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

This paper investigates the number of zeros of sixth-degree diagonal equations \sum _k=1^nx_k^6=0 over finite fields. By applying Dirichlet characters, the analytic properties of Gauss sums, and the combinatorial properties of Jacobi sums, we derive an explicit expression for the number of their zeros. The work extends the study of zeros of diagonal equations from the cubic and quartic cases to the sixth-degree case, thereby advancing the development of high-degree diagonal equations over finite fields and providing a reference for subsequent research.