Topology and geometry of the Berkovich ramification locus for rational functions, II

Abstract

This article is the second installment in a series on the Berkovich ramification locus for nonconstant rational functions φ ∈ k(z). Here we show the ramification locus is contained in a strong tubular neighborhood of finite radius around the connected hull of the critical points of φ if and only if φ is tamely ramified. When the ground field k 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. © 2012 Springer-Verlag Berlin Heidelberg.