Fourier transform of algebraic measures

TL;DR

This paper discusses the Fourier transform of algebraic measures defined by polynomial phase functions over local fields, highlighting its smoothness properties on certain open subsets and providing a simplified proof of key results.

Contribution

It offers a simplified proof of the smoothness of the Fourier transform of algebraic measures, avoiding complex D-module and model theory techniques.

Findings

Fourier transform is smooth on a Zariski-open conic subset of the dual space.

The proof simplifies existing arguments by using resolution of singularities.

The results apply to functions of the form $ ext{ψ}(P(x))$ over local fields.

Abstract

These are notes of a talk based on the work arXiv:1212.3630 joint with A. Aizenbud. Let V be a finite-dimensional vector space over a local field F of characteristic 0. Let f be a function on V of the form $f(x)= \psi (P(x))$, where P is a polynomial on V and $\psi$ is a nontrivial additive character of F. Then it is clear that the Fourier transform of f is well-defined as a distribution on $V^*$. Due to J.Bernstein, Hrushovski-Kazhdan, and Cluckers-Loeser, it is known that the Fourier transform is smooth on a non-empty Zariski-open conic subset of $V^*$. The goal of these notes is to sketch a proof of this result (and some related ones), which is very simple modulo resolution of singularities (the existing proofs use D-module theory in the Archimedean case and model theory in the non-Archimedian one).