Geodesic
Wilf’s conjecture and Macaulay’s theorem
Journal of the European Mathematical Society, Tome 20 (2018) no. 9, pp. 2105-2129 Cet article a éte moissonné depuis la source EMS Press

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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 
@article{JEMS_2018_20_9_a1,
     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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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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