Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 313 / 3 / i / Solution

For a generic target value $y\in {\mathbb{CP}}^1$, its finite preimages solveWhen $m\ne n$ this has $\max (m,n)$ roots after the missing roots or poles at infinity are counted; when $m=n$, a generic $y$ again gives degree $m$. Holomorphic maps preserve orientation at regular preimages, so every local sign is positive. Hence