Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: When is the Fourier transform of an elementary function elementary?   Let V be a finite dimensional vector space over a local field. Let us say
that a complex function on V is elementary if it is a product of the additive
character of a rational function Q on V and multiplicative characters of
polynomials on V.
  In this paper we study when the Fourier transform of an elementary function
is elementary. If Q has a nonzero Hessian, a necessary condition for this is
that the Legendre transform Q_* of Q is rational. The basic example is a
nondegenerate quadratic form. We study such functions Q, give examples, and
find all of them such that both Q and Q_* are of the form f(x)/t, where f is a
cubic form in many variables (the simplest case after quadratic forms). It
turns out that this classification is closely related to Zak's classification
of Severi varieties.
  The second half of the paper is devoted to finding and classifying elementary
functions with elementary Fourier transforms when Q is a fixed function with
rational Q_*. We consider the simplest case when Q is a monomial, and classify
combinations of multiplicative characters that can arise. The answer (for real
and complex fields) is given in terms of exact covering systems. We also
describe examples related to prehomogeneous vector spaces. Finally, we consider
examples over p-adic fields, and in particular give a local proof of an
integral formula of D.K. that could previously be proved only by a global
method.

6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom