The Frobenius problem for repunit numerical semigroups

Rosales, J. C.; Branco, M. B.; Torrão, D.

doi:10.1007/s11139-015-9719-3

Cited by 25 publications

(27 citation statements)

References 13 publications

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“…For our purposes, it will be enough to assume that d is relatively prime to a and c (otherwise, we can divide by an adequate factor to reduce to this case). This covers the sequences considered in [1,2,3,4,5,8,9] of the form ca n − d, and give us enough generality to unify some ideas, have many new examples and results.…”

Section: Introductionmentioning

confidence: 98%

“…This way, we can associate a family of submonoids of N with a given sequence of positive integers. In recent works on numerical semigroups [1,2,3,4,5,8,9], a particular generating sequence (in some cases a a family of generating sequences) (x n ) n≥1 is given, and for the associated numerical semigroups S n they find 1. the minimal generating set and the embedding dimension of S n ; 2. the Apéry set Ap(S n , x n ), the Frobenius number and genus of S n ; 3. the pseudo-Frobenius numbers and the type of S n (see [6] for the definitions and related concepts).…”

Section: Introductionmentioning

confidence: 99%

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The Frobenius problem for numerical semigroups generated by sequences that satisfy a linear recurrence relation

Arias¹,

Borja²

2021

Preprint

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Consider a sequence of positive integers of the form ca n − d, n ≥ 1, where a, c and d are positive integers, a > 1. For each n ≥ 1, let S n be the submonoid of N generated by s j = ca n+j − d, with j ∈ N. We obtain a numerical semigroup (1/e)S n by dividing every element of S n by e = gcd(S n ).We characterize the embedding dimension of S n and describe a method to find the minimal generating set of S n . We also show how to find the maximum element of the Apéry set Ap(S n , s 0 ), characterize the elements of Ap(S n , s 0 ), and use these results to compute the Frobenius number of the numerical semigroup (1/e)S n , where e = gcd(S n ).

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

The Frobenius problem for numerical semigroups generated by sequences that satisfy a linear recurrence relation

Arias¹,

Borja²

2021

Preprint

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“…The submonoids of N generated by A are studied in detail in [3] as a generalization of the numerical semigroups introduced by D. Torrão et al in [13] and [14]; in this context, Corollary 2 provides a minimal presentation of the submonoid of N generated by a 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Branco¹,

Isabel²,

Ojeda³

2021

Preprint

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Let $a, b$ and $n > 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n > 3$, the ideal $I$ has a unique minimal system of generators if and only if $a < b-1$.

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“…The submonoids of N generated by A are studied in detail in [5] as a generalization of the numerical semigroups introduced by D. Torrão et al (see [6,7]); in this context, Corollary 2 provides a minimal presentation of the submonoid of N generated by a 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

2021

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The Frobenius problem for repunit numerical semigroups

Cited by 25 publications

References 13 publications

The Frobenius problem for numerical semigroups generated by sequences that satisfy a linear recurrence relation

The Frobenius problem for numerical semigroups generated by sequences that satisfy a linear recurrence relation

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

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