Journal of the European Mathematical Society

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Volume 20, Issue 9, 2018, pp. 2105–2129
DOI: 10.4171/JEMS/807

Published online: 2018-06-06

Wilf’s conjecture and Macaulay’s theorem

Shalom Eliahou[1]

(1) Université du Littoral Côte d’Opale, Calais, France

Let $S \subseteq {\mathbb N}$ be a numerical semigroup with multiplicity $m=\mathrm {min}(S\setminus \{0\})$, conductor $c=\mathrm {max}({\mathbb N} \setminus S)+1$ 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.

Keywords: Numerical semigroup, Wilf conjecture, Apéry element, graded algebra, Hilbert function, binomial representation, sumset

Eliahou Shalom: Wilf’s conjecture and Macaulay’s theorem. J. Eur. Math. Soc. 20 (2018), 2105-2129. doi: 10.4171/JEMS/807