On the Waring Problem for Matrices over Finite Fields

Abstract

We prove that if $k$ is a positive integer, then for every finite field ${\mathbb{F}}_q$ of cardinality $q\ne 2$ and for every positive integer $n$ such that $q^n>(k-1{)}^4$, every $n\times n$ matrix over ${\mathbb{F}}_q$ can be expressed as a sum of two $k$-th powers.

V2: A major improvement over the previous version. We revised some of the computational methods, which allowed us to simplify and correct the proofs, as well as improve the main result