An effective van der Corput-type method in higher dimensions

TL;DR

This paper introduces a novel van der Corput-type method for higher-dimensional oscillatory integrals using birational maps between toric varieties and Euclidean space, leading to a new derivation of Varchenko's asymptotic expansion.

Contribution

It presents a new approach leveraging birational maps to extend van der Corput analysis to higher dimensions, providing a more elegant derivation of asymptotic formulas.

Findings

Effective higher-dimensional van der Corput analysis

Elegant derivation of Varchenko's asymptotic expansion

Potential applications to oscillatory integral problems

Abstract

Using the birational map between a smooth toric variety (adapted to the phase function of the oscillatory integral) and $\mathbb{R}^n\textbackslash\{0\}$, we can effectively carry out the van der Corput-type analysis in higher dimensions. This allows us to give an elegant derivation of the leading term in Varchenko's asymptotic expansion \cite{Var76}. We expect that this observation may have further applications to other problems involving oscillatory integrals.