Model theoretic properties of metric valued fields

TL;DR

This paper explores the model theoretic properties of metric valued fields, establishing their elementary class, bi-interpretability of projective spaces, and stability characteristics of their algebraic closures.

Contribution

It introduces the elementary class of metric valued fields, proves bi-interpretability of projective spaces, and characterizes the stability and dependence of their algebraic closures.

Findings

The class of metric valued fields is elementary with theory MVF.

Projective spaces over these fields are bi-interpretable.

Algebraically closed metric valued fields have a strictly stable theory.

Abstract

We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic. For technical reasons we prefer to consider not the valued field $(K,|{\cdot}|)$ directly, but rather the associated projective spaces $K\bP^n$, as bounded metric structures. We show that the class of (projective spaces over) metric valued fields is elementary, with theory $MVF$, and that the projective spaces $\bP^n$ and $\bP^m$ are bi\"interpretable for every $n,m \geq 1$. The theory $MVF$ admits a model completion $ACMVF$, the theory of algebraically closed metric valued fields (with a non trivial valuation). This theory is strictly stable (even up to perturbation). Similarly, we show that the theory of real closed metric valued fields, $RCMVF$, is the model companion of the theory of formally real metric valued fields, and that it is dependent.