Trace and discriminant criteria for a matrix to be a sum of sixth and eighth powers of matrices

TL;DR

This paper investigates conditions under which matrices over a commutative ring can be expressed as sums of sixth and eighth powers, extending previous trace and discriminant criteria to these specific powers.

Contribution

It establishes new trace and discriminant criteria for matrices to be sums of sixth and eighth powers, including cases with composite, non-prime-power exponents.

Findings

Derived trace criteria for sixth and eighth power sums.

Established discriminant criteria for matrices over algebraic number rings.

Extended previous results to new composite exponents.

Abstract

In this paper, we shall be considering the Waring's problem for matrices. One version of the problem involves writing an $n \times n$ matrix over a commutative ring $R$ with unity as a sum of $k$-th powers of matrices over $R.$ This study is motivated by the interesting results of Carlitz, Newman, Vaserstein, Griffin, Krusemeyer, Richman etc. obtained earlier in this direction. The results are for the case $n \geq k \geq 2$ in terms of the trace of the matrix. For $n < k,$ it was shown by Katre, Garge that it is enough to work with the special case $n = 2$ and $k \geq 3.$ The cases $3 \leq k \leq 5$ and $k = 7$ were settled in earlier results. There was no case of a composite, non-prime-power $k$ occuring above. In this paper, we will find the trace criteria for a matrix to be a sum of sixth (a composite non-prime power) and eighth powers of matrices over a commutative ring $R$ with unity. An elegant discriminant criterion was obtained by Katre and Khule earlier in the special case of an order in an algebraic number field $\mathcal{O}.$ We will derive here similar discriminant criteria for every matrix over $\mathcal{O}$ to be a sum of sixth and eighth powers of matrices over $\mathcal{O}.$