The number of solutions of certain diagonal equations over finite fields (Q1201729)

Using character sums, an expression for the number \(N\) of solutions \((x_ 1,\dots,x_ s)\in GF(q)^ s\) of the equation \(a_ 1x_ 1^ d+\dots+a_ s x_ s^ d=b\) (\(a_ j\in GF(q)^*\), \(b\in GF(q)\)) is given. The result is used to determine \(N\) if \(q\) is a square and \(d\) is a special divisor of \(q-1\). Examples of projective varieties over \(GF(q)\) attaining Weil-Deligne bound are deduced. For even \(q\) and any divisor \(d\) of \(q-1\), the number \(N\) of solutions of \(x_ 1^ d+\dots+x_ s^ d=0\) is expressed by means of the weight distribution of an irreducible cyclic code.