Submission date: 13 November 2014. Revised version: 15 July 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 KxK^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, 12J12

Keywords and phrases: Cell decomposition, p-minimal, p-optimal, dimension of definable sets

Full text: arXiv 1412.2571: pdf, ps. Version of 13 November 2014: pdf, ps, dvi.