Cell decomposition for semi-affine structures on p-adic fields

We use cell decomposition techniques to study additive reducts of p- adic fields. We consider a very general class of fields, including fields with infinite residue fields, which we study using a multi-sorted language. The results are used to obtain cell decomposition results for the case of finite residue fields. We do not require fields to be Henselian, and we allow them to be of any characteristic.Comment: 22 page

Integration and Cell Decomposition in PP-minimal Structures

We show that the class of $\mathcal{L}$-constructible functions is closed under integration for any $P$-minimal expansion of a $p$-adic field $(K,\mathcal{L})$. This generalizes results previously known for semi-algebraic and sub-analytic structures. As part of the proof, we obtain a weak version of cell decomposition and function preparation for $P$-minimal structures, a result which is independent of the existence of Skolem functions. %The result is obtained from weak versions of cell decomposition and function preparation which we prove for general $P$-minimal structures. A direct corollary is that Denef's results on the rationality of Poincar\'e series hold in any $P$-minimal expansion of a $p$-adic field $(K,\mathcal{L})$.Comment: 22 page

Cell Decomposition for semibounded p-adic sets

We study a reduct L\ast of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the L\ast-definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K,L\ast) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multi- plication can only be defined on bounded sets, and we consider the existence of definable Skolem functions.Comment: 20 page

Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We give a definition, in the ring language, of Z_p inside Q_p and of F_p[[t]] inside F_p((t)), which works uniformly for all $p$ and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Z_p inside Q_p uniformly for all p. For any fixed finite extension of Q_p, we give an existential formula and a universal formula in the ring language which define the valuation ring