Eliahou number, Wilf function and concentration of a numerical semigroup

TL;DR

This paper investigates properties of numerical semigroups, focusing on the Wilf function, Eliahou number, and concentration, providing new bounds, examples, and conditions that support the Wilf conjecture.

Contribution

It introduces new estimates for the Wilf function, characterizes conditions for negative Eliahou numbers, and defines highly dense semigroups that satisfy the Wilf conjecture.

Findings

Minimal positive Wilf function value related to concentration

Conditions for negative Eliahou number involving multiplicity and Wilf function

Highly dense semigroups satisfy the Wilf conjecture

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.