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.A numerical semigroup is a subset S ⊆ N closed under addition, containing 0 and of finite complement in N. The elements of N \ S are called the gaps of S. The largest gap is denoted F(S) = max(N \ S) and is called the Frobenius number of S. The integer c(S) = F(S) + 1 is known as the conductor of S. It satisfies c(S) + N ⊆ S and is minimal for that property. The number of gaps g(S) = |N \ S| is known as the genus of S, and the smallest nonzero element m(S) = min(S \ {0}) as the multiplicity of S.Every numerical semigroup S is finitely generated, i.e. is of the form S = a 1 , . . . , a n = Na 1 + · · · + Na n for suitable globally coprime integers a 1 , . . . , a n . The least number n of generators of S is denoted e = e(S) and is called the embedding dimension of S.

On a question of Eliahou and a conjecture of Wilf

“…We refer to the number E(S ) = |P ∩ L||L| − q|D q | + ρ as the Eliahou number of S . Eliahou [5] proved the following result, which relates E(S ) to the conjecture of Wilf. Figure 2 represents all of them.…”

“…This section is dedicated to the introduction of some terminology and to fix some notation. Most of it is borrowed from Eliahou's paper [5]. From the same paper we borrow a convenient partition of the integers.…”

Near-misses in Wilf’s conjecture

“…In particular, we have P ∩ L = P 1 ∪ · · · ∪ P q−1 = P \ P q . The following formula appears in [5].…”

“…So far, it has only been settled for a few families of numerical semigroups, including the five independent cases |P| ≤ 3, |L| ≤ 4, m ≤ 8, g ≤ 60 and q ≤ 3. See [1,2,4,5,6,7,8,9,14] for more details.…”

“…The number W 0 (S). Denote S q = [c, c + m − 1] and D q = D ∩ S q = S q \ P. We may now define the closely related number W 0 (S) introduced in [5]. It involves |P ∩ L| rather than |P| as in W (S), as well as D q and the numbers q, ρ given by (1).…”