The number of solutions of generalized Markoff-Hurwitz-type equations over finite fields

Abstract

Let Nq denote the number of solutions of the generalized Markoff-Hurwitz-type equations (a1x1m1+a2x2m2+…+anxnmn)k=cx1k1 x2k2…xtkt over the finite field $\mathbb{F}$ q, where n ≥ 2, mi, k, kj and t ≥ n are positive integers, ai, c∈ $\mathbb{F}$ q*, for 1 ≤ i ≤ n and 1 ≤ j ≤ t.Recently, some researches considered the above equation with k=k1=…=kt=1 and obtained some generalizations of Carlitz's results.In this paper, we determine Nq explicitly in some other cases.This extends the previous conclusions.