399 research outputs found
Integration and Cell Decomposition in -minimal Structures
We show that the class of -constructible functions is closed
under integration for any -minimal expansion of a -adic field
. 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 -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 -minimal structures. A direct corollary is that
Denef's results on the rationality of Poincar\'e series hold in any -minimal
expansion of a -adic field .Comment: 22 page
Integrability of oscillatory functions on local fields: transfer principles
For oscillatory functions on local fields coming from motivic exponential
functions, we show that integrability over implies integrability over
for large , and vice versa. More generally, the integrability
only depends on the isomorphism class of the residue field of the local field,
once the characteristic of the residue field is large enough. This principle
yields general local integrability results for Harish-Chandra characters in
positive characteristic as we show in other work. Transfer principles for
related conditions such as boundedness and local integrability are also
obtained. The proofs rely on a thorough study of loci of integrability, to
which we give a geometric meaning by relating them to zero loci of functions of
a specific kind.Comment: 44 page
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
Representing Scott sets in algebraic settings
We prove that for every Scott set there are -saturated real closed
fields and models of Presburger arithmetic
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