Igusa's \(p\)-adic local zeta function associated to a polynomial mapping and a polynomial integration measure (Q431149)

In this article, the author gives an explicit formula, in terms of Newton polyhedra, for Igusa's \(p\)-adic zeta function associated to a strongly non-degenerated polynomial mapping and a polynomial integration measure. The known notions are generalized by considering alternative polynomial integration measure on \(\mathbb{Z}_p^n\), i.e. measures of the form \(|g(x)||dx|\), with \(g\) a polynomial in \(\mathbb{Z}_p[x_1,\dots,x_n]\). The obtained formula is based on and is a generalization of the known formulas of Denef and Hoornaert (for the case of a single non-degenerated polynomial), of Howald, Mustaţă and Yuen (for the case of a monomial ideal) and of Veys and Zúñiga-Galindo (for a non-degenerated polynomial mapping).