On Dimensions, Standard Part Maps, and $p$-Adically Closed Fields

Abstract

The aim of this paper is to study the dimensions and standard part maps between the field of $p$-adic numbers ${\mathbb{Q}}_p$ and its elementary extension $K$ in the language of rings $L_r$. We show that for any $K$-definable set $X\subseteq K^m$, ${\text{dim}}_K(X)\ge {\text{dim}}_{{\mathbb{Q}}_p}(X\cap {{\mathbb{Q}}_p}^m)$. Let $V\subseteq K$ be convex hull of $K$ over ${\mathbb{Q}}_p$, and $\text{\st}: V\rightarrow {{\mathbb Q}_p}$ be the standard part map. We show that for any $K$-definable function $f:K^m\to K$, there is definable subset $D\subseteq {{\mathbb{Q}}_p}^m$ such that ${{\mathbb{Q}}_p}^m\backslash D$ has no interior, and for all $x\in D$, either $f(x)\in V$ and $\text{st}(f({\text{st}}^{-1}(x)))$ is constant, or $f({\text{st}}^{-1}(x))\cap V=\varnothing$. We also prove that ${\text{dim}}_K(X)\ge {\text{dim}}_{{\mathbb{Q}}_p}(\text{st}(X\cap V^m))$ for every definable $X\subseteq K^m$.

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• Nippy proofs of p-adic results of Delon and Yao