The wave front set of the Fourier transform of algebraic measures

TL;DR

This paper proves that the Fourier transform of the absolute value of a polynomial over a local field is smooth on an open dense set, using resolution of singularities, and provides explicit bounds for this set.

Contribution

It introduces a new method based on Hironaka's desingularization to analyze the Fourier transform of algebraic measures, offering uniform results for both Archimedean and non-Archimedean cases.

Findings

Fourier transform is smooth on an open dense set

Explicit bounds on the smoothness set are provided

Method applies to various algebraic measures

Abstract

We study the Fourier transform of the absolute value of a polynomial on a finite-dimensional vector space over a local field of characteristic 0. We prove that this transform is smooth on an open dense set. We prove this result for the Archimedean and the non-Archimedean case in a uniform way. The Archimedean case was proved in [Ber]. The non-Archimedean case was proved in [HK] and [CL]. Our method is different from those described in [Ber,HK,CL]. It is based on Hironaka's desingularization theorem, unlike [Ber] which is based on the theory of D-modules and [HK,CL] which is based on model theory. Our method also gives bounds on the open dense set where the Fourier transform is smooth. These bounds are explicit in terms of resolution of singularities. We also prove the same result on the Fourier transform of other measures of algebraic origins.