An effective van der Corput-type method in higher dimensions

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.

💡 Research Summary

The paper addresses the problem of obtaining optimal decay estimates for high‑dimensional oscillatory integrals of the form
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