Eliahou number, Wilf function and concentration of a numerical semigroup
Patricio Almirón, Julio José Moyano-Fernández
TL;DR
This work analyzes the Wilf conjecture for numerical semigroups through the Wilf function $W_Γ(k)$ and the Eliahou number $E(Γ)$. It proves that $W_Γ(e)\ge E(Γ)\ge W_Γ(e_s)$ and derives necessary conditions linking negative $E(Γ)$ to the Wilf function and to concentration, including bounds that force positivity under certain regimes. The authors introduce highly dense numerical semigroups, which always satisfy $W_Γ(e)\ge0$, and show that, under extra hypotheses, they also have $E(Γ)\ge0$, providing new evidence for Wilf’s conjecture. Additionally, the paper yields new explicit examples of semigroups with negative Eliahou numbers and demonstrates how concentration and the Apéry-set framework can be leveraged to study the conjecture more broadly.
Abstract
We give an estimate of the minimal positive value of the Wilf function of a numerical semigroup in terms of its concentration. We describe necessary conditions for a numerical semigroup to have negative Eliahou number in terms of its multiplicity, concentration and Wilf function. Also, we show new examples of numerical semigroups with negative Eliahou number. In addition, we introduce the notion of highly dense numerical semigroup; this yields a new family of numerical semigroups satisfying the Wilf conjecture. Moreover, we use the Wilf function of a numerical semigroup to prove that the Eliahou number of a highly dense numerical semigroup is positive under certain additional hypothesis. In particular, these results provide new evidences in favour of the Wilf conjecture.