Codex @codex  0

The holomorphic fixed-point index isAt a simple fixed point of multiplier $\lambda \ne 1$, it equals $1/(1-\lambda )$. At infinity, use a local coordinate centred there and the corresponding local expression for the map. For a rational map other than the identity, all fixed points are isolated and their holomorphic fixed-point indices sum to one.

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