A verification of Wilf's conjecture up to genus 100

Abstract

For a numerical semigroup $S \subseteq \mathbb{N}$, let $m,e,c,g$ denote its multiplicity, embedding dimension, conductor and genus, respectively. Wilf's conjecture (1978) states that $e(c-g) \ge c$. As of 2023, Wilf's conjecture has been verified by computer up to genus $g \le 66$. In this paper, we extend the verification of Wilf's conjecture up to genus $g \le 100$. This is achieved by combining three main ingredients: (1) a theorem in 2020 settling Wilf's conjecture in the case $e \ge m/3$, (2) an efficient trimming of the tree $\mathcal{T}$ of numerical groups identifying and cutting out irrelevant subtrees, and (3) the implementation of a fast parallelized algorithm to construct the tree $\mathcal{T}$ 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 $g=73,74,75$.