Isolated factorizations and their applications in simplicial affine semigroups

García-Sánchez, Pedro A.; Herrera-Poyatos, Andrés

doi:10.1142/s0219498820500826

Cited by 6 publications

(10 citation statements)

References 30 publications

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“…Proof. We prove that if S verifies (10), then S is a complete intersection numerical semigroup; the other implication also holds, as we have just seen. Recall that P S (x) = (1 − x) H S (x) is a polynomial and that by (1) we have P S (1) = 1.…”

Section: Complete Intersection Numerical Semigroupssupporting

confidence: 57%

“…Let S be a numerical semigroup. Then S is a complete intersection numerical semigroup if and only if H S satisfies (10).…”

Section: Complete Intersection Numerical Semigroupsmentioning

confidence: 99%

“…In what follows, S is a numerical semigroup. We say that S is Betti-sorted if Betti(S) is totally ordered with respect to ≤ S , and that S is Betti-divisible if Betti(S) is totally ordered with respect to the divisibility order in N. These two families of numerical semigroups were introduced in [10], where the authors showed that they are complete intersections (see Section 2.4 for a definition).…”

Section: Introductionmentioning

confidence: 99%

“…The third family we consider is that of numerical semigroups with a unique Betti element, which is obviously a subset of each of the two previous families. This family was studied in [10] and [11].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Complete Intersection Numerical Semigroupssupporting

confidence: 57%

“…Let S be a numerical semigroup. Then S is a complete intersection numerical semigroup if and only if H S satisfies (10).…”

Section: Complete Intersection Numerical Semigroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cyclotomic exponent sequences of numerical semigroups

Ciolan¹,

García-Sánchez²,

Herrera-Poyatos³

et al. 2021

Preprint

Self Cite

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show abstract

“…Cyclotomic numerical semigroups have some interesting properties. Indeed in [11] the concept of cyclotomic numerical semigroup is conjectured to coincide with that of complete intersection, the latter being the topic of many publications (see [22] and the references given therein).…”

Section: 7mentioning

confidence: 99%

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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On robustness and related properties on toric ideals

García-Marco

Tatakis²

2022

J Algebr Comb

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Isolated factorizations and their applications in simplicial affine semigroups

Cited by 6 publications

References 30 publications

Cyclotomic exponent sequences of numerical semigroups

Cyclotomic exponent sequences of numerical semigroups

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

On robustness and related properties on toric ideals

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