Rectilinearization of semi-algebraic p-adic sets and Denef's rationality of Poincare series

摘要

In [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105-114], it is shown that a p-adic semi-algebraic set can be partitioned in such a way that each part is semi-algebraically isomorphic to a Cartesian product Pi(l)(i=1) R-(k) where the sets R-(k) are very basic subsets of Q(p). It is suggested in [R. Cluckers, Classification of semi-algebraic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001) 105-114] that this result can be adapted to become useful to p-adic integration theory, by controlling the Jacobians of the occurring isomorphisms. In this paper we show that the isomorphisms can be chosen in Such a way that the valuations of their Jacobians equal the valuations of products of coordinate functions, hence obtaining a kind of explicit p-adic resolution of singularities for semi-algebraic p-adic functions. We do this by restricting the used isomorphisms to a few specific types of functions, and by controlling the order in which they appear. This leads to an alternative proof of the rationality of the Poincare series associated to the p-adic points on a variety, as proven by Denef in [J. Denef, The rationality of the Poincare series associated to the p-adic points on a variety, Invent. Math. 77 (1984) 1-23]. (C) 2008 Elsevier Inc. All rights reserved.