Cell decomposition and classification of definable sets in p-optimal fields

by L. Darnière and I. Halupczok

Journal of Symbolic Logic 82 (2017), no. 1, 120-136

Submitted on November 13, 2014.
Re-submitted on July 02, 2015.

Abstract

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-optimal fields satisfying the Extreme Value Property (a property which in particular holds in fields which are elementarily equivalent to a p-adic one). For such fields K, we prove that every definable subset of K×K^d whose fibers are inverse images by the valuation of subsets of the value group, are semi-algebraic. Combining the two we get a preparation theorem for definable functions on p-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are isomorphic iff they have the same dimension.

Mathematics Subject Classification

03C07 Basic properties of first-order languages and structures
12J12 Formally p-adic fields

Electronic version of the paper

Version February 2016 (19 pages) pdf