On the number of simultaneous solutions of certain diagonal equations over finite fields

Abstract: In this paper we obtain explicit estimates and existence results on the number of $\mathbb{F}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.