Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude

TL;DR

This paper investigates the asymptotic behavior of oscillatory integrals at infinity using Newton polyhedra, focusing on cases where the amplitude vanishes at critical points, and explores related properties of local zeta functions.

Contribution

It introduces a detailed analysis connecting Newton polyhedra with the asymptotics of oscillatory integrals, especially when the amplitude has zeros at critical points.

Findings

Characterization of asymptotic behavior using Newton polyhedra

Analysis of poles of local zeta functions in this context

Insights into the influence of amplitude zeros on integral behavior

Abstract

The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.