Title: Fourier Transform of Algebraic Measures

Abstract: We study the Fourier transform of an absolute value of a polynomial on a finite-dimensional vector space. We prove that this transform is smooth on an open dense set.

Our proof is based on Hironaka’s desingularization theorem and on the study of the wave front set of the  Fourier Transform. Our method suits, both the Archimedean and the non-Archimedean case. We also give some bounds on the open dense set where the Fourier transform is smooth and more generally on its  wave front set. These bounds are explicit in terms of resolution of singularities.

We also prove the same result on Fourier transform of other measures of algebraic origins.

Similar (but less general and explicit) results was proven earlier by Bernstein, Cluckers- Loeser and Hrushovski-Kazhdan using diffident  methods.