Wilf's conjecture for numerical semigroups

TL;DR

This paper proves Wilf's conjecture for certain classes of numerical semigroups by establishing specific inequalities involving the Apéry set and parameters like multiplicity and embedding dimension.

Contribution

It introduces new sufficient conditions under which Wilf's conjecture holds for numerical semigroups, extending known cases.

Findings

Wilf's conjecture holds if $w_{m-1} \\geq w_1 + w_\alpha$ and $(2+\frac{\alpha-3}{q})\nu \geq m$.

The conjecture is valid when $ (2+\frac{1}{q})\nu \geq m$, $m-\nu \leq 5$, or $m=9$.

Proves the conjecture for cases where $w_{m-1} \geq w_{\alpha-1} + w_{\alpha}$ and $(\frac{\alpha+3}{3})\nu \geq m$.

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 $1<\alpha<m-1$.