PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Cyclotomic factors of necklace polynomials

Volume 204 / 2022

Trevor Hyde Acta Arithmetica 204 (2022), 287-316 MSC: Primary 11C08; Secondary 11T22, 11T24, 52C35, 68R15. DOI: 10.4064/aa210304-11-5 Published online: 25 August 2022


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 $\Phi _m(x)$ divides both $M_d(x)$ and $\Phi _d(x) - 1$. We explain these cyclotomic factors of $M_d(x)$ and $\Phi _d(x) - 1$ in terms of what we call the $d$th necklace operator. Finally, we show how these cyclotomic factors correspond to certain hyperplane arrangements in finite abelian groups.


  • Trevor HydeDepartment of Mathematics
    University of Chicago
    Chicago, IL 60637, USA

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image