## 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

### MSC:

 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

### Keywords:

definability; Poincaré series; cell decomposition

### Citations:

Zbl 0561.00004; Zbl 0537.12011