Topology and Geometry of the Berkovich Ramification Locus for Rational Functions, II

TL;DR

This paper investigates the structure of the Berkovich ramification locus for rational functions, establishing conditions under which it is contained within a tubular neighborhood of critical points and providing applications to non-Archimedean analysis.

Contribution

It characterizes when the ramification locus is contained in a tubular neighborhood based on tameness and introduces a new version of Rolle's theorem for rational functions.

Findings

Ramification locus contained in tubular neighborhood iff tamely ramified

Bound depends only on residue characteristic in characteristic zero

New version of Rolle's theorem for non-Archimedean analysis

Abstract

This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions f: P^1 -> P^1. Here we show the ramification locus of f is contained in a strong tubular neighborhood of finite radius around the connected hull of the critical points if and only if f is tamely ramified at all of its critical points. When the ground field has characteristic zero, this bound may be chosen to depend only on the residue characteristic. We give two applications to classical non-Archimedean analysis, including a new version of Rolle's theorem for rational functions.