Wilf's conjecture and Macaulay's theorem
Combinatorics
2021-08-19 v1
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.
Cite
@article{arxiv.1703.01761,
title = {Wilf's conjecture and Macaulay's theorem},
author = {S Eliahou},
journal= {arXiv preprint arXiv:1703.01761},
year = {2021}
}