Toric resolution of singularities in a certain class of $C^{\infty}$ functions and asymptotic analysis of oscillatory integrals

TL;DR

This paper extends Varchenko's analysis of oscillatory integrals from real analytic phases to a specific class of smooth ($C^{ olinebreak}^ olinebreak ext{infty}$) functions using toric resolution of singularities, and studies related zeta functions.

Contribution

It introduces a toric resolution method for a class of $C^{ olinebreak}^ olinebreak ext{infty}$ functions, generalizing previous analytic results to broader smooth functions.

Findings

Generalized asymptotic behavior of oscillatory integrals for $C^{ olinebreak}^ olinebreak ext{infty}$ phases.

Analyzed poles of local zeta functions in the new setting.

Established properties of singularities and their resolutions in the smooth function class.

Abstract

In a seminal work of A. N. Varchenko, the behavior at infinity of oscillatory integrals with real analytic phase is precisely investigated by using the theory of toric varieties based on the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize his results to the case that the phase is contained in a certain class of $C^{\infty}$ functions. The key in our analysis is a toric resolution of singularities in the above class of $C^{\infty}$ functions. 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.