Let Fqnβ denote the finite field with qn elements. In this
paper we determine the number of Fqnβ-rational points of the
affine Artin-Schreier curve given by yqβy=x(xqiβx)βΞ» and of the
Artin-Schreier hypersurface yqβy=βj=1rβajβxjβ(xjqijβββxjβ)βΞ». Moreover in both cases, we show that the
Weil bound is attained only in the case where the trace of Ξ»βFqnβ over Fqβ is zero. We use quadratic forms and permutation
matrices to determine the number of affine rational points of these curves and
hypersurfaces