Bounds for Exponential Sums over Finite Fields

In this paper, bounds for exponential sums associated to polynomial ƒ defined over finite fields are given. We introduce the q-ary weight of the degrees of the polynomial ƒ and the bound is expressed in particular in terms of these when the usual expression depends on the degree. To do this, we associate to ƒ a transformed polynomial FR, we apply the Deligne bound to this polynomial, and we extend the results established by O. Moreno and P. Kumar for other polynomial families ƒ and for fields of any characteristic. The bound obtained here improves the famous Weil bound in several cases. In the same way, using results of A. Adolphson and S. Sperber involving Newton polyhedra, bounds for diagonal polynomials are also given in terms of q-ary weight of their degrees, when the underlying field is a quadratic extension of q. Finally, we apply the previous result to obtain a bound of the number of zeros of diagonal polynomials.