308 research outputs found
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 [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.Comment: 40 page
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