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On the evaluation of certain p-adic integrals. (English) Zbl 0597.12021
Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1983-84, Prog. Math. 59, 25-47 (1985).
[For the entire collection see Zbl 0561.00004.]
One can attach to a polynomial f in n variables with coefficients in \({\mathbb{Z}}_ p\) two Poincaré series P and \(\tilde P.\) According to Igusa \(\tilde P\) is a rational function. In an earlier paper [Invent. Math. 77, 1-23 (1984; Zbl 0537.12011)] the author proves that both P and \(\tilde P\) are rational functions. The method is based upon a representation of the Poincaré series as an integral over \({\mathbb{Q}}^ n_ p\) and a suitable ”cell decomposition” of \({\mathbb{Q}}^ n_ p\). In the present paper this cell decomposition is used to derive two new results. Both give an explicit formula for a certain integral over \({\mathbb{Q}}^ n_ p\) showing the dependence on the parameters.
In the proofs and statements one uses notions ”from logic”: definable function, definable set, \(\tau\)-definable.
Reviewer: M.van der Put

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
13H15 Multiplicity theory and related topics
14G20 Local ground fields in algebraic geometry
03C10 Quantifier elimination, model completeness and related topics
12L12 Model theory of fields