The Number of Solutions of Certain Equations over Finite Fields

Abstract

Let s be a positive integer, p be an odd prime, [see formula in PDF], and let [see formula in PDF] be a finite field of q elements. Let [see formula in PDF] be the number of solutions of the following equations: [see formula in PDF][see formula in PDF] over the finite field [see formula in PDF], with [see formula in PDF][see formula in PDF][see formula in PDF] and [see formula in PDF][see formula in PDF] are positive integers. In this paper, we find formulas for [see formula in PDF] when there is a positive integer l such that dD|([see formula in PDF]) , where [see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF][see formula in PDF] And we determine [see formula in PDF] explicitly under certain cases. This extends Markoff-Hurwitz-type equations over finite field.