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I am trying to calculate the degree of a map $f:\mathbb CP^1\rightarrow \mathbb CP^1$ such that $f([x:y])=[x^8+y^8:x^8-y^8]$. The $[x:y]$ are homogenous coordinates.

I have been trying to claculate this for long, but all my proofs seem incorrect - so I think I probably do not understand some basic things.

My definition of a degree of a map $f$ between two connected and oriented manifolds is the one using the notion of counting the times where $Df(x)$ preserves orientaion for $x$ in the preimage of a regular value.

My first attempt was to prove that the $x^8+1 \over x^8-1$ has the same degree as $1+$$2 \over x^8-1$, and by that to deduce that the degree of $f$ is $-8$ using the properties of homogenous coordinates. But I got stuck and it made me think that don't even understand well the notion of homogenous coordinates.

My second attempt was to look at it in a more intuitve way, and to simply count the times the orientation is preserved for any regular value. But then I noticed that for differet regular values I get different results, which of course should not happen.

This made me understand that I do not understnad correctly the concept of a degree of a map on $\mathbb CP^1\ $ nor the concept of homogenous coordinates, so any help in understanding how to sole this problem would be appreciated.

MasterJ
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1 Answers1

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Consider the identification $\mathbb C\hookrightarrow\mathbb C \mathbb P^1: z\mapsto (z:1)$.
The map $f$ restricts to $f_0:U\to V: z\mapsto \frac {z^8+1}{z^8-1}$, where $U=\mathbb C\setminus\{e^{\frac {2ik\pi}{8}}\vert k=0,\cdots,7\}$ and $V=\mathbb C\setminus \{1\}$.
The only critical point in $U$ for $f_0$ is $z=0$, hence the only critical value of $f_0$ is $f_0(0)=-1$ so that in particular $0$ is a regular value of $f_0$.
But then notice that the equation $f_0(z)=0$ is equivalent to $z^8+1=0$ and has $8$ solutions.
All these solutions are to be counted with a positive sign because $f_0$, being holomorphic, preserves the orientation.

Conclusion: the map $f$ has degree $8.$

Georges Elencwajg
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