Research on a Class of Equations over Finite Fields

Let F_q stand for the finite field of odd characteristic p with q elements ( q = pⁿ,n∈N ) and F_q^* denote the set of all the nonzero elements of F_q. In this paper, by using the augmented degree matrix and the result given by Cao, we obtain a formula for the number of rational points of the following equation over \({F_q}:f({x_1},{x_2},...{x_n}) = {({a_1}{x_1}{x_2} + {a_1}{x_2}x_3^d + {a_{n - 1}}{x_{n - 1}}x_1^d + {a_n}{x_n}x_1^d)^\lambda } - bx_1^{{d_1}}x_n^{{d_n}}...x_n^{{d_n}},with{a_i},b \in F_q^*,n \geqslant 2,\lambda > 0\) being positive integers, and d, d_i being nonnegative integers for 1 in. This technique can be applied to the polynomials of the form h₁^λ=h₂ with λ being positive integer and \({h_1},{h_2} \in {F_q}{x_1},{x_2},...{x_n}]\). It extends the results of the Markoff- Hurwitz-type equations.