Abstract
Let Fq be the finite field of order q. Let Nq denote the number of solutions to the generalized Markoff-Hurwitz-type equationx1m1+x2m2+⋯+xnmn=bx1t1x2t2⋯xntn with mi,ti∈Z>0 and b∈Fq⁎. Carlitz proposed the problem of finding an explicit formula for Nq for the special form. Let m=m1⋯mn. Cao proved that Nq=qn−1+(−1)n−1 if gcd(∑i=1ntim/mi−m,q−1)=1. In this paper, we obtain an explicit formula for Nq under certain case when gcd(∑i=1ntim/mi−m,q−1)>1. In particular, for the case of mi=ti=2(i=1,…,n), if either gcd(n−1,q−1)=1 or 2n≡4(modq−1), the formula for Nq can be easily deduced. This generalizes Cao's result as well as partially solves Carlitz's problem.
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