On faces of the Kunz cone and the numerical semigroups within them

2023/09/14 by Levi Borevitz, Borevitz, Levi, Tara Gomes +15
Decision Sciences · Mathematics · #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #Scheduling and Timetabling Solutions

paper · pdf · doi:10.48550/arxiv.2309.07793

openalex publication_date 2023/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A numerical semigroup is a cofinite subset of the non-negative integers that is closed under addition and contains 0. Each numerical semigroup S with fixed smallest positive element m corresponds to an integer point in a rational polyhedral cone \mathcal C_m, called the Kunz cone. Moreover, numerical semigroups corresponding to points in the same face F ⊆ \mathcal C_m are known to share many properties, such as the number of minimal generators. In this work, we classify which faces of \mathcal C_m contain points corresponding to numerical semigroups. Additionally, we obtain sharp bounds on the number of minimal generators of S in terms of the dimension of the face of \mathcal C_m containing the point corresponding to S.

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