Matrices over commutative rings as sums of $k$-th powers

Abstract:

In this paper, for $n,k \ge 2,$ and $R$ a commutative and associative ring with unity, we give necessary and sufficient trace conditions for an $n \times n$ matrix over $R$ to be a sum of $k$-th powers of matrices over $R$. We also prove that for $n\ge m \ge 1$, if every $m \times m$ matrix over $R$ is a sum of $k$-th powers of matrices over $R$, then so is every $n \times n$ matrix. As concrete examples, we prove a discriminant criterion for every $n \times n$ matrix over an order $R$ in an algebraic number field to be a sum of cubes and fourth powers of matrices over $R$. We also show that if $q$ is a prime and $n \geq 2$, then every $n \times n$ matrix over the ring ${\mathcal O}$ of integers of a quadratic number field $K$ is a sum of $q$-th powers of matrices over ${\mathcal O}$ if and only if $q$ is coprime to the discriminant of $K$.