Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 157 / 4 / a / Solution

The holomorphic fixed-point index, or residue index, isFor a simple fixed point with multiplier $\lambda \ne 1$ it equals $1/(1-\lambda )$. Three distinct fixed points of a quadratic rational map are simple, and the rational fixed-point formula givesIf, say, ${\lambda }_0{\lambda }_1=1$, thenThe remaining term would have to be zero, which is impossible. Thus the multiplier relation for three distinct fixed points of a quadratic rational map is