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The Apéry set for numerical semigroups of embedding dimension 4

Neil Steinburg and Brittney Griffith
Vol. 18 (2025), No. 5, 767–777
DOI: 10.2140/involve.2025.18.767
Abstract

The Apéry set is a useful tool to understanding many invariants of a numerical semigroup. We give formulas for the Apéry set, as well as Frobenius number and genus of a numerical semigroup with embedding dimension 4 or less under certain conditions. In particular, we will consider numerical semigroups of the form ⟨m,m + n,m + an,m + abn⟩.

Keywords
numerical semigroup, Frobenius number, Apéry set, Wilf's conjecture
Mathematical Subject Classification
Primary: 05E16, 05E40
Secondary: 13H10
Milestones
Received: 7 July 2023
Revised: 11 April 2024
Accepted: 22 April 2024
Published: 12 November 2025

Communicated by Scott T. Chapman
Authors
Neil Steinburg
Department of Mathematics
South Dakota School of Mines and Technology
Rapid City, SD
United States
Brittney Griffith
South Dakota School of Mines and Technology
Rapid City, SD
United States
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