The wave front set of the Fourier transform of algebraic measures

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 [Ber1]. The non-Archimedean case was proved in [HK] and [CL1], [CL2]. Our method is different from those described in [Ber1], [HK], [CL1], [CL2]. It is based on Hironaka’s desingularization theorem, unlike [Ber1] which is based on the theory of D-modules and [HK], [CL1], [CL2] which is based on model theory. Our method also gives bounds on the open dense set where the Fourier transform is smooth and, moreover, on the wave front set of the Fourier transform. These bounds are explicit in terms of resolution of singularities and field-independent. We also prove the same results on the Fourier transform of more general measures of algebraic origins.