Coefficients and higher order derivatives of cyclotomic polynomials: old and new

TL;DR

This paper surveys formulas for cyclotomic polynomial coefficients, introduces a unified approach using Bell polynomials, and applies these results to distinguish non-cyclotomic polynomials, confirming a conjecture about numerical semigroups.

Contribution

It provides a comprehensive survey of coefficient formulas, introduces a new unified method using Bell polynomials, and proves the existence of certain non-cyclotomic semigroups, confirming a conjecture.

Findings

Unified approach to derive coefficient formulas

Shorter proofs for known formulas

Existence of non-cyclotomic semigroups with specific properties

Abstract

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers $n$. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach to derive some of them, leading also to shorter proofs as a by-product. In particular, we show that some of the formulas have a more elegant reinterpretation in terms of Bell polynomials. This approach amounts to computing the logarithmic derivatives of $\Phi_n$ at certain points. Furthermore, we show that the logarithmic derivatives at $\pm 1$ of any Kronecker polynomial (a monic product of cyclotomic polynomials and a monomial) satisfy a family of linear equations whose coefficients are Stirling numbers of the second kind. We apply these equations to show that certain polynomials are not Kronecker. In particular, we infer that for every $k\ge 4$ there exists a symmetric numerical semigroup with embedding dimension $k$ and Frobenius number $2k+1$ that is not cyclotomic, thus establishing a conjecture of Alexandru Ciolan, Pedro Garc\'ia-S\'anchez and the second author. In an appendix Pedro Garc\'ia-S\'anchez shows that for every $k\ge 4$ there exists a symmetric non-cyclotomic numerical semigroup having Frobenius number $2k+1.$