A verification of Wilf's conjecture up to genus 100

2023/10/11 by Manuel Delgado, Delgado, Manuel, Shalom Eliahou +3 · 4 citations
Mathematics · #Commutative Algebra and Its Applications #FOS: Mathematics #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2310.07742

openalex publication_date 2023/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

For a numerical semigroup S ⊆ ℕ, let m,e,c,g denote its multiplicity, embedding dimension, conductor and genus, respectively. Wilf's conjecture (1978) states that e(c-g) ≥ c. As of 2023, Wilf's conjecture has been verified by computer up to genus g ≤ 66. In this paper, we extend the verification of Wilf's conjecture up to genus g ≤ 100. This is achieved by combining three main ingredients: (1) a theorem in 2020 settling Wilf's conjecture in the case e ≥ m/3, (2) an efficient trimming of the tree T of numerical groups identifying and cutting out irrelevant subtrees, and (3) the implementation of a fast parallelized algorithm to construct the tree T up to a given genus. We further push the verification of Wilf's conjecture up to genus 120 in the particular case where m divides c. Finally, we unlock three previously unknown values of the number n_g of numerical semigroups of genus g, namely for g=73,74,75.

Cited by

• Algorithms and results on multiparameter counting of numerical semigroups
• Families of Eliahou semigroups linked to Farey intervals
• Numerical semigroup tree: type-representation
• Ordinarization numbers of numerical semigroups

Related

• Trimming the numerical semigroups tree to probe Wilf's conjecture to higher genus
• Exploring the tree of numerical semigroups
• Wilf's conjecture for numerical semigroups with large second generator
• Wilf's conjecture in fixed multiplicity
• Wilf's conjecture for numerical semigroups