论文标题
矩阵的痕量和判别标准是矩阵的第六和第八大幂的总和
Trace and discriminant criteria for a matrix to be a sum of sixth and eighth powers of matrices
论文作者
论文摘要
在本文中，我们将考虑矩阵的警告问题。该问题的一个版本涉及在$ R上的$ k $ r $上写下$ n \ times n $矩阵。$ k $ - 矩阵的总和。结果是针对矩阵的跟踪的情况。 对于$ n <k，$由Katre显示，Garge表明，与特殊情况$ n = 2 $和$ k \ geq 3一起工作是足够的。上面没有发生复合的，非Prime-Power $ K $的情况。在本文中，我们将发现矩阵的痕量标准是第六级（复合非选动功率）和矩阵的第八级，而不是统一的通勤环$ r $。 Katre和Khule早些时候在代数数字字段$ \ Mathcal {o}的特殊情况下获得了一个优雅的判别标准。$我们将在这里为$ \ nathcal {o} $ ighth和eighth powers $ ynation $ y的每个矩阵提供类似的歧视标准。
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 $\mathcal{O}.$
