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Title: Rectilinearization of semi-algebraic p-adic sets and Denef's rationality of Poincare series
Authors: Cluckers, Raf *
Leenknegt, Eva * # ×
Issue Date: Jul-2008
Publisher: Academic press inc elsevier science
Series Title: Journal of Number Theory vol:128 issue:7 pages:2185-2197
Article number: doi:10.1016/j.jnt.2007.10.013
Abstract: 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.
URI: 
ISSN: 0022-314X
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Algebra Section
Mathematics - miscellaneous
* (joint) first author
× corresponding author
# (joint) last author

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