Wilf's conjecture for numerical semigroups

Abstract

Let $S\subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m$, embedding dimension $\nu$ and conductor $c=f+1=qm-\rho$ for some $q,\rho\in\mathbb{N}$ with $\rho<m$. Let Ap$(S,m) = \{w\_0<w_1 < \ldots < w_{m-1}\}$ 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 $1<\alpha<m-1$, 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 $1<\alpha<m-1$.