Waring Problem for Matrices over Finite Fields

We prove that for all integers $k\ge 1$, $q\ge (k-1{)}^4+6k$, and $m\ge 1$, every matrix in $M_m({\mathbb{F}}_q)$ is a sum of two kth powers: $M_m({\mathbb{F}}_q)=\{A^k+B^k\mathrm{\mid }A,B\in M_m({\mathbb{F}}_q)\}$. We further generalize and refine this result in the cases when both $B$ and $C$ can be chosen to be invertible, cyclic, or split semisimple, when $k$ is coprime to $p$, or when $m$ is sufficiently large. We also give a criterion for the Waring problem in terms of stabilizers.