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 K 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. At the Edinburgh meeting on the model theory of valued fields in May 1999, Luc Bélair posed the question of whether there is a definable bijection between the set of p-adic integers and the set of p-adic integers with one point removed. At the same meeting, Jan Denef asked what is the Grothendieck ring of the p-adic numbers, as did Jan Kraj́ıček independently in [6]. A general introduction to Grothendieck rings of logical structures was recently given in [7] and in [DL2, par. 3.7]. Calculations of non-trivial Grothendieck rings and related topics such as motivic integration can be found in [4] and [3]. The logical notion of the Grothendieck ring of a structure is analogous to that of the Grothendieck ring in the context of algebraic K-theory and has analogous elementary properties (see [9]). Here we recall the definition. Definition 1. Let M be a structure and Def(M) the set of definable subsets ofM for every positive integer n. For anyX,Y ∈ Def(M), write X ∼= Y iff there is a definable bijection (an isomorphism) from X to Y . Let F be the free abelian group whose generators are isomorphism classes ⌊X⌋ with X ∈ Def(M) (so ⌊X⌋ = ⌊Y ⌋ if and only if X ∼= Y ) and let E be the subgroup generated by all expressions ⌊X⌋+⌊Y ⌋−⌊X ∪Y ⌋−⌊X ∩Y ⌋ with X,Y ∈ Def(M). Then the Grothendieck group of M is the quotient group F/E. Write [X] for the image of X ∈ Def(M) in F/E. The Grothendieck group has a natural structure as a ring with multiplication induced by [X] · [Y ] = [X ×Y ] for X,Y ∈ Def(M). We call this ring the Grothendieck ring K0(M) of M. It is easy to see that the above questions are related: the Grothendieck ring is trivial if and only if there is a definable bijection between M and itself minus a point for some k, which happens if and only if the ∗ Research Assistant of the Fund for Scientific Research – Flanders (Belgium)(F.W.O.)