Model Theoretic Properties of Metric Valued Fields

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, |·|) directly, but rather the associated projective spaces KP, as bounded metric structures. We show that the class of (projective spaces over) metric valued fields is elementary, with theory MV F , and that the projective spaces Pn and Pm are biïnterpretable for every n,m ≥ 1. The theory MV F admits a model completion ACMV F , 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, RCMV F , is the model companion of the theory of formally real metric valued fields, and that it is dependent. 1. The theory of metric valued fields Let us recall some terminology from Berkovich [Ber90]. A semi-normed ring is a unital commutative ring R equipped with a mapping |·| : R → R such that (i) |1| = 1, (ii) |xy| ≤ |x||y|, (iii) |x+ y| ≤ |x|+ |y|. If |x| = 0 =⇒ x = 0 then |·| is a norm. A semi-norm is multiplicative if |xy| = |x||y|. A multiplicative norm is also called a valuation. Thus, a valued field is equipped with a natural metric structure d(x, y) = |x− y|. In some contexts, a valuation is allowed to take values in Γ∪{0} where (Γ, ·) is an arbitrary ordered Abelian group and 0 < Γ, but this will not be the case in the present text. When we wish to make this explicit we shall refer to our fields as metric valued fields. If K is a complete valued field then either K ∈ {R,C} and |·| is the usual absolute value to some power (in which case |·| is Archimedean) or |x + y| ≤ |x| ∨ |y| (|·| is non Archimedean, or ultra-metric). From a model theoretic point of view, Archimedean valued fields, being locally compact, resemble finite structures of classical logic and are thus far less interesting than their ultra-metric counterparts. On the other hand, while everything we do here applies to arbitrary valued fields, including Archimedean ones, restricting our attention to the ultra-metric case does allow us many simplifications. Thus, with very little loss of generality, we shall only consider ultra-metric valued fields. Convention 1.1. Throughout, unless explicitly stated otherwise, by a valued field we mean a non Archimedean one. The valuation is said to be trivial if |x| = 1 for every x 6= 0. It is discrete if the image of |·| on K is discrete. Clearly every trivial valuation is discrete. On the other hand, a non trivial valuation on an algebraically (or separably) closed field cannot be discrete. A non trivially valued field is unbounded as a metric space, and therefore does not fit in the framework of standard bounded continuous logic. One device we use quite often with Banach space structures (Banach spaces, Banach lattices, and so on) is to restrict our attention to the structure formed by the closed unit ball. This approach may seem natural for valued fields as well, since the unit ball is simply the corresponding 2000 Mathematics Subject Classification. 03C90 ; 03C60 ; 03C64.