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It's number 23 in chapter 2-4(The tangent plane). The Problem is,

Let $P: C \rightarrow C$ be the complex polynomial

$P(\phi)=a_0\phi^n +a_1\phi^{n-1}+ \cdots +a_n $, $a_0 \not= 0$, $a_i \in {C}$

Denote by $\pi_N$ the stereographic projection of $S^2 $ from the north pole $N=(0,0,1)$ onto $R^2$. $F:S^2 \rightarrow S^2 $ is given by

$F(p)= \pi_N^-1 \circ P \circ \pi_N (p) , \text{ if } p \in S^2 - \{ N \}$,

$F(N)=N$

Prove the map $F:S^2 \rightarrow S^2 $ has only a finite number of critical points.

I have no idea at all. I've thought that critical point means it's a local maximum or local minimum.. Help me please. And it's my first time to leave a question, let me know if I've done something wrong!

user773114
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  • Critical points are points where the differential of $F$ not surjective. So you need to compute differential in coordinates, which is jacobian of $F$ in charts, and look for places where the jacobi have zero determinant. – Kelvin Lois Jun 03 '20 at 00:52
  • In fact, I don't know how to do it. $dF_p$ is a map from tangent plane to a tangent plane, and so to get jacobi, I need to represent the differntial in the basis of $T_{F(p)} S$.. How can I do that? – user773114 Jun 05 '20 at 14:57
  • Yes. Choose a chart, which is clear that we should choose chart with strereographic map $\pi_N$, so now $T_pS^2$ is spaned by $\partial/\partial x^1, \partial/\partial x^2$, with $x^1=x$ and $x^2=y$. Therefore we need to compute matrix $\partial \hat{F}^i/\partial x^j$, with $\hat{F} = \pi_N^{-1} \circ F \circ \pi_N = $. – Kelvin Lois Jun 06 '20 at 02:19
  • I got it. Thankyou so much!!! – user773114 Jun 07 '20 at 17:18

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