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

TL;DR

This paper derives explicit formulas for Igusa's p-adic local zeta functions associated with polynomial mappings and measures, using Newton polyhedra, extending previous results to more general cases.

Contribution

It provides a generalized explicit formula for Igusa's local zeta functions involving polynomial mappings and measures, expanding on prior work with new non-degeneracy conditions.

Findings

Explicit formulas in terms of Newton polyhedra

Applicable to polynomial mappings and polynomial measures

Generalizes previous results by Denef, Hoornaert, Howald, Veys, and Zuniga-Galindo

Abstract

For p prime, we give an explicit formula for Igusa's local zeta function associated to a polynomial mapping f=(f_1,...,f_t): Q_p^n -> Q_p^t, with f_1,...,f_t in Z_p[x_1,...,x_n], and an integration measure on 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 of Denef - Hoornaert, Howald et al., and Veys - Zuniga-Galindo.