Newton polyhedra and the poles of Igusa's local zeta function
For a prime \(p\), let \(\mathbb{Q}_p\) denote the field of \(p\)-adic numbers and \(\mathbb{Z}_p\), the ring of \(p\)-adic integers. Let \(\chi\) be a character of \(\mathbb{Z}_p\), the set of units of \(\mathbb{Z}_p\). Let \(f(x)= f(x_1,\dots, x_n)\in \mathbb{Z}_p[x_1,\dots, x_n]\). Igusa's local zeta-function \(Z_f(s)\) is directly related to the number of solution of the congruence \(f(x)\equiv 0\bmod p^m\), \(m= 1,2,3,\dots\)\ . Using resolutions of singularities, \textit{J.-I. Igusa} [Trans. Am. Math. Soc. 245, 419--429 (1978; Zbl 0401.12013)] proved that \(Z_f(s,\chi)\) is a rational function of \(p^{-s}\). \textit{J. Denef} [Invent. Math. 77, 1--23 (1984; Zbl 0537.12011)] obtained an entirely different proof using \(p\)-adic cell decomposition. In this paper, a very explicit formula has been obtained for \(Z_f(s,\chi)\) where \(f\) is non-degenerate over the finite field \(F_p\) with respect to all the faces of its Newton polyhedron. This study was first started by \textit{B. Lichtin} and \textit{D. Meuser} [Compos. Math. 55, 313--332 (1985; Zbl 0606.14022)] for polynomials in two variables. Using the formula for \(Z_f(s,\chi)\) a set of possible poles for \(Z_f(s,\chi)\) together with upper bounds for their orders have also been obtained. In particular, it gives information on the largest real pole of \(Z_f(s)\) and its order. Moreover, the formula implies that \(Z_f(s)\) has always at least one real pole.
scientific article; zbMATH DE number 2031097