Waring Problem for matrices over finite local rings

Abstract

This paper addresses the matrix Waring problem for matrices over finite principal local rings. Let ${\mathcal{O}}_{\ell }$ be a finite principal local ring of length $\ell$ with the maximal ideal $\mathfrak{m}$ and the residue field ${\mathbb{F}}_q={\mathcal{O}}_{\ell }/\mathfrak{m}$. When $-1$ is a $k$-th power in ${\mathbb{F}}_q$ and the characteristic of ${\mathbb{F}}_q$ does not divide $k$, we show that for sufficiently large $q$, any matrix in $M_n({\mathcal{O}}_{\ell })$ can be expressed as a sum of two $k$-th powers. Furthermore, we establish that these two conditions are strictly necessary for the result to hold in general.