On the number of zeros of sixth-degree diagonal equation over finite fields

The number of zeros of sixth-degree diagonal equation \displaystyle\sum _k=1^nx_k^6=0 over finite fields is investigated in this paper． Application of Dirichlet characters, analytic properties of Gauss sums, and combinatorial properties of Jacobi sums enabled derivation of an explicit expression for the number of their zeros． This work extends the study of zeros of diagonal equations from the cubic and quartic cases to the sixth-degree case, thereby advances development of high-degree diagonal equations over finite fields and provides a reference for subsequent research．