Cyclotomic factors of necklace polynomials

Abstract

We observe that the necklace polynomials $M_d(x) = \frac{1}{d}\sum_{e\mid d}\mu(e)x^{d/e}$ are highly reducible over $\mathbb{Q}$ with many cyclotomic factors. Furthermore, the sequence $\Phi_d(x) - 1$ of shifted cyclotomic polynomials exhibits a qualitatively similar phenomenon, and it is often the case that $M_d(x)$ and $\Phi_d(x) - 1$ have many common cyclotomic factors. We explain these cyclotomic factors of $M_d(x)$ and $\Phi_d(x) - 1$ in terms of what we call the \emph{$d$th necklace operator}. Finally, we show how these cyclotomic factors correspond to certain hyperplane arrangements in finite abelian groups.