On properties of Weil sums over finite fields and finite abelian groups

We develop an approach involving new parameters of polynomials for estimating exponential sums. The reduced Weil bound is proved, which is stronger than the Weil bound (we mean the constant at $q^{1/2}$). The proof is based on a new partition of all polynomials into the equivalence classes such that the Weil sum in each class is constant. For an arbitrary finite abelian group, we describe functions which are analogous to polynomials over a field and consider the Weil sums for these functions.
The research was supported by the Russian Foundation for Basic Research, grants 99–01–00929 and 99–01–00941.