Waring Problem for Matrices over Finite Fields

We prove that for all integers $k \geq 1$, $q\ge (k-1)^4+ 6k$, and $m \geq 1$, every matrix in $M_m(\mathbb F_q)$ is a sum of two kth powers: $M_m(\mathbb F_q)=\{A^k+B^k|A,B\in M_m(\mathbb F_q)\}$. 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.