Igusa’s p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure

Abstract

For p prime, we give an explicit formula for Igusa’s local zeta function associated to a polynomial mapping ${{\bf f} = (f_1, \ldots, f_t) : {\bf Q}_p^{n} \to {\bf Q}_p^{t}}$ , with ${f_1, \ldots, f_t \in {\bf Z}_p[x_1, \ldots, x_n]}$ , and an integration measure on ${{\bf Z}_p^{n}}$ of the form ${|g(x)||dx|}$ , with g another polynomial in Z _p [x ₁, . . ., x _n ]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F _p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results in Denef and Hoornaert (J Number Theory 89(1):31–64, 2001), Howald et al. (Proc Am Math Soc 135(11):3425–3433, 2007) and Veys and Zúñiga-Galindo (Trans Am Math Soc 360(4):2205–2227, 2008).