Numerical Semigroups Generated by Concatenation of Arithmetic Sequences

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

doi:10.48550/arxiv.1802.02564

Cited by 3 publications

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“…, n+(e−3)d, for suitable positive integers m, n, d. It is clearly a concatenation of two arithmetic sequences with the same common difference d. Moreover, with suitable condition on these integers we can make sure that this concatenation can not be completed to a complete arithmetic sequence with first term m and common difference d. We feel that semigroups generated by such sequences would give us a model for creating examples of numerical semigroups whose cardinality of a minimal presentation and also perhaps the higher Betti numbers are not bounded functions of the embedding dimension e. This would then generalize examples defined by Bresinsky [3], [5], in arbitrary embedding dimension. In light of this study and [4], it seems that one requires a non-linear dependence between the integers m and e in order to obtain the right generalization of Bresinsky's examples.…”

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Almost Maximal Numerical Semigroups formed by Concatenation of Arithmetic Sequences

Mehta¹,

Saha²,

Sengupta³

2018

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confidence: 94%

Almost Maximal Numerical Semigroups formed by Concatenation of Arithmetic Sequences

Mehta¹,

Saha²,

Sengupta³

2018

Preprint

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“…However, some special types of numerical semigroups have received a large amount of attention, and the invariants of these numerical semigroups are often well-understood. This includes numerical semigroups with small embedding dimensions [2,8,16], numerical semigroups generated by arithmetic progressions [6,10], and d-symmetric numerical semigroups [15]. For example, the following very fundamental result is due to Sylvester.…”

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confidence: 99%

On the genus of a quotient of a numerical semigroup

et al. 2019

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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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Numerical Semigroups Generated by Concatenation of Arithmetic Sequences

Abstract: We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviours.

Cited by 3 publications

References 10 publications

Almost Maximal Numerical Semigroups formed by Concatenation of Arithmetic Sequences

Almost Maximal Numerical Semigroups formed by Concatenation of Arithmetic Sequences

On the genus of a quotient of a numerical semigroup

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

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