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

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

2011-04-05

arxiv.org/abs/1104.0943v2

Mathematics

Number Theory

22 pages; major expository overhaul from the first version; added an alternate definition of the visible ramification that is

Scientific paper

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.

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