Submission date: 15 July 2020

Abstract:

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

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 2007.07521: pdf, ps.