Integration of positive constructible functions against Euler characteristic and dimension

Following recent work of R. Cluckers and F. Loeser [Fonctions constructible et integration motivic I, C. R. Math. Acad. Sci. Paris 339 (2004) 411 - 416] on motivic integration, we develop a direct image formalism for positive constructible functions in the globally subanalytic context. This formalism is generalized to arbitrary first-order logic models and is illustrated by several examples on the p-adics, on the Presburger structure and on o-minimal expansions of groups. Furthermore, within this formalism, we define the Radon transform and prove the corresponding inversion formula.Comment: To appear in Journal of Pure and Applied Algebra; 8 page

Definable group extensions in semi-bounded o-minimal structures

In this note we show: Let ℛ = 〈 R, <, +, 0,...〉 be a semi-bounded (respectively, linear) o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([2]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈 Rm, +〉.FCT Financiamento Base 2008 - USFL/1/209; FCT grant SFRH/BPD/35000/200

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

Coverings by open cells

We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.Comment: 17 pages, revised versio

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

Product cones in dense pairs

LetM=〈M,<,+,...〉be an o-minimal expansion of an ordered group, andP⊆Ma dense set such thatcertain tameness conditions hold. We introduce the notion of aproduct conein ̃M=〈M,P〉, and prove: ifMexpands a real closed field, then ̃Madmits a product cone decomposition. IfMis linear, then it does not. Inparticular, we settle a question from [10]