399 research outputs found

    Integration and Cell Decomposition in PP-minimal Structures

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    We show that the class of L\mathcal{L}-constructible functions is closed under integration for any PP-minimal expansion of a pp-adic field (K,L)(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 PP-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 PP-minimal structures. A direct corollary is that Denef's results on the rationality of Poincar\'e series hold in any PP-minimal expansion of a pp-adic field (K,L)(K,\mathcal{L}).Comment: 22 page

    Integrability of oscillatory functions on local fields: transfer principles

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    For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over QpnQ_p^n implies integrability over Fp((t))nF_p ((t))^n for large pp, 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

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    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

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    We prove that for every Scott set SS there are SS-saturated real closed fields and models of Presburger arithmetic
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