On the Waring Problem for Matrices over Finite Fields

TL;DR

This paper establishes conditions under which matrices over finite fields can be expressed as sums of a small number of k-th powers, extending the Waring problem to matrix algebra over finite fields.

Contribution

It proves that matrices over finite fields can be represented as sums of three or two k-th powers under specific size and field conditions, advancing the understanding of the Waring problem in matrix settings.

Findings

Matrices over finite fields can be expressed as sums of three k-th powers when q^n > (k-1)^4.

For n ≥ 7 and k < q, matrices can be written as sums of two k-th powers.

The results extend classical Waring problem results to matrix algebra over finite fields.

Abstract

We prove that if $k$ is a positive integer then for every finite field $\mathbb{F}$ of cardinality $q\neq 2$ and for every positive integer $n$ such that $q^n>(k-1)^4$, every $n\times n$ matrix over $\mathbb{F}$ can be expressed as a sum of three $k$-th powers. Moreover, if $n\geq 7$ and $k<q$, every $n\times n$ matrix over $\mathbb{F}$ can be written as a sum of two $k$-th powers.