On the logarithmic terms in the asymptotic expansion of integrals

TL;DR

This paper investigates the asymptotic expansion of integrals over vanishing cycles in singularity theory, identifying conditions for precisely determining the leading logarithmic term in the expansion.

Contribution

It provides a method to exactly compute the leading logarithmic term in the asymptotic expansion for a class of integrals associated with Newton non-degenerate singularities.

Findings

Exact determination of the leading logarithmic term under specific conditions

Characterization of the asymptotic behavior of integrals in singularity theory

Application to integrals in the Milnor fibration context

Abstract

We consider integrals over vanishing cycles in the Milnor fibration of an isolated singularity defined by a Newton non-degenerate function. We single out a condition where the leading logarithmic term of the expansion of the integral into a logarithmic sum can be determined exactly.