Submission date: 12 March 2024

Abstract:

This paper deals with Hensel minimal structures on non-trivially valued fields K. The main aim is to establish the following two properties of closed 0-definable subsets A in the affine spaces K^n. Every such subset A is the zero locus of a continuous 0-definable function f:K^n → K, and there exists a 0-definable retraction r: K^n → A. While the former property is a non-Archimedean counterpart of the one from o-minimal geometry, the former does not hold in real geometry in general. The proofs make use of a model-theoretic compactness argument and ubiquity of clopen sets in non-Archimedean geometry.

Mathematics Subject Classification: 03C65, 03C98, 12J25

Keywords and phrases:

Full text arXiv 2403.08039: pdf, ps.