We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of $(1-\zeta )$, where ζ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of a graph-adjacency matrix, which is a crucial ingredient for the proofs of our main results.

中文翻译：

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单位根埃尔米特矩阵及其特征多项式

我们研究单位根埃尔米特矩阵的光谱条件。我们的主要结果是由以下幂生成的此类矩阵模理想的特征多项式的残差类别数量的推测性上限。$（1-\delta ）$，其中z是单位根。我们还证明了 Harary 和 Schwenk 关于图邻接矩阵幂迹关系的经典结果的推广，这是证明我们主要结果的关键要素。