On oscillatory integrals associated to phase functions with degenerate singular points

TL;DR

This paper derives asymptotic expansions for oscillatory integrals involving multivariable phase functions with degenerate singular points, including specific types like $A_k$, $E_6$, and $E_8$, using one-variable results.

Contribution

It introduces a method to obtain asymptotic expansions for complex multivariable oscillatory integrals with degenerate singularities, expanding the understanding of such integrals.

Findings

Asymptotic expansions for phase functions with degenerate singular points.

Explicit expansions for $A_k$, $E_6$, and $E_8$-type germs.

Extension of one-variable results to multivariable cases.

Abstract

In this note, by using the result in one variable, we obtain asymptotic expansions of oscillatory integrals for certain multivariable phase functions with {\bf degenerate} singular points. Moreover by using this result, we have asymptotic expansions of oscillatory integrals with phase function of type $A_{k}$, $E_6$, $E_8$-function germs.