Geodesic
Wilf’s conjecture and Macaulay’s theorem
Journal of the European Mathematical Society, Tome 20 (2018) no. 9, pp. 2105-2129

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DOI

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.
DOI : 10.4171/jems/807
Classification : 05-XX, 11-XX, 13-XX, 20-XX
Keywords: Numerical semigroup, Wilf conjecture, Apéry element, graded algebra, Hilbert function, binomial representation, sumset 
Shalom Eliahou. Wilf’s conjecture and Macaulay’s theorem. Journal of the European Mathematical Society, Tome 20 (2018) no. 9, pp. 2105-2129. doi: 10.4171/jems/807
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     author = {Shalom Eliahou},
     title = {Wilf{\textquoteright}s conjecture and {Macaulay{\textquoteright}s} theorem},
     journal = {Journal of the European Mathematical Society},
     pages = {2105--2129},
     year = {2018},
     volume = {20},
     number = {9},
     doi = {10.4171/jems/807},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/807/}
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