Waring Problem for matrices over finite local rings

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.