Matrix Waring Problem

Abstract

We prove that for all integers $k\ge 1$, there exists a constant $C_k$ depending only on $k$, such that for all $q>C_k$, and for $n=1,2$ every matrix in $M_n({\mathbb{F}}_q)$ is a sum of two $k$th powers and for all $n\ge 3$ every matrix in $M_n({\mathbb{F}}_q)$ is a sum of at most three $k$th powers.