Cyclotomic factors of necklace polynomials

2018/11/21 by Hyde, Trevor
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1811.08601

Abstract

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

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