Ramification loci of non-archimedean cubic rational functions

TL;DR

This paper classifies the shapes of ramification loci for cubic rational functions over non-archimedean fields, identifying two main configurations and providing explicit forms and calculations.

Contribution

It systematically enumerates all possible forms of cubic rational functions and determines their Berkovich ramification loci shapes.

Findings

Two main shapes of ramification loci identified

Explicit forms of cubic rational functions provided

Calculations of ramification loci for each form included

Abstract

For a cubic rational function with coefficients in a non-archimedean field $K$ whose residue characteristic is $0$ or greater than $3$, there are $2$ possibilities for the shape of its Berkovich ramification locus, considered as an endomorphism of the Berkovich projective line: one is the connected hull of all the critical points, and the other is consisting of $2$ disjoint segments. In this paper, we list up all the possible forms of cubic rational functions and calculate their ramification loci.