Newton polyhedra and Igusa's local zeta function
The authors consider Igusa's local zeta function \(Z_f(s) = \int_{{\mathbb Z}_p} |f(x)|^s |d x|\) associated with a polynomial \(f(x_1,\dots,x_n)\in{\mathbb Z}_p[x_1,\dots,x_n]\). For \(f(0)=0\) and \(f\) non-degenerate over the finite field \({\mathbb F}_p\) with respect to all the faces of its Newton polyhedron \(\Gamma(f)\), they obtain an explicit formula for \(Z_f(s)\). Among applications we mention the study of the largest real pole different from \(-1\) of Igusa's local zeta function. The paper also contains examples and explicit computations.
scientific article; zbMATH DE number 1749022
scientific article