{\RRx}{\mathbb{R}[x]} +\newcommand{\ZZi}{\mathbb{Z}[i]} +\newcommand{\ZZom}{\mathbb{Z}[\omega]} +\newcommand{\ZZpx}{\mathbb{Z}_p[x]} +\newcommand{\ZZx}{\mathbb{Z}[x]} + +\newcommand{\dual}[1]{#1^{*}} +\newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} +\newcommand{\M}[1]{\mathcal{M}_{#1}\left(\KK\right)} +\newcommand{\nsg}{\mathrel{\unlhd}} +\renewcommand{\vec}[1]{\underline{#1}} + +\newcommand{\hatpi}{\hat{\pi}} +\newcommand{\hatpip}{\hat{\pi}_p} + +\theoremstyle{definition} + +\newtheorem{corollary}{Corollario}[section] +\newtheorem{definition}{Definizione}[section] +\newtheorem{example}{Esempio}[section] +\newtheorem{exercise}{Esercizio}[section] +\newtheorem{lemma}{Lemma}[section] +\newtheorem*{note}{Osservazione} +\newtheorem{proposition}{Proposizione}[section] +\newtheorem{theorem}{Teorema}[section] + +\DeclareMathOperator{\existsone}{\exists !} +\DeclareMathOperator{\Imm}{Imm} +\DeclareMathOperator{\Ker}{Ker} +\DeclareMathOperator{\MCD}{MCD} +\DeclareMathOperator{\mcm}{mcm} +\DeclareMathOperator{\tr}{tr} + +\let\oldemptyset\emptyset +\let\emptyset\varnothing + +\let\oldcirc\circ +\let\circ\undefined +\DeclareMathOperator{\circ}{\oldcirc} + +\let\oldexists\exists +\let\exists\undefined +\DeclareMathOperator{\exists}{\oldexists} + +\let\oldforall\forall +\let\forall\undefined +\DeclareMathOperator{\forall}{\oldforall} + +\let\oldnexists\nexists +\let\nexists\undefined +\DeclareMathOperator{\nexists}{\oldnexists} + +\let\oldland\land +\let\land\undefined +\DeclareMathOperator{\land}{\oldland} + +\let\oldlnot\lnot +\let\lnot\undefined +\DeclareMathOperator{\lnot}{\oldlnot} + +\let\oldlor\lor +\let\lor\undefined +\DeclareMathOperator{\lor}{\oldlor} + +\setlength\parindent{0pt} + +\title{Proprietà fondamentali di $\ZZi$, \\ $\ZZx$, $\ZZpx$ e $\QQx$} +\author{Gabriel Antonio Videtta} +\date{\today} + +\begin{document} + +\maketitle + +\tableofcontents + +\section{Irriducibili e corollari di aritmetica in $\ZZi$} + +Come già dimostrato, $\ZZi$ è un anello euclideo con la seguente +funzione grado: + +\[ g : \ZZi \setminuszero \to \ZZ,\, a+bi \mapsto \norm{a+bi}^2.\] + +A partire da questo preconcetto è possibile dimostrare un teorema +importante in aritmetica, il \nameref{th:teorema_natale}, +che discende direttamente come corollario di un teorema più +generale riguardante $\ZZi$. + +\subsection{Il teorema di Natale di Fermat e gli irriducibili in $\ZZi$} + +\begin{lemma} + \label{lem:riducibile_due_quadrati} + Sia $p$ un numero primo riducibile in $\ZZi$, allora $p$ + può essere scritto come somma di due quadrati in $\ZZ$. +\end{lemma} + +\begin{proof} + Se $p$ è riducibile in $\ZZi$, allora esistono $a+bi$ e + $c+di$ appartenenti a $\ZZi \setminus \ZZi^*$ tali che $p=(a+bi)(c+di)$. \\ + + Impiegando le proprietà dell'operazione di coniugio si + ottiene la seguente equazione: + + \[ \overline{p}=p=(a-bi)(c-di) \implies p^2=p \overline{p} = (a^2+b^2)(c^2+d^2). \] + + Dal momento che $a+bi$ e $c+di$ non sono invertibili, + i valori della funzione grado calcolati in essi sono strettamente + maggiori del valore assunto nell'unità, ovverosia: + + \[ a^2+b^2>1, \qquad c^2+d^2>1. \] + + Allora devono per forza valere le seguenti equazioni: + + \[ p=a^2+b^2, \qquad p=c^2+d^2, \] + + da cui la tesi. +\end{proof} + +\begin{lemma} + \label{lem:quadrato_mod_4} + Sia $p$ un numero primo tale che $p \equiv 1 \pmod4$. Allora + esiste un $x \in \ZZ$ tale che $p \mid x^2+1$. +\end{lemma} + +\begin{proof} + Per il \textit{Teorema di Wilson}, $(p-1)! \equiv -1 \pmod p$. + Attraverso varie manipolazioni algebriche si ottiene: + + \[-1 \equiv 1 \cdots \frac{p-1}{2} \cdot \frac{p+1}{2} \cdots (p-1) \equiv 1 \cdots \frac{p-1}{2} \left(-\frac{p-1}{2}\right) \cdots (-1) \equiv\] + + \[ \equiv (-1)^{\frac{p-1}{2}} \left(\left( \frac{p-1}{2} \right)!\right)^2 \equiv + \left(\left( \frac{p-1}{2} \right)!\right)^2 \pmod p, + \] + + \vskip 0.1in + + da cui con $x = \left( \frac{p-1}{2} \right)!$ si verifica la + tesi. +\end{proof} + +\begin{theorem} + \label{th:primo_1_mod_4_riducibile} + Sia $p$ un numero primo tale che $p \equiv 1 \pmod4$. Allora + $p$ è riducibile in $\ZZi$. +\end{theorem} + +\begin{proof} + Per il \textit{Lemma \ref{lem:quadrato_mod_4}}, si ha che esiste + un $x \in \ZZ$ tale che $p \mid x^2+1$. Se $p$ fosse irriducibile, + dacché $\ZZi$ è un PID in quanto euclideo, $p$ sarebbe anche un + primo di $\ZZi$. Dal momento che $x^2+1=(x+i)(x-i)$, $p$ dovrebbe + dividere almeno uno di questi due fattori. \\ + + Senza perdità di generalità, si ponga che $p \mid (x+i)$. Allora + $\exists a+bi \in \ZZi \mid x+i=(a+bi)p$. Uguagliando le parti + immaginarie si ottiene $bp=1$, che non ammette soluzioni, \Lightning{}. Pertanto $p$ è riducibile. +\end{proof} + +\begin{corollary}[\textit{Teorema di Natale di Fermat}] + \label{th:teorema_natale} + Sia $p$ un numero primo tale che $p \equiv 1 \pmod4$. Allora + $p$ è somma di due quadrati in $\ZZ$. +\end{corollary} + +\begin{proof} + Per il \textit{Teorema \ref{th:primo_1_mod_4_riducibile}}, + $p$ è riducibile in $\ZZi$. In quanto riducibile in $\ZZi$, per + il \textit{Lemma \ref{lem:riducibile_due_quadrati}}, $p$ è allora + somma di due quadrati. +\end{proof} + +\begin{theorem} + \label{th:primo_-1_mod_4_irriducibile} + Sia $p$ un numero primo tale che $p \equiv -1 \pmod4$. Allora + $p$ è irriducibile in $\ZZi$. +\end{theorem} + +\begin{proof} + Se $p$ fosse riducibile in + $\ZZi$, per il \nameref{th:teorema_natale} esisterebbero $a$ e $b$ + in $\ZZ$ tali che $p=a^2+b^2$. Dal momento che $p$ è dispari, + possiamo supporre, senza perdità di generalità, che + $a$ sia pari e che $b$ sia dispari. Pertanto $a^2 \equiv 0 \pmod 4$ e $b^2 \equiv 1 \pmod 4$, dacché sono uno pari e l'altro dispari\footnote{Infatti, $0^2 \equiv 0 + \pmod4$, $1^2 \equiv 1 \pmod4$, $2^2 \equiv 4 \equiv 0 \pmod 4$, + $3^2 \equiv 9 \equiv 1 \pmod 4$.}. Tuttavia la congruenza + $a^2+b^2 \equiv 1 \equiv -1 \pmod4$ non è mai soddisfatta, + \Lightning{}. Pertanto $p$ può essere solo irriducibile. +\end{proof} + +\begin{note} + Si osserva che $2=(1+i)(1-i)$. Dal momento che $\norm{1+i}^2= + \norm{1-i}^2=2\neq1$, si deduce che nessuno dei due fattori + è invertibile. Pertanto $2$ non è irriducibile. +\end{note} + +\begin{proposition} + \label{prop:irriducibili_zz_zzi} + Gli unici primi $p \in \ZZ$ irriducibili in $\ZZi$ sono i primi $p$ tali + che $p \equiv -1 \pmod4$. +\end{proposition} + +\begin{proof} + Per l'osservazione precedente, $2$ non è irriducibile in $\ZZi$, + così come i primi congrui a $1$ in modulo $4$, + per il \textit{Teorema \ref{th:primo_1_mod_4_riducibile}}. Al + contrario i primi $p$ congrui a $-1$ in modulo $4$ sono + irriducibili, per il \textit{Teorema \ref{th:primo_-1_mod_4_irriducibile}}, da cui la tesi. +\end{proof} + +\begin{theorem} + $z \in \ZZi$ è irriducibile se e solo se $z$ è un associato di un $k \in \ZZ$ tale che $k \equiv -1 \pmod 4$, o se $\norm{z}^2$ è primo. +\end{theorem} + +\begin{proof} Si dimostrano le due implicazioni separatamente. \\ + + ($\implies$)\; Sia $z \in \ZZi$ irriducibile. Chiaramente + $z \mid z \overline{z} = g(z)$. Dacché $\ZZ$ è un UFD, + $g(z)$ può decomporsi in un prodotto di primi $q_1q_2\cdots q_n$. + Dal momento che $\ZZi$ è un PID, in quanto anello euclideo, + $z$ deve dividere uno dei primi della fattorizzazione di + $g(z)$. Si assuma che tale primo sia $q_i$. Allora esiste + un $w \in \ZZi$ tale che $q_i=wz$. \\ + + Se $w \in \ZZi^*$, si + deduce che $z$ è un associato di $q_i$. Dal momento che + $z$ è irriducibile, $q_i$, che è suo associato, è a sua + volta irriducibile. Allora, per la \textit{Proposizione \ref{prop:irriducibili_zz_zzi}}, $q_i \equiv -1 \pmod4$. + \\ + + Altrimenti, se $w$ non è invertibile, si ha che $g(w)>g(1)$, + ossia che $\norm{w}^2>1$. Inoltre in quanto irriducibile, anche + $z$ non è invertibile, e quindi + $g(z)>g(1) \implies \norm{z}^2>1$. Dalla proprietà + moltiplicativa + del modulo si ricava $q_i^2 = \norm{q_i}^2 = \norm{w}^2 \norm{z}^2$, + da cui necessariamente consegue che: + + \[ \norm{w}^2=q_i, \quad \norm{z}^2=q_i, \] + + attraverso cui si verifica l'implicazione. \\ + + ($\,\Longleftarrow\,\,$)\; Se $k \in \ZZ$ e $k \equiv -1 \pmod4$, per + il \textit{Teorema \ref{th:primo_-1_mod_4_irriducibile}}, $k$ è + irriducibile. Allora in quanto suo associato, anche $z$ è irriducibile. \\ + + Altrimenti, se $\norm{z}^2$ è un primo $p$, si ponga + $z=ab$ con $a$ e $b \in \ZZi$. Per la proprietà moltiplicativa + del modulo, $p = \norm{z}^2 = \norm{ab}^2 = \norm{a}^2\norm{b}^2$. + Tuttavia questo implica che uno tra $\norm{a}^2$ e $\norm{b}^2$ + sia pari a $1$, ossia che uno tra $a$ e $b$ sia invertibile, + dacché $g(1)=1$. Pertanto $z$ è in ogni caso irriducibile. +\end{proof} + +Infine si enuncia un'ultima identità inerente all'aritmetica, ma +strettamente collegata a $\ZZi$. + +\subsection{L'identità di Brahmagupta-Fibonacci} + +\begin{proposition}[\textit{Identità di Brahmagupta-Fibonacci}] + \label{prop:fibonacci} + Il prodotto di due somme di quadrati è ancora una + somma di quadrati. In particolare: + + \[ (a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2. \] +\end{proposition} + +\begin{proof} + La dimostrazione altro non è che una banale verifica + algebrica. Ciononostante è possibile risalire a questa + identità in via alternativa mediante l'uso + del modulo dei numeri complessi. \\ + + Siano $z_1=a+bi$, $z_2=c+di \in \CC$. Allora, per le proprietà + del modulo dei numeri complessi: + + \begin{equation} + \label{eq:modulo_z} + \norm{z_1}\norm{z_2}=\norm{z_1z_2}. + \end{equation} + + + Computando il prodotto tra $z_1$ e $z_2$ si ottiene: + + \[ z_1z_2 = (ac-bd) + (ad+bc)i, \] + + da cui a sua volta si ricava: + + \[ \norm{z_1z_2} = \sqrt{(ac-bd)^2 + (ad+bc)^2}, \] + + assieme a: + + \[ \norm{z_1}=\sqrt{a^2+b^2}, \quad \norm{z_2}=\sqrt{c^2+d^2}. \] + + Infine, da \eqref{eq:modulo_z}, elevando al quadrato, si deduce l'identità + presentata: + + \begin{multline*} + \sqrt{a^2+b^2}\sqrt{c^2+d^2}=\sqrt{(ac-bd)^2 + (ad+bc)^2} \implies (a^2+b^2)(c^2+d^2)= \\ (ac-bd)^2+(ad+bc)^2. + \end{multline*} +\end{proof} + +\begin{example} + Si consideri $65=5 \cdot 13$. Dal momento che sia $5$ + che $13$ sono congrui a $1$ in modulo $4$, sappiamo + già si possono scrivere entrambi come somme di due + quadrati. Allora, dall'\nameref{prop:fibonacci}, + anche $65$ è somma di due quadrati. \\ + + Infatti $5=2^2+1^2$ e $13=3^2+2^2$. Pertanto + $65=5\cdot 13=(2\cdot3-1\cdot2)^2 + (2\cdot2+1\cdot3)^2=4^2+7^2$. +\end{example} + +\section{Irriducibilità in $\ZZx$ e in $\QQx$} + +\subsection{Criterio di Eisenstein e proiezione in $\ZZpx$} + +Prima di studiare le irriducibilità in $\ZZ$, si guarda +alle irriducibilità nei vari campi finiti $\ZZp$, con +$p$ primo. Questo metodo presenta un vantaggio da non +sottovalutare: in $\ZZp$ per ogni grado $n$ esiste un +numero finito di polinomi monici\footnote{Si prendono in + considerazione solo i polinomi monici dal momento che vale + l'equivalenza degli associati: se $a$ divide $b$, allora + tutti gli associati di $a$ dividono $b$. $\ZZp$ è infatti + un campo, e quindi $\ZZpx$ è un anello euclideo.} -- in particolare, $p^n$ -- +e quindi per un polinomio di grado $d$ è sufficiente controllare +che questo non sia prodotto di tali polinomi monici per +$1 \leq n < d$. \\ + +In modo preliminare, si definisce un omomorfismo fondamentale. + +\begin{definition} + Sia il seguente l'\textbf{omomorfismo di proiezione} da + $\ZZ$ in $\ZZp$: + + \[ \hatpip : \ZZx \to \ZZpx,\, a_n x^n + \ldots + a_0 \mapsto [a_n]_p \, x^n + \ldots + [a_0]_p. \] +\end{definition} + +\begin{note} + Si dimostra facilmente che $\hatpi$ è un omomorfismo di anelli. + Innanzitutto, $\hatpi(1) = [1]_p$. Vale chiaramente la linearità: + + \begin{multline*} + \hatpip(a_n x^n + \ldots + a_0) + \hatpip(b_n x^n + \ldots + b_0) = [a_n]_p \, x^n + \ldots + [b_n]_p \, x^n + \ldots = \\ + = [a_n+b_n]_p \, x^n + \ldots = \hatpip(a_n x^n + \ldots + a_0 + b_n x^n + \ldots + b_0). + \end{multline*} + + Infine vale anche la moltiplicatività: + + \begin{multline*} + \hatpip(a_n x^n + \ldots + a_0) \hatpip(b_n x^n + \ldots + b_0) = ([a_n]_p \, x^n + \ldots)([b_n]_p \, x^n + \ldots) = \\ + = \sum_{i=0}^n \sum_{j+k=i} [a_j]_p \, [b_k]_p \, x^i + = \sum_{i=0}^n \sum_{j+k=i} [a_j b_k]_p \, x^i + = \hatpip\left(\sum_{i=0}^n \sum_{j+k=i} a_j b_k x^i\right) = \\ + =\hatpip\left((a_n x^n + \ldots + a_0)(b_n x^n + \ldots + b_0)\right). + \end{multline*} +\end{note} + +Prima di enunciare un teorema che si rivelerà +importante nel determinare l'irriducibilità di un +polinomio in $\ZZx$, si enuncia una definizione che +verrà ripresa anche in seguito + +\begin{definition} + Un polinomio $a_n x^n + \ldots + a_0 \in \ZZx$ si dice + \textbf{primitivo} se $\MCD(a_n, \ldots, a_0)=1$. +\end{definition} + +\begin{theorem} + \label{th:proiezione_irriducibilità} + Sia $p$ un primo. Sia $f(x) = a_n x^n + \ldots \in \ZZx$ + primitivo. Se $p \nmid a_n$ e + $\hatpip(f(x))$ è irriducibile in $\ZZpx$, allora anche $f(x)$ lo + è in $\ZZx$. +\end{theorem} + +\begin{proof} + Si dimostra la tesi contronominalmente. Sia $f(x) = + a_nx^n + \ldots \in \ZZ[x]$ primitivo e riducibile, con + $p \nmid a_n$. Dal momento che $f(x)$ è riducibile, esistono + $g(x)$, $h(x)$ non invertibili tali che $f(x)=g(x)h(x)$. \\ + + Si dimostra che $\deg g(x) \geq 1$. Se infatti fosse nullo, + $g(x)$ dovrebbe o essere uguale a $\pm 1$ -- assurdo, dal + momento che $g(x)$ non è invertibile, \Lightning{} -- o + essere una costante non invertibile. Tuttavia, nell'ultimo + caso, risulterebbe che $f(x)$ non è primitivo, poiché + $g(x)$ dividerebbe ogni coefficiente del polinomio. + Analogamente anche $\deg h(x) \geq 1$. \\ + + Si consideri ora $\hatpip(f(x))=\hatpip(g(x))\hatpip(h(x))$. + Dal momento che $p \nmid a_n$, il grado di $f(x)$ rimane costante + sotto l'operazione di omomorfismo, ossia $\deg \hatpip(f(x)) = + \deg f(x)$. \\ + + Inoltre, poiché nessuno dei fattori di $f(x)$ è nullo, $\deg f(x) = \deg g(x) + + \deg h(x)$. Da questa considerazione si deduce che anche i + gradi di $g(x)$ e $h(x)$ non devono calare, altrimenti si + avrebbe che $\deg \hatpip(f(x)) < \deg f(x)$, \Lightning{}. + Allora $\deg \hatpip(g(x)) = \deg g(x) \geq 1$, + $\deg \hatpip(h(x)) = \deg h(x) \geq 1$. \\ + + Poiché $\deg \hatpip(g(x))$ e $\deg \hatpip(h(x))$ sono + dunque entrambi non nulli, $\hatpip(g(x))$ e $\hatpip(h(x))$ + non sono invertibili\footnote{Si ricorda che $\ZZpx$ + è un anello euclideo. Pertanto, non avere lo stesso grado + dell'unità equivale a non essere invertibili.}. Quindi + $f(x)$ è prodotto di non invertibili, ed è dunque riducibile. + +\end{proof} + +\begin{theorem}[\textit{Criterio di Eisenstein}] + \label{th:eisenstein} + Sia $p$ un primo. + Sia $f(x) = a_n x^n + \ldots + a_0 \in \ZZx$ primitivo tale che: + + \begin{enumerate}[ (1)] + \item $p \nmid a_n$, + \item $p \mid a_i$, $\forall i \neq n$, + \item $p^2 \nmid a_0$. + \end{enumerate} + + Allora $f(x)$ è irriducibile in $\ZZx$. +\end{theorem} + +\begin{proof} + Si ponga $f(x)$ riducibile e sia pertanto $f(x)=g(x)h(x)$ con + $g(x)$ e $h(x)$ non invertibili. Analogamente a come visto + per il \textit{Teorema \ref{th:proiezione_irriducibilità}}, si + desume che $\deg g(x)$, $\deg h(x) \geq 1$. \\ + + Si applica l'omomorfismo di proiezione in $\ZZpx$: + + \[ \hatpip(f(x))=\underbrace{[a_n]_p}_{\neq 0} x_n, \] + + da cui si deduce che $\deg \hatpip(f(x)) = \deg f(x)$. \\ + + Dal momento che $\hatpip(f(x))=\hatpip(g(x))\hatpip(h(x))$ e + che $\ZZpx$, in quanto campo, è un dominio, + necessariamente sia $\hatpip(g(x))$ che $\hatpip(h(x))$ + sono dei monomi. \\ + + Inoltre, sempre in modo analogo a come visto per il \textit{Teorema + \ref{th:proiezione_irriducibilità}}, sia $\deg \hatpip(g(x))$ + che $\deg \hatpip(h(x))$ sono maggiori o uguali ad $1$. \\ + + Combinando questo risultato col fatto che questi due fattori + sono monomi, si desume che + $\hatpip(g(x))$ e $\hatpip(h(x))$ sono monomi di grado positivo. + Quindi $p$ deve dividere entrambi i termini noti di $g(x)$ e + $h(x)$, e in particolare $p^2$ deve dividere il loro prodotto, + ossia $a_0$. Tuttavia questo è un assurdo, \Lightning{}. +\end{proof} + +\begin{note} + Si consideri $x^k-2$, per $k \geq 1$. + Per il \nameref{th:eisenstein}, + considerando come primo $p=2$, si verifica che + $x^k-2$ è sempre irriducibile. Pertanto, per ogni + grado di un polinomio esiste almeno un irriducibile -- + a differenza di come invece avviene in $\RRx$ o in $\CCx$. +\end{note} + +\begin{theorem} + Sia $f(x) \in \ZZx$ primitivo e sia $a \in \ZZ$. Allora $f(x)$ è + irriducibile se e solo se $f(x+a)$ è irriducibile. +\end{theorem} + +\begin{proof} + Si dimostra una sola implicazione, dal momento che l'implicazione + contraria consegue dalle stesse considerazioni poste + studiando prima $f(x+a)$ e poi $f(x)$. \\ + + Sia $f(x)=a(x)b(x)$ riducibile, con $a(x)$, $b(x) \in \ZZx$ non + invertibili. Come già visto per il \textit{Teorema + \ref{th:proiezione_irriducibilità}}, $\deg a(x)$, $\deg b(x) \geq 1$. \\ + + Allora chiaramente $f(x+a)=g(x+a)h(x+a)$, con $\deg g(x+a) = + \deg g(x) \geq 1$, $\deg h(x+a) = \deg h(x) \geq 1$. Pertanto + $f(x+a)$ continua a essere riducibile, da cui la tesi. +\end{proof} + +\begin{example} + Si consideri $f(x) = x^{p-1}+\ldots+x^2+x+1 \in \ZZx$, dove + tutti i coefficienti del polinomio sono $1$. Si verifica che: + + \[ f(x+1)=\frac{(x+1)^p-1}x = p+\binom{p}{2}x+\ldots+x^{p-1}. \] + + Allora, per il \nameref{th:eisenstein} con $p$, $f(x+1)$ è + irriducibile. Pertanto anche $f(x)$ lo è. +\end{example} + +\subsection{Alcuni irriducibili di $\ZZ_2[x]$} + +Tra tutti gli anelli $\ZZpx$, $\ZZ_2[x]$ ricopre sicuramente +un ruolo fondamentale, dal momento che è il meno costoso +computazionalmente da analizzare, dacché $\ZZ_2$ consta +di soli due elementi. Pertanto si computano adesso gli +irriducibili di $\ZZ_2[x]$ fino al quarto grado incluso, a meno +di associati. \\ + +Sicuramente $x$ e $x+1$ sono irriducibili, dal momento che sono di +primo grado. I polinomi di secondo grado devono dunque essere +prodotto di questi polinomi, e pertanto devono avere o $0$ o +$1$ come radice: si verifica quindi che $x^2+x+1$ è l'unico +polinomio di secondo grado irriducibile. \\ + +Per il terzo grado vale ancora lo stesso principio, per cui +$x^3+x^2+1$ e $x^3+x+1$ sono gli unici irriducibili di tale grado. +Infine, per il quarto grado, i polinomi riducibili soddisfano +una qualsiasi delle seguenti proprietà: + +\begin{itemize} + \item $0$ e $1$ sono radici del polinomio, + \item il polinomio è prodotto di due polinomi irriducibili di + secondo grado. +\end{itemize} + +Si escludono pertanto dagli irriducibili i polinomi non omogenei -- +che hanno sicuramente $0$ come radice --, e i polinomi con $1$ come +radice, ossia $x^4+x^3+x+1$,\ \ +$x^4+x^3+x^2+1$, e $x^4+x^2+x+1$. Si esclude anche +$(x^2+x+1)^2 = x^4+x^2+1$. Pertanto gli unici irriducibili di +grado quattro sono $x^4+x^3+x^2+x+1$,\ \ $x^4+x^3+1$,\ \ $x^4+x+1$. \\ + +Tutti questi irriducibili sono raccolti nella seguente tabella: + +\begin{itemize} + \item (grado 1) $x$, $x+1$, + \item (grado 2) $x^2+x+1$, + \item (grado 3) $x^3+x^2+1$, $x^3+x+1$, + \item (grado 4) $x^4+x^3+x^2+x+1$,\ \ $x^4+x^3+1$,\ \ $x^4+x+1$. +\end{itemize} + +\begin{example} + Il polinomio $51x^3+11x^2+1 \in \ZZx$ è primitivo dal momento + che $\MCD(51, 11, 1)=1$. Inoltre, poiché $\hatpi_2(51x^3+11x^2+1)= + x^3+x+1$ è irriducibile, si deduce che anche $51x^3+11x^2+1$ lo + è per il \textit{Teorema \ref{th:proiezione_irriducibilità}}. +\end{example} + +\subsection{Teorema delle radici razionali e lemma di Gauss} + +Si enunciano in questa sezione i teoremi più importanti per +lo studio dell'irriducibilità dei polinomi in $\QQx$ e +in $\ZZx$, a partire dai due teoremi più importanti: il +classico \nameref{th:radici_razionali} e il \nameref{th:lemma_gauss}, +che si pone da ponte tra l'analisi dell'irriducibilità in $\ZZx$ e +quella in $\QQx$. + +\begin{theorem}[\textit{Teorema delle radici razionali}] + \label{th:radici_razionali} + Sia $f(x) = a_n x^n + \ldots + a_0 \in \ZZx$. Abbia $f(x)$ + una radice razionale. Allora, detta tale radice $\frac{p}{q}$, già ridotta ai minimi termini, questa è tale che: + + \begin{enumerate}[ (i.)] + \item $p \mid a_0$, + \item $q \mid a_n$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Poiché $\frac{p}{q}$ è radice, $f\left(\frac{p}{q}\right)=0$, e + quindi si ricava che: + + \[ a_n \left( \frac{p}{q} \right)^n + \ldots + a_0 = 0 \implies + a_n p^n = -q( \ldots + a_0 q^{n-1}). \] + + \vskip 0.1in + + Quindi $q \mid a_n p^n$. Dal momento che $\MCD(p, q)=1$, si + deduce che $q \mid a_n$. \\ + + Analogamente si ricava che: + + \[ a_0 q^n = -p(a_n p^{n-1} + \ldots). \] + + \vskip 0.1in + + Pertanto, per lo stesso motivo espresso in precedenza, + $p \mid a_0$, da cui la tesi. +\end{proof} + +\begin{theorem}[\textit{Lemma di Gauss}] + \label{th:lemma_gauss} + Il prodotto di due polinomi primitivi in $\ZZx$ è anch'esso primitivo. +\end{theorem} + +\begin{proof} + Siano $g(x) = a_m x^m + \ldots + a_0$ e $h(x) = b^n x^n + \ldots + b_0$ due polinomi primitivi in $\ZZx$. Si assuma che $f(x)=g(x)h(x)$ + non sia primitivo. Allora esiste un $p$ primo che divide tutti i + coefficienti di $f(x)$. \\ + + Siano $a_s$ e $b_t$ i più piccoli coefficienti non divisibili + da $p$ dei rispettivi polinomi. Questi sicuramente esistono, + altrimenti $p$ dividerebbe tutti i coefficienti, e quindi + o $g(x)$ o $h(x)$ non sarebbe primitivo, \Lightning{}. \\ + + Si consideri il coefficiente di $x^{s+t}$ di $f(x)$: + + \[c_{s+t} = \sum_{j+k=s+t} a_j b_k = \underbrace{a_0 b_{s+t} + a_1 b_{s+t-1} + \ldots}_{\equiv \, 0 \pmod p} + a_s b_t + \underbrace{a_{s+1}b_{t-1} + \ldots}_{\equiv \, 0 \pmod p},\] + + dal momento che $p \mid c_{s+t}$, si deduce che $p$ deve dividere + anche $a_sb_t$, ossia uno tra $a_s$ e $b_t$, che è assurdo, \Lightning{}. Quindi $f(x)$ è primitivo. + +\end{proof} + +\begin{theorem}[\textit{Secondo lemma di Gauss}] + \label{th:lemma_gauss_2} + Sia $f(x) \in \ZZx$. Allora $f(x)$ è irriducibile in $\ZZx$ + se e solo se $f(x)$ è irriducibile in $\QQx$ ed è primitivo. +\end{theorem} + +\begin{proof} Si dimostrano le due implicazioni separatamente. \\ + + ($\implies$)\; Si dimostra l'implicazione contronominalmente, + ossia mostrando che se $f(x)$ non è primitivo o se è + riducibile in $\QQx$, allora $f(x)$ è riducibile in $\ZZx$. \\ + + Se $f(x)$ non è primitivo, allora + $f(x)$ è riducibile in $\ZZx$. Sia quindi $f(x)$ primitivo + e riducibile in $\QQx$, con $f(x)=g(x)h(x)$, + $g(x)$, $h(x) \in \QQx \setminus \QQx^*$. \\ + + Si descrivano $g(x)$ e $h(x)$ nel seguente modo: + + \[ g(x)=\frac{p_m}{q_m} x^m + \ldots + \frac{p_0}{q_0}, \quad \MCD(p_i, q_i)=1 \; \forall 0 \leq i \leq m, \] + + \[ h(x)=\frac{s_n}{t_n} x^n + \ldots + \frac{s_0}{t_0}, \quad + \MCD(s_i, t_i)=1 \; \forall 0 \leq i \leq n. \] + + \vskip 0.1in + + Si definiscano inoltre le seguenti costanti: + + \[ \alpha = \frac{\mcm(q_m, \ldots, q_0)}{\MCD(p_m, \ldots, p_0)}, \quad \beta = \frac{\mcm(t_n, \ldots, t_0)}{\MCD(s_n, \ldots, s_0)}. \] + + \vskip 0.1in + + Si verifica che sia $\hat{g}(x)=\alpha g(x)$ che + $\hat{h}(x)=\beta h(x)$ appartengono a $\ZZx$ e che entrambi + sono primitivi. Pertanto $\hat{g}(x) \hat{h}(x) \in \ZZx$. \\ + + Si descriva $f(x)$ nel seguente modo: + + \[ f(x)=a_k x^k + \ldots + a_0, \quad \MCD(a_k,\ldots,a_0)=1. \] + + \vskip 0.1in + + Sia $\alpha \beta = \frac{p}{q}$ con $\MCD(p,q)=1$, allora: + + \[\hat{g}(x) \hat{h}(x) = \alpha \beta f(x) = \frac{p}{q} (a_k x^k + \ldots + a_0), \] + + da cui, per far sì che $\hat{g}(x) \hat{h}(x)$ appartenga + a $\ZZx$, $q$ deve necessariamente dividere tutti i + coefficienti di $f(x)$. Tuttavia $f(x)$ è primitivo, e quindi + $q=\pm 1$. Pertanto $\alpha \beta = \pm p \in \ZZ$. \\ + + Infine, per il \nameref{th:lemma_gauss}, $\alpha \beta f(x)$ + è primitivo, da cui $\alpha \beta = \pm 1$. Quindi + $f(x) = \pm \hat{g}(x) \hat{h}(x)$ è riducibile. \\ + + ($\,\Longleftarrow\,\,$)\; Se $f(x)$ è irriducibile in $\QQx$ + ed è primitivo, sicuramente $f(x)$ è irriducibile anche in + $\ZZx$. Infatti, se esiste una fattorizzazione in + irriducibili in $\ZZx$, essa non include alcuna costante + moltiplicativa dal momento che $f(x)$ è primitivo, e quindi + esisterebbe una fattorizzazione in irriducibili anche in $\QQx$. +\end{proof} + +\end{document} diff --git a/Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.html b/Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.html new file mode 100644 index 0000000..4325b95 --- /dev/null +++ b/Algebra/Notebook/1. Algoritmo di rappresentazione dei polinomi simmetrici/Algoritmo di rappresentazione dei polinomi simmetrici negli e_i.html @@ -0,0 +1,13884 @@ + + + + +Algoritmo di rappresentazione dei polinomi simmetrici negli e_i + + + + + + + + + + + + + + + + + + + + + + +