On the number of rational points on Artin-Schreier hypersurfaces

Abstract

Let Fqk denote the finite field with qk elements. In this work, we study the number of Fqk-rational points of the affine hypersurfaces given by yq−y=a1x1d1+…+asxsds+b. We prove that if b∈Fqk has a nonzero trace over Fq, then Weil's bound cannot be achieved and that a sharp bound can be explicitly provided. In the case Trqk/q(b)=0, 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 Fqk-rational points can be obtained.