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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.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.

For each k ¿ 2, we exhibit inÿnite families of prime k-component links with Jones polynomial equal to that of the k-component unlink.

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In this paper, we formulate and prove linear analogues of results concerning matchings in groups. A matching in a group G is a bijection ϕ between two finite subsets A, B of G with the property, motivated by old questions on symmetric tensors, that aϕ(a) / ∈ A for all a ∈ A. Necessary and sufficient conditions on G, ensuring the existence of matchings under appropriate hypotheses, are known. Here we consider a similar question in a linear setting.Given a skew field extension K ⊂ L, where K commutative and central in L, we introduce analogous notions of matchings between finite-dimensional K -subspaces A, B of L, and obtain existence criteria similar to those in the group setting. Our tools mix additive number theory, combinatorics and algebra.

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Let I ⊂ K[x, y] be a monomial ideal. How small can µ(I 2 ) be in terms of µ(I)? It has been expected that the inequality µ(I 2 ) > µ(I) should hold whenever µ(I) ≥ 2. Here we disprove this expectation and provide a somewhat surprising answer to the above question.Theorem 1.1. For every integer m ≥ 5, there exists a monomial ideal I ⊂ K[x, y] such that µ(I) = m and µ(I 2 ) = 9.Moreover, this result is best possible for m ≥ 6.Theorem 1.2. Let I ⊂ K[x, y] be a monomial ideal. If µ(I) ≥ 6 then µ(I 2 ) ≥ 9.Here are some notation to be used throughout. We denote by M the set of monomials in K[x, y], i.e. M = {x i y j | i, j ∈ N}.As usual, we view M as partially ordered by divisibility. For a monomial ideal J ⊂ K[x, y], we denote by G(J) its unique minimal system of monomial generators. It is well known that G(J) is of cardinality µ(J) and consists of all monomials in J which are minimal under divisibility, i.e. G(J) = M ∩ J \ M ∩ J M * where M * = M \ {1}.