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arXiv · 2607.11093

Existence of a far-flung Gorenstein numerical semigroup attaining the Herzog--Kumashiro--Stamate bound

Akihiro Sugawara
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Abstract

We address a question posed by Herzog, Kumashiro, and Stamate regarding the upper bound for the multiplicity of any far-flung Gorenstein numerical semigroup. We answer this question in the affirmative by presenting a method for constructing certain far-flung Gorenstein numerical semigroups with maximal reduced type. Furthermore, using this method, we consider a question raised by Herzog, Hibi, and Stamate. More precisely, for any integer $t\ge5$, we construct a certain far-flung Gorenstein numerical semigroup with type $t$, which is a counterexample to the question.

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Akihiro Sugawara. 2026-07-13. Existence of a far-flung Gorenstein numerical semigroup attaining the Herzog--Kumashiro--Stamate bound. https://arxiv.org/abs/2607.11093

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