Stable domination and independence in algebraically closed valued fields

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries

Residue field domination in some henselian valued fields

We generalize previous results about stable domination and residue field domination to henselian valued fields of equicharacteristic 0 with bounded Galois group, and we provide an alternate characterization of stable domination in algebraically closed valued fields for types over parameters in the field sort

Grothendieck rings of Z-valued fields

We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K² to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point

Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries

It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in K of certain definable R-submodules of K (for all n 1). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts