Annales scientifiques de l ecole normale superieure vol:39 issue:4 pages:535-568
The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over F-q((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et integration motivique I, Comptes rendus de l'Academie des Sciences 339 (2004) 411-416]. (C) 2006 Elsevier Masson SAS.