On Symmetric But Not Cyclotomic Numerical Semigroups

Sawhney, Mehtaab; Stoner, David

doi:10.1137/17m1138479

Cited by 7 publications

(7 citation statements)

References 7 publications

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“…In Section 6 we present an example of such an application. We show that 1 − x + x k − x 2k−1 + x 2k is not Kronecker for every k ≥ 4 (this was proven independently using a different method by Sawhney and Stoner [45]). This leads to the following result.…”

Section: Introductionmentioning

confidence: 76%

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

Herrera-Poyatos

Moree

2021

Expositiones Mathematicae

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The n th cyclotomic polynomial Φn(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 Φn at certain points. Furthermore, we show that the logarithmic derivatives at ±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 ≥ 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 García-Sánchez and the second author. In an appendix Pedro García-Sánchez shows that for every k ≥ 4 there exists a symmetric non-cyclotomic numerical semigroup having Frobenius number 2k + 1. 1 arXiv:1805.05207v2 [math.NT] 22 Aug 2018Proof. The result follows on invoking (5.5) and noting that for every positive integer j we haved|n J j (d) − J j (n) = (2 j − 1)(n j − J j (n)).The following corollary is a consequence of the previous identity and its proof is similar to that of Theorem 5.8.

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Section: Introductionmentioning

confidence: 76%

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

Herrera-Poyatos

Moree

2021

Expositiones Mathematicae

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“…Since, by assumption, we have e j = 0 for every gap j of S with 2 ≤ j < g, the series expansion of g−1 j=m(S) (1 − x j ) e j is of the form s∈S a s x s , and so in particular the coefficient of x g is zero. On comparing the coefficients of x g in both sides of (18), we obtain 0 = −e g , a contradiction. Proof.…”

Section: Cyclotomic Exponent Sequences Gaps and Minimal Generatorsmentioning

confidence: 88%

“…Therefore, every cyclotomic numerical semigroup is symmetric. The converse is generally not true; for instance, it can be shown that for every positive integer e ≥ 4, there exists a numerical semigroup of embedding dimension e that is symmetric but not cyclotomic [12,18]. Proposition 2.4 raises the question whether one can classify cyclotomic numerical semigroups for which P S decomposes into a small number of irreducible factors.…”

Section: Preliminariesmentioning

confidence: 99%

Cyclotomic exponent sequences of numerical semigroups

Ciolan¹,

García-Sánchez²,

Herrera-Poyatos³

et al. 2021

Preprint

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We study the cyclotomic exponent sequence of a numerical semigroup S, and we compute its values at the gaps of S, the elements of S with unique representations in terms of minimal generators, and the Betti elements b ∈ S for which the set {a ∈ Betti(S) : a ≤S b} is totally ordered with respect to ≤S (we write a ≤S b whenever a − b ∈ S, with a, b ∈ S). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, García-Sánchez and Moree (2016) stating that S is cyclotomic if and only if it is a complete intersection.

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“…In fact, for any odd integer F with F ≥ 9, there is a numerical semigroup with Frobenius number F that is symmetric and non-cyclotomic. This was proven independently by García-Sánchez (in the appendix of [7]), Herrera-Poyatos and Moree [7] and Sawhney and Stoner [13]. In the latter two papers it is shown (by quite different methods) that, for every k ≥ 5, the polynomial…”

Section: Introductionmentioning

confidence: 87%

“…Consequently, n = pq or n = 2pq. From (13) and the equality P S (x) ≡ Φ n (x) (mod x 2 ), we obtain µ(n) = 1 and so n = pq and hence S = p, q . Theorem 3.2.…”

Section: Cyclotomic Numerical Semigroups Of Polynomial Lengthmentioning

confidence: 93%

Cyclotomic numerical semigroup polynomials with at most two irreducible factors

Borzì,

Herrera-Poyatos,

Moree

2021

Preprint

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A numerical semigroup S is cyclotomic if its semigroup polynomial P S is a product of cyclotomic polynomials. The number of irreducible factors of P S (with multiplicity) is the polynomial length ℓ(S) of S. We show that a cyclotomic numerical semigroup is complete intersection if ℓ(S) ≤ 2. This establishes a particular case of a conjecture of Ciolan, García-Sánchez and Moree (2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between ℓ(S) and the embedding dimension of S.

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On Symmetric But Not Cyclotomic Numerical Semigroups

Cited by 7 publications

References 7 publications

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

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

Cyclotomic exponent sequences of numerical semigroups

Cyclotomic numerical semigroup polynomials with at most two irreducible factors

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