Wilf's conjecture for numerical semigroups

Mariam Dhayni

Published 2016-10-27Version 1

Let $S\subseteq \N$ be a numerical semigroup with multiplicity $m$, embedding dimension $\nu$ and conductor $c=f+1=qm-\rho$ for some $q,\rho\in\N$ with $\rho\textless{}m$. Let Ap$(S,m) = \{w\_0\textless{}w\_1 \textless{} \ldots \textless{}w\_{m-1}\}$ be the Ap\'ery set of $S$. The aim of this paper is to prove Wilf's Conjecture in some special cases. First, we prove that if $w\_{m-1}\geq w\_1+w\_{\alpha}$ and $(2+\frac{\alpha-3}{q})\nu\geq m$ for some $1\textless{}\alpha\textless{}m-1$, then $S$ satisfies Wilf's Conjecture. Then, we prove the conjecture in the following cases: $(2+\frac{1}{q})\nu\geq m$, $m-\nu\leq 5$ and $m=9$. Finally, the conjecture is proved if $w\_{m-1}\geq w\_{\alpha-1}+w\_{\alpha}$ and $(\frac{\alpha+3}{3})\nu\geq m$ for some $1\textless{}\alpha\textless{}m-1$.