Matrix Waring Problem

TL;DR

This paper establishes bounds on expressing matrices over finite fields as sums of kth powers, showing that for large enough fields, matrices can be represented as sums of two or three kth powers depending on their size.

Contribution

It proves the existence of a universal constant C_k ensuring all matrices over sufficiently large finite fields are sums of at most three kth powers.

Findings

Matrices in 1x1 and 2x2 over large fields are sums of two kth powers.

Matrices in larger sizes are sums of at most three kth powers.

The constant C_k depends only on k and guarantees the representation for sufficiently large fields.

Abstract

We prove that for all integers $k \geq 1$, 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 $n \geq 3$ every matrix in $M_n(\mathbb{F}_q)$ is a sum of at most three $k$th powers.