Almost Maximal Numerical Semigroups formed by Concatenation of Arithmetic Sequences

Mehta, Ranjana K.; Saha, Joydip; Sengupta, Indranath

doi:10.48550/arxiv.1805.08972

Cited by 2 publications

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“…We calculate the Apéry set in order to prove that the cardinality of minimal presentation is a bounded function of the embedding dimension e and in turn give an affirmative answer to the question on the boundedness of the number of minimal relations for a symmetric numerical semigroup. Numerical semigroups with the property "multiplicity= embedding di-mension+1" has been studied before in [12], where it was proved that the minimal number of generators for the defining ideal of this class of numerical semigroups is a bounded function of e. In [8] we have examined numerical semigroups formed by concatenation together with the condition "multiplicity= embedding dimension+1"; we call this the almost maximal concatenation. We have explicitly calculated the Apéry set, the Frobenius number.…”

Section: Concatenation Of Arithmetic Sequencesmentioning

confidence: 99%

Numerical Semigroups Generated by Concatenation of Arithmetic Sequences

Mehta¹,

Saha²,

Sengupta³

2018

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Section: Concatenation Of Arithmetic Sequencesmentioning

confidence: 99%

Numerical Semigroups Generated by Concatenation of Arithmetic Sequences

Mehta¹,

Saha²,

Sengupta³

2018

Preprint

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“…What is certainly interesting is that n 1 + n 2 = n 3 + n 4 , for the sequence of integers n = (n 1 , n 2 , n 3 , n 4 ) given by Bresinsky. We have initiated a study of numerical semigroups defined by a sequence of integers formed by concatenation of two arithmetic sequences and we believe that such semigroups with correct conditions would finally give us a good model of numerical semigroups in arbitrary embedding dimension with unbounded Betti numbers; see [10], [11].…”

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Betti numbers of Bresinsky's curves in $\mathbb{A}^{4}$

Mehta,

Saha,

Sengupta

2018

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Bresinsky defined a class of monomial curves in A 4 with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behaviour of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for the defining ideal of this class of curves. BRESINSKY'S EXAMPLESLet r ≥ 3 and n 1 , . . . , n r be positive integers with gcd(n 1 , . . . , n r ) = 1. Let us assume that the numbers n 1 , . . . , n r generate the numerical semigroupminimally, that is if n i = r j=1 z j n j for some non-negative integers z j , then z j = 0 for all j = i and zwhere k is a field. Let p(n 1 , . . . , n r ) = ker(η). Let β i (p(n 1 , . . . , n r )) denote the i-th Betti number of the ideal p(n 1 , . . . , n r ). Therefore, β 1 (p(n 1 , . . . , n r )) denotes the minimal number of generators p(n 1 , . . . , n r ). For a given r ≥ 3, let β i (r) = sup(β i (p(n 1 , . . . , n r ))), where sup is taken over all the sequences of positive integers n 1 , . . . , n r . Herzog [8] proved that β 1 (3) is 3 and it follows easily that β 2 (3) is a finite integer as well. Bresinsky inextensively studied relations among the generators n 1 , . . . , n r of the numerical semigroup defined by these integers. It was proved in [2] and [3] respectively that, for r = 4 and for certain cases in r = 5, the symmetry condition on the semigroup generated by n 1 , . . . , n r imposes an upper bound on the first Betti number β 1 (p(n 1 , . . . , n r )). This remains an open question in general whether symmetry condition on the numerical semigroup generated by n 1 , . . . , n r imposes an upper bound on β 1 (p(n 1 , . . . , n r )). Bresinsky [1] 2010 Mathematics Subject Classification. Primary 13C40, 13P10.

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Almost Maximal Numerical Semigroups formed by Concatenation of Arithmetic Sequences

Abstract: Given integer e ≥ 4, we have constructed a class of symmetric numerical semigroups of embedding dimension e and proved that the cardinality of a minimal presentation of the semigroup is a bounded function of the embedding dimension e. This generalizes the examples given by J.C. Rosales.

Cited by 2 publications

References 7 publications

Numerical Semigroups Generated by Concatenation of Arithmetic Sequences

Numerical Semigroups Generated by Concatenation of Arithmetic Sequences

Betti numbers of Bresinsky's curves in $\mathbb{A}^{4}$

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