Model theoretic properties of metric valued fields

Itaï Ben Yaacov (1)

1 ICJ - Institut Camille Jordan

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 biïnterpretable 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.

Keywords: valued field; real closed field; metric structure

Domains: Logic [math.LO]

Itaï Ben Yaacov. Model theoretic properties of metric valued fields. 2009. ⟨hal-00407559v1⟩