Rectilinearization of semi-algebraic p-adic sets and Denef’s rationality of Poincaré series
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چکیده
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 ∏l i=1 R(k) where the sets R(k) are very basic subsets of Qp . It is suggested in [R. Cluckers, Classification of semi-algebraic sets up to semialgebraic 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 Poincaré series associated to the p-adic points on a variety, as proven by Denef in [J. Denef, The rationality of the Poincaré series associated to the p-adic points on a variety, Invent. Math. 77 (1984) 1–23]. © 2008 Elsevier Inc. All rights reserved.
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تاریخ انتشار 2007