The degree of a map between complex projective lines

Question

Let $P$ and $Q$ be complex polynomials such that $\deg P=p$, $\deg Q=q$ and $\gcd(P,Q)=1$.

How can I:

• show that $F(z)=\frac{P(z)}{Q(z)}$ defines a smooth map $\mathbb{C}P^1\to\mathbb{C}P^1$?
• compute a degree of $F$?

Answer 1

You can homogeneize the map: write $P(z)=\sum_{i=0}^p a_i z^i$, $Q(z)=\sum_{i=0}^q b_i z_i$, $d=\max\{p,q\}$ and define $$\begin{array}{rcl} \mathbb{P}^1&\to & \mathbb{P}^1\\ [z:w]&\mapsto & [\sum_{i=0}^p a_i z^iw^{d-i}:\sum_{i=0}^q b_i z^iw^{d-i}] \end{array}$$ This is a holomorphic map from $\mathbb{P}^1$ (or $\mathbb{C}P^1$) to itself and which restricts to your map on the subset where $w=1$. The degree is equal to $d$.