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differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex +++ b/Corsi/Geometria e topologia differenziale/Scheda/sections/4-varieta-e-teoria-del-grado.tex @@ -2194,6 +2194,129 @@ Segue immediatamente dal Lemma \ref{lem:grado_zk}. \end{proof} - \subsection{Lemma di Hopf} + \subsection{Lemma di Hopf e teorema fondamentale dell'algebra} + \begin{lemma}[di Hopf] \label{lem:hopf} + Sia $X \subseteq \RR^m$ una $m$-varietà compatta con bordo. Sia + $v : X \to \RR^m$ un campo vettoriale con zeri isolati e $\restr{v}{\partial X}$ + mai nullo. \smallskip + + Allora: + \[ + \boxed{\sum_{z \in v\inv(0)} \ind(v, z) = \deg\left(\bigrestr{\frac{v}{\norm{v}}}{\partial X} : \partial X \to S^{m-1} \right).} + \] + \end{lemma} + + \begin{proof} + Poiché gli zeri di $v$ sono isolati, possiamo togliere a $X$ una famiglia di dischi disgiunti $\{B_{\eps_i}(z_i)\}$ contenenti tali zeri. + Allora, poiché tali dischi sono interni, l'orientazione indotta su $\partial W$ sarà: + \[ + \partial W = \partial X \cup \bigsqcup_i -\partial B_{\eps_i}(z_i). + \] + Quindi, rispettando le orientazioni si ottiene: + \begin{multline*} + \deg\left(\bigrestr{\frac{v}{\norm{v}}}{\partial W} : \partial W \to S^{m-1} \right) = \\ + \deg\left(\bigrestr{\frac{v}{\norm{v}}}{\partial X} : \partial X \to S^{m-1} \right) - \sum_{z \in v\inv(0)} \ind(v, z). + \end{multline*} + Tuttavia, poiché $\bigrestr{\frac{v}{\norm{v}}}{\partial W}$ è la restrizione di $\frac{v}{\norm{v}} : W \to S^{m-1}$ sul bordo, per il Lemma + \ref{lem:grado_mappa_estendibile}, si ha anche: + \[ + \deg\left(\bigrestr{\frac{v}{\norm{v}}}{\partial W} : \partial W \to S^{m-1} \right) = 0, + \] + da cui segue immediatamente la tesi. + \end{proof} + + \begin{theorem}[fondamentale dell'algebra] + Sia $p(z) \in \CC[z]$ con $\deg(p) = n$. Allora: + \[ + \boxed{\deg(p) = \sum_{z_0 \in p\inv(0)} \mult(p, z_0),} + \] + dove $\mult$ indica la molteplicità algebrica di uno zero in un polinomio. + \end{theorem} + + \begin{proof} + Dal momento che $p(x)$ non può avere più di $n$ zeri, questi sono sicuramente + isolati e possiamo prendere inoltre una palla $B_r(0) \subseteq \CC$ con + $p\inv(0) \subseteq B_r(0)$. Possiamo allora applicare il Lemma \ref{lem:hopf} + su $p : \overline{B_r(0)} \to \CC$ e ottenere: + \[ + \deg\left(\frac{p}{\norm{p}} : \partial B_r(0) \to S^1 \right) = \sum_{z_0 \in p\inv(0)} \ind(p, z_0). + \] + Mostriamo che il termine a sinistra coincide con $\deg(p)$ (1), e che $\ind(p, z_0)$ coincide + con $\mult(p, z_0)$ (2), ottenendo infine la tesi. + + \begin{enumerate}[(1)] + \item Supponiamo che $\deg(p)$ sia $n$ e che $p(z)$ sia dunque della seguente forma: + \[ + p(z) = a_n z^n + \underbrace{a_{n-1} z^{n-1} + \ldots + a_0}_{\mathclap{g(z)}}, + \] + dove $g(z) \defeq p(z) - a_n z^n$. \smallskip + + Osserviamo che: + \begin{equation} \tag{*} + \lim_{\abs{z} \to \infty} \abs{\frac{g(z)}{z^n}} = 0. + \end{equation} + Una volta posto $p_t(z) = a_n z^n + t g(z)$, si ottiene: + \[ + \abs{\frac{p_t(z)}{z^n}} \geq \abs{a_n} - t \abs{\frac{g(z)}{z^n}}. + \] + Allora, per (*), possiamo scegliere $r$ sufficientemente grande in modo tale che si verifichi sempre: + \[ + \abs{\frac{p_t(z)}{z^n}} > 0, \quad z \in \partial B_r(0). + \] + In particolare, $p_t(z)$ non si annulla su $\partial B_r(0)$. Possiamo + allora considerare l'omotopia indotta da $\frac{p_t(z)}{\abs{p_t(z)}}$. Osserviamo che + $p_0(z) = a_n z^n$ e che $p_1(z) = p(z)$. Per il Teorema + \ref{thm:fondamentale_grado_intero}, si ha allora: + \[ + \begin{split} + \deg\left( \frac{p(z)}{\abs{p(z)}} : \partial B_r(0) \to S^1 \right) = \hspace{2cm} \\ + \hspace{2cm} \deg\left( \frac{a_n}{\abs{a_n}} \frac{z^n}{\abs{z^n}} : \partial B_r(0) \to S^1 \right), + \end{split} + \] + a cui, applicando il Lemma \ref{lem:grado_zk} e il fatto secondo cui la moltiplicazione per una costante di + fase è isotopa all'identità (vd. dimostrazione del Lemma \ref{lem:grado_zk}), si ottiene facilmente che: + \[ + \deg\left( \frac{p(z)}{\abs{p(z)}} : \partial B_r(0) \to S^1 \right) = n = \deg(p). + \] + + \item Sia $z_0$ uno zero di $p(z)$. Allora $p(z)$ si scrive come: + \[ + p(z) = (z - z_0)^\ell q(z), + \] + per un qualche polinomio $q(z) \in \CC[z]$, dove $\ell = \mult(p, z_0)$. Entro + una certa palla di raggio $\eps$ centrata in $z_0$, $q(z)$ non ha alcuno zero. Se + consideriamo la mappa $f : S^1 \to \partial B_{z_0}(\eps)$ tale per cui + $f(z) = z_0 + \eps z$, che preserva l'orientazione, allora si ha: + \[ + \ind(p, z_0) = \deg\left(\frac{p \circ f}{\norm{p \circ f}} : \partial S^1 \to S^1 \right). + \] + Osserviamo che: + \[ + \frac{p \circ f}{\norm{p \circ f}}\big(z\big) = \frac{z^\ell q(z_0 + \eps z)}{\abs{q(z_0 + \eps z)}}. + \] + Possiamo definire un'omotopia $H : S^1 \times [0, 1] \to S^1$ tale per cui: + \[ + H_t(z) = \frac{z^\ell q(z_0 + t \eps z)}{\abs{q(z_0 + t \eps z)}}, + \] + che porta $z^\ell \frac{q(z_0)}{\abs{q(z_0)}}$ in $\frac{p \circ f}{\norm{p \circ f}}$. Quindi, + per il Teorema \ref{thm:fondamentale_grado_intero}, si ha: + \[ + \begin{split} + \deg\left(z^\ell \frac{q(z_0)}{\abs{q(z_0)}} : S^1 \to S^1\right) = \hspace{2cm} \\ + \hspace{2cm} \deg\left(\frac{p \circ f}{\norm{p \circ f}} : \partial S^1 \to S^1 \right). + \end{split} + \] + Come visto nella dimostrazione del Lemma \ref{lem:grado_zk}, la moltiplicazione per elemento di $S^1$ è isotopa all'identità, e dunque: + \[ + \deg\left(z^\ell \frac{q(z_0)}{\abs{q(z_0)}} : S^1 \to S^1\right) = \deg(z^\ell : S^1 \to S^1) = \ell, + \] + da cui segue, combinando i pezzi, che: + \[ + \ind(p, z_0) = \ell. + \] + \end{enumerate} + \end{proof} + + \subsection{Teorema di Poincaré-Hopf} \end{multicols*}