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We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.
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In an extended abstract Ressayre considered real closed exponential fields and integer parts that respect the exponential function. He outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayres construction, which becomes canonical once we fix the real closed exponential field, a residue field section, and a well ordering of the field. The procedure is constructible over these objects; each step looks effective, but may require many steps. We produce an example of an exponential field $R$ with a residue field $k$ and a well ordering $<$ such that $D^c(R)$ is low and $k$ and $<$ are $Delta^0_3$, and Ressayres construction cannot be completed in $L_{omega_1^{CK}}$.
The text is based on notes from a class entitled {em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from cite{hhmcrelle}, cite{hhm} and cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {em generic reparametrization}. I also try to bring out the relation to the geometry of cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
After recalling the definition of Zilber fields, and the main conjecture behind them, we prove that Zilber fields of cardinality up to the continuum have involutions, i.e., automorphisms of order two analogous to complex conjugation on (C,exp). Moreover, we also prove that for continuum cardinality there is an involution whose fixed field, as a real closed field, is isomorphic to the field of real numbers, and such that the kernel is exactly 2{pi}iZ, answering a question of Zilber, Kirby, Macintyre and Onshuus. The proof is obtained with an explicit construction of a Zilber field with the required properties. As further applications of this technique, we also classify the exponential subfields of Zilber fields, and we produce some exponential fields with involutions such that the exponential function is order-preserving, or even continuous, and all of the axioms of Zilber fields are satisfied except for the strong exponential-algebraic closure, which gets replaced by some weaker axioms.
We answer two open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable subset X of M^n, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.
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