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Quelques résultats recents concernant les fonctions d’Igusa. (Some recent results concerning the Igusa functions). (French) Zbl 0644.12006
Sémin. Théor. Nombres, Univ. Bordeaux I 1986-1987, Exp. No. 25, 10 p. (1987).
Let K be a local field of characteristic 0, let $$F\in K[x_ 1,...,x_ n]$$ and let $$| dx|$$ be a Haar measure on $$K^ n$$. When $$\Phi$$ is a locally constant function with compact support, the Igusa function associated to F is defined by $$Z_{\Phi,K}(s)=\iint_{K^ n}\Phi | F|^ s| dx|$$. - In the archimedean case the author had proven that the poles of $$Z_{\Phi,K}$$ are in the form s-n with s a root of the Bernstein polynomial b associated to F.
Now in the non archimedean case, such a link between the poles of $$Z_{\Phi,K}$$ and the roots of b does not exist. However, when $$n=2$$, the author shows that a pole of $$Z_{\Phi,K}$$ has a real part which is a root of b. He then considers possible generalizations of this theorem when $$n>2$$, and works on technical results about it.
Reviewer: A.Escassut

##### MSC:
 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 12J25 Non-Archimedean valued fields 11S40 Zeta functions and $$L$$-functions