Waring Problem for Matrices over Finite Fields

Krishna Kishore, Adrian Vasiu, Sailun Zhan

Published 2023-06-11Version 1

Let $k$ be a natural number. The Waring problem for square matrices over finite fields, solved by Kishore and Singh, asked for the existence of a constant $C(k)$ such that for all finite fields $\mathbb F_q$ with $q\ge C(k)$, each matrix $A\in M_n(\mathbb F_q)$ with $n\ge 2$ is a sum $A=B^k+C^k$ of two $k$-th powers. We present a new proof of this result with the explicit bound $C(k)=(k-1)^4+6k$ and we generalize and refine this result in many cases, including those when $B$ and $C$ are required to be (invertible) (split) semisimple or invertible cyclic, when $k$ is assumed to be coprime to $p$ (the improved bound being $k^3-3k^2+3k$), or when $n$ is big enough.