A verification of Wilf's conjecture up to genus 100

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 ng of numerical semigroups of genus g, namely for g=73,74,75.