On diagonal equations over finite fields

Let F-q be a finite field with q = p(n) elements. In this paper, we study the number of solutions of equations of the form a(1)x(1)(d1) + ... + a(s)x(s)(ds) = b with x(i) is an element of F-pti , where a(i), b is an element of F-q and t(i)vertical bar n for all i = 1, ... , s. In our main results, we employ results on quadratic forms to give an explicit formula for the number of solutions of diagonal equations with restricted solution sets satisfying certain natural restrictions on the exponents. As a consequence, we present conditions for the existence of solutions. In the second part of the paper, we focus on the case t(1) = ... = t(s) = n. A classic well-known result from Weil yields a bound for such number of solutions. In the case d(1) = ... = d(s), we present necessary and sufficient conditions for the number of solutions of a diagonal equation being maximal and minimal with respect to Weil's bound. In particular, we completely characterize maximal and minimal Fermat type curves. We also discuss further questions concerning equations and present some open problems. (C) 2021 Elsevier Inc. All rights reserved.