Model theoretic properties of metric valued fields

Details Model theoretic properties of metric valued fields Model theoretic properties of metric valued fields

2009-07-27

arxiv.org/abs/0907.4560v1

Mathematics

Logic

Scientific paper

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.

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