Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields

Let $N$ denote the number of solutions to the generalized Markoff-Hurwitz-type equation $$(a_1{X}_1^m+\cdots +a_n{X}_n^m+a{)}^k=b{X}_1\cdots X_n$$ over the finite field ${\mathbb{F}}_q$, where $m,k$ are positive integers, and $a,b,a_i\in {\mathbb{F}}_q^{\ast }$ for $i=1,\dots ,n$, with $k,m\ge 2$ and $n\ge 3$. Using techniques from algebraic geometry, we provide an estimate for $N$ and establish conditions under which the equation admits solutions where all $X_i$ are nonzero.