Waring problem for triangular matrix algebra

Abstract

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 ( F-q) consisting of upper triangular matrices over a finite field. We prove that for all integers k, n >= 1, there exists a constant C ( k, n ), such that for all q > C ( k, n ), every matrix in T-n ( F-q) is a sum of three k -th powers. Moreover, if - 1 is k -th power in F-q , then for all q > C ( k, n ), every matrix in T-n ( 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.