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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

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