# Item

ITEM ACTIONSEXPORT

Released

Journal Article

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

##### External Resource

https://doi.org/10.1016/j.exmath.2019.07.003

(Publisher version)

https://doi.org/10.48550/arXiv.1805.05207

(Preprint)

##### Fulltext (restricted access)

There are currently no full texts shared for your IP range.

##### Fulltext (public)

There are no public fulltexts stored in PuRe

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Herrera-Poyatos, A., & Moree, P. (2021). Coefficients and higher order derivatives
of cyclotomic polynomials: old and new.* Expositiones Mathematicae,* *39*(3),
309-343. doi:10.1016/j.exmath.2019.07.003.

Cite as: https://hdl.handle.net/21.11116/0000-0006-9DAF-5

##### 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 $2k+1.$

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 $2k+1.$