Graded algebras with cyclotomic Hilbert series

Borzì, Alessio; D’Alì, Alessio

doi:10.1016/j.jpaa.2021.106764

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…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. This and similar questions are addressed in [5], where the authors show, for example, that P S = Φ n if and only if n = pq and S = p, q for distinct prime numbers p and q.…”

Section: Preliminariesmentioning

confidence: 80%

“…Section 2.1 is exceptional in that it is not needed for the rest of the paper. Its purpose is to show that the original definition of a cyclotomic numerical semigroup S, given in [7] through saying that P S admits a factorization into cyclotomic polynomials as in (5), is equivalent with the definition used here.…”

Section: Preliminariesmentioning

confidence: 99%

“…By (4) it then follows that P S (x) can be written as in (5), where a priori some of the integers h d may be negative. As the complex zeros of the cyclotomic polynomials are all different, this would lead to P S having a pole, contradicting the fact that P S is a polynomial.…”

Section: Preliminariesmentioning

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: Preliminariesmentioning

confidence: 80%

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Cyclotomic exponent sequences of numerical semigroups

Ciolan¹,

García-Sánchez²,

Herrera-Poyatos³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

Cyclotomic numerical semigroup polynomials with at most two irreducible factors

2021

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A numerical semigroup S is cyclotomic if its semigroup polynomial $$\mathrm {P}_S$$ P S is a product of cyclotomic polynomials. The number of irreducible factors of $$\mathrm {P}_S$$ P S (with multiplicity) is the polynomial length $$\ell (S)$$ ℓ ( S ) of S. We show that a cyclotomic numerical semigroup is complete intersection if $$\ell (S)\le 2$$ ℓ ( S ) ≤ 2 . This establishes a particular case of a conjecture of Ciolan et al. (SIAM J Discrete Math 30(2):650–668, 2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between $$\ell (S)$$ ℓ ( S ) and the embedding dimension of S.

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The h -vectors of the edge rings of a special family of graphs

Higashitani

Shibu

2023

Communications in Algebra

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Graded algebras with cyclotomic Hilbert series

Cited by 3 publications

References 11 publications

Cyclotomic exponent sequences of numerical semigroups

Cyclotomic exponent sequences of numerical semigroups

Cyclotomic numerical semigroup polynomials with at most two irreducible factors

The h -vectors of the edge rings of a special family of graphs

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