2020/07/15 by Pablo Cubides Kovacsics, Kovacsics, Pablo Cubides, Françoise Delon +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2007.07521
openalex publication_date 2020/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\colon U → ΓK (where ΓK 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×ΓKn 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.