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Journal Articles Journal of the European Mathematical Society Year : 2018

Wilf's conjecture and Macaulay's theorem

Abstract

Let S ⊆ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ 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 ≤ 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) = |N \ 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.
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Dates and versions

hal-01482758 , version 1 (03-03-2017)

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S Eliahou. Wilf's conjecture and Macaulay's theorem. Journal of the European Mathematical Society, 2018, 20, pp.2105-2129. ⟨10.4171/JEMS/807⟩. ⟨hal-01482758⟩
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