Definable completeness of P-minimal fields and applications

Abstract

We show that every definable nested family of closed and bounded subsets of a [Formula: see text]-minimal field [Formula: see text] has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that [Formula: see text]-minimal fields satisfy the “extreme value property”: for every closed and bounded subset [Formula: see text] and every interpretable continuous function [Formula: see text] (where [Formula: see text] denotes the value group), [Formula: see text] admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of [Formula: see text] is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every [Formula: see text]-minimal field is polynomially bounded. The second one characterizes those [Formula: see text]-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues.