Local Zeta Functions for Non-degenerate Laurent Polynomials Over p-adic Fields

TL;DR

This paper investigates local zeta functions associated with non-degenerate Laurent polynomials over p-adic fields, deriving asymptotic expansions for p-adic oscillatory integrals based on the poles of twisted Igusa-type zeta functions.

Contribution

It introduces new asymptotic expansion results for p-adic oscillatory integrals linked to Laurent polynomials, controlled by their local zeta function poles.

Findings

Existence of two distinct asymptotic expansions for p-adic oscillatory integrals.

Asymptotic behavior is governed by the poles of twisted local zeta functions.

Results apply to non-degenerate Laurent polynomials over p-adic fields.

Abstract

In this article, we study local zeta functions attached to Laurent polynomials over p-adic fields, which are non-degenerate with respect to their Newton polytopes at infinity. As an application we obtain asymptotic expansions for p-adic oscillatory integrals attached to Laurent polynomials. We show the existence of two different asymptotic expansions for p-adic oscillatory integrals, one when the absolute value of the parameter approaches infinity, the other when the absolute value of the parameter approaches zero. These two asymptotic expansions are controlled by the poles of twisted local zeta functions of Igusa type.