Submission date: 30 June 2022

Abstract:

We prove the nonarchimedean counterpart of a real inequality involving the metric entropy and measure geometric invariants V_i, called Vitushkin's variations. Our inequality is based on a new convenient partial preorder on the set of constructible motivic functions, extending the one considered by R. Cluckers and F. Loeser in Constructible motivic functions and motivic integration, Invent. Math., 173 (2008). We introduce, using motivic integration theory and the notion of riso-triviality, nonarchimedean substitutes of the Vitushkin variations V_i, and in particular of the number V_0 of connected components. We also prove the nonarchimedean global Cauchy-Crofton formula for definable sets of dimension d, relating V_d and the motivic measure in dimension d.

Mathematics Subject Classification: 14B05, 14B07, 03C60, 03C98

Keywords and phrases:

Full text arXiv 2206.15412: pdf, ps.