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

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 $\mathcal{O}.$