A graph-theoretic approach to Wilf's conjecture

This talk concerns numerical semigroups, i.e. cofinite submonoids $S$ of $\mathbb{N}$. Wilf's conjecture (1987) on numerical semigroups $S$ states that $e \cdot \ell \ge c$, where $e$ is the embedding dimension of $S$ and $\ell$ is the number of elements of $S$ which are smaller than its conductor $c$. We shall present the main ideas of a recently published proof of Wilf's conjecture in the particular case $e \ge m/3$, where $m$ is the smallest nonzero element of $S$. Asymptotically, most numerical semigroups seem to satisfy $e \ge m/3$ as the genus goes to infinity. The proof to be presented consists in attaching a certain graph $G$ to $S$ and analysing its properties.

Date and Venue

Start Date
Venue
Online seminar
End Date

Speaker

Shalom Eliahou

Speaker's Institution

Université du Littoral Côte d'Opale

Files

Area

Algebra, Combinatorics and Number Theory