Matrices over commutative rings as sum of higher powers

TL;DR

This paper extends existing trace conditions for matrices over commutative rings to determine when they can be expressed as sums of higher powers, specifically for powers 9 through 16.

Contribution

It introduces new trace conditions for matrices to be written as sums of k-th powers for k=9 to 16, expanding previous results for smaller powers.

Findings

Derived trace conditions for k=9 to 16.

Extended the understanding of Waring's problem for matrices.

Provided criteria applicable to matrices over commutative rings.

Abstract

On the Waring's problems for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as a sum of $k$-th powers with $k=2,3,4,5,6,7,8$ in several literatures. In this paper, we provide the similar conditions for matrices written as a sum of $k$-th powers with $k=9,10,11,12,13,14,15,16$.