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\newtheorem{lem}[thm]{��������}
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\begin{document}
 \parindent=0cm
  \section*{Applications}

  {\bf Fundamental theorem of algebra}  :

  \textit{$f(z)=z^n+a_{n-1}z^{n-1}+\cdot\cdot\cdot+a_1z+a_0=0$ has at least one
  zero.($z,a_i\in\mathbf{C}$)}\\



\begin{proof}
���� $|a_{n-1}|+|a_{n-2}|+\cdot\cdot\cdot+|a_0|<1$ �̶��
�����ص� ����. �ֳ��ϸ�, $z=\lambda x(\lambda >0) $ ���� ����
$f(z)$ �� $\lambda^n$���� ������

$x^n+\frac{a_{n-1}}{\lambda}x^{n-1}+\cdot\cdot\cdot+\frac{a_1}{\lambda^{n-1}}x+\frac{a_0}{\lambda^n}=0$

�� �ǹǷ�,
$|\frac{a_{n-1}}{\lam}|+\cdot\cdot\cdot+|\frac{a_0}{\lam^n}|<1$ ��
�����ϴ� ū $\lam$ �� ������ �ȴ�.\\

���� $f(z)$�� ���� ������ �ʴ´ٰ� ��������. �׷���
$f:\textbf{C}\rightarrow\textbf{C}\setminus\{0\}$ �̰�,\\$k(z)=z^n
: S^1\rightarrow S^1$ �� ���� $f|_{S^1} \simeq k :
S^1(\subset\textbf{C})\rightarrow\textbf{C}\setminus \{0\}$ ����
������.

Define
$F(z,t)=z^n+t(a_{n-1}z^{n-1}+\cdot\cdot\cdot+a_0),\hspace{2em}0\leq
t\leq 1$.

���� $F$�� $\mathbf{C}\setminus \{0\}$ ���� �������� ������.
$0\leq t\leq 1 \,\,,\,\, |z|=1$ �̰� ������
$|a_{n-1}|+|a_{n-2}|+\cdot\cdot\cdot+|a_0|<1$���� �����Ͽ����Ƿ�

$|F(z,t)|\geq|z|^n\,\,-\,\,t\,|\,\,a_{n-1}z^{n-1}+\cdot\cdot\cdot
a_0\,\,|$

$\hspace{3.7em}\geq|z|^n\,\,-\,\,t(\,|a_{n-1}z^{n-1}|+\cdot\cdot\cdot+|a_0|\,)$

$\hspace{3.7em}=1-t(|a_{n-1}|+|a_{n-2}|+\cdot\cdot\cdot|a_0|)$

$\hspace{3.7em}>0 $

�̰�, �̴� $f|_{S^1}$ �� $k$ ���� homotopy�� �ش�. \\

���� ���� diagram �� ���캸��,



$\hspace{2em}\hspace{2em}f\hspace{4.2em}p$

$S^1\hookrightarrow\textbf{C}\rightarrow
\textbf{C}\setminus\{0\}\rightarrow S^1\,\,\,\,\,\,\,\,\,\,$
where $\,\,\,\,\,\,p:z\mapsto \frac{z}{|z|}$.\\



$S^1$���� $\textbf{C}\setminus\{0\}$ ���� ���� map $f|_{S^1}$��
$k$ �� ���� $f|_{S^1} \simeq k$ �̰�, $p\circ k=k$ �̹Ƿ� $p\circ
f|_{S^1}=g$ ��� ������ �� $g=p\circ f|_{S^1}\simeq p\circ k=k$
�̴�. ���� $g_{\sharp}=k_{\sharp}:\pi_1(S^1)\rightarrow
\pi_1(S^1)$ �ε�, $f$�� $D^2$�� extend �ǹǷ� $g$ ����
���������̴�. �� $g_{\sharp}=0$ �ε� ����
$k_{\sharp}:\textbf{Z}\rightarrow\textbf{Z}$ �� n�� ���ִ�
homomorphism�̾ �̴� ����� �ȴ�.


\end{proof}

\begin{thm}

\textit{(Existence of "����")}

 Let  $f:{D}^{2} \rightarrow
\textbf{R}^{2}$ be a non-vanishing continuous vector field. Then
$\exists$ a point of $S^1$ where f is directly inward, i.e.,
$f(x)\cdot x<0$.(Similarly $\exists$ a point of $S^1$ where f is
directly outward, i.e., $f(x)\cdot x>0$)
\end{thm}
\begin{proof}
inward point�� ���ٰ� ��������. ��   $f(x)\cdot x\geq 0$, $\forall
x\in S^1$ �̶�� �� ��

Define $F(x,t):S^1\times I \rightarrow
\textbf{R}^2\setminus\{0\}\,\,$ by $\,\,F(x,t)=tf(x) + (1-t)x$.

(1) $F$ �� �� ���ǵǾ����� ������. ��, F�� 0�� ���� ������ ������.

���� $tf(x)+(1-t)x=0$�̶��,\\
$\,|\,tf(x)+(1-t)x|^2=t^2|f(x)|^2+2tx(1-t)f(x)+(t-1)^2|x|^2=0$ \\�� �����ϰ�, ���⼭ ��
���� ��� 0���� ũ�ų� �����Ƿ� ���  0�� �Ǿ���Ѵ�.
$|x|=1$�̹Ƿ� $t=1$�̰�, ���� $f(x)=0$ �̹Ƿ� �̴� $f$��
non-vanishing �̶�µ��� ����̴�. �׸��� $F(x,0)=x, F(x,1)=f(x)$
�̹Ƿ� $F$�� $x$�� $f(x)$ ������ homotopy�� �ش�.\\

(2) $\hspace{6.5em}F\hspace{4.8em}p$

$\hspace{5em}S^1\times I\rightarrow
\textbf{R}^2\setminus\{0\}\rightarrow S^1$\\

�̹Ƿ� $p\circ inclusion=id \simeq p\circ f : S^1\rightarrow S^1$
�̴�. ������ $f$�� ���� $D^2$�� extension�� �����ϹǷ� $p\circ f$
���� �����ϰ� ����  $(p\circ f)_{\sharp}=0$ �̴�. ��
$id=0:\textbf{Z}\rightarrow \textbf{Z}$�̹Ƿ� �̴� ����̴�.

"outward point"�� ��쿡�� $f$��� $-f$�� ���� �ȴ�.


\end{proof}
\begin{thm}


$\nexists$ non-vanishing continuous vector field on $S^2$.

\end{thm}

\begin{proof}
���� nonvanishing vector field�� $S^2$���� �����Ѵٰ� ��������.
$S^2$���� �ϱ� $N$������ vector $v$�� �����ϸ� �����̶�� ���ǿ���
$v$�� ���� ���� vector�� ������ $N$�� ������ �ٹ��� �����Ѵ�.

���� $S^2$�� $\textbf{R}^2$�� stereographic projection�� �Ѵ�.
�׷��� $D^2$���� nonvanishing vector field�� ��µ� �̰���
�ڱ��ڽ��� ���̷� ������ ������� unit vector field��  $f$�� �θ�
���� �׸����� $f$�� $z$�� $z^2$ ���� ������ map�� homotopic����
�� �� �ְ�, ���� $f_{\sharp}$�� $\times 2$�� �־����� map�̴�.
������ $f$�� $f|_{S^1}:S^1\rightarrow S^1$���� $D^2$ ��
Ȯ��ǹǷ�  $f_{\sharp}=0$ �̰�, �̴� ����̴�.\\

**�׸�3**

\end{proof}








\end{document}
