Submission date: 24 February 2020

Abstract:

The aim of this paper is to study the dimensions and standard part maps between the field of p-adic numbers ℚ_p and its elementary extension K in the language of rings L_r. We show that for any K-definable set X ⊆ K^m, dim_K(X) ≥ dim_{ℚ_p}(X ∩ ℚ_p^m). Let V ⊆ K be convex hull of K over ℚ_p, and st: V → ℚ_p be the standard part map. We show that for any K-definable function f:K^m → K, there is definable subset D ⊆ ℚ_p^m such that ℚ_p^m ∖ D has no interior, and for all x in D, either f(x) in V and st(f(st^{-1}(x))) is constant, or f(st^{-1}(x)) ∩ V=∅. We also prove that dim_K(X) ≥ dim_{ℚ_p}(st(X ∩ V^m)) for every definable X ⊆ K^m.

Mathematics Subject Classification: 03C60 30G06 03H05

Keywords and phrases:

Full text arXiv 2002.10117: pdf, ps.