On the number of rational points on artin-schreier hypersurfaces

Abstract

Let denote the finite field with elements. In this work, we study the number of -rational points of the affine hypersurfaces given by . We prove that if has a nonzero trace over , then Weil's bound cannot be achieved and that a sharp bound can be explicitly provided. In the case , we give necessary and sufficient conditions for Weil's bound to be attained. Furthermore, we present several new cases in which formulas for the number of -rational points can be obtained.