Waring Problem For Triangular Matrix Algebra

The Matrix Waring problem is if we can write every matrix as a sum of $k$-th powers. Here, we look at the same problem for triangular matrix algebra $T_n({\mathbb{F}}_q)$ consisting of upper triangular matrices over a finite field. We prove that for all integers $k,n\ge 1$, there exists a constant $\mathcal{C}(k,n)$, such that for all $q>\mathcal{C}(k,n)$, every matrix in $T_n({\mathbb{F}}_q)$ is a sum of three $k$-th powers. Moreover, if $-1$ is $k$-th power in ${\mathbb{F}}_q$, then for all $q>\mathcal{C}(k,n)$, every matrix in $T_n({\mathbb{F}}_q)$ is a sum of two $k$-th powers. We make use of Lang-Weil estimates about the number of solutions of equations over finite fields to achieve the desired results.