Exponential sums as discrete fourier transform with invariant phase functions

Abstract

We give estimates for exponential sums over finite fields in several variables. We study the case where the phase is either quadratic or more generally invariant under the action of a finite group. The bounds obtained are better than the general ones; they imply some estimates for certain sums in one variable, and for the number of solutions of the trace equation Tr(xd + vx)=0. In an appendix we discuss the link between exponential sums and bent functions.