Wilf’s conjecture and Macaulay’s theorem

Abstract

Let $S\subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m=\min (S\setminus \{0\})$, conductor $c=\max (\mathbb{N}\setminus S)+1$ and minimally generated by $e$ elements. Let $L$ be the set of elements of $S$ which are smaller than $c$. Wilf conjectured in 1978 that $|L|$ is bounded below by $c/e$. We show here that if $c\le 3m$, then $S$ satisfies Wilf's conjecture. Combined with a recent result of Zhai, this implies that the conjecture is asymptotically true as the genus $g(S)=|\mathbb{N}\setminus S|$ goes to infinity. One main tool in this paper is a classical theorem of Macaulay on the growth of Hilbert functions of standard graded algebras.