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.

23 pages. Completely rewritten with new, stronger results based on the originally observed phenomenon. The second half of the original manuscript was split off into arXiv:2011.05572