Bounds for invariants of numerical semigroups and Wilf’s conjecture

Abstract

Given coprime positive integers g_1< ldots < g_e, the Frobenius number F=F(g_1,ldots ,g_e) is the largest integer not representable as a linear combination of g_1,ldots ,g_e with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that F+1 le e n. We provide bounds for g_1 and for the type of the numerical semigroup S=langle g_1,ldots ,g_e rangle in function of e and n, and use these bounds to prove that F+1 le q e n, where q= Bigg lceil frac{F+1}{g_1} Bigg rceil , and F+1 le e n^2. Finally, we give an alternative, simpler proof for the Wilf conjecture if the numerical semigroup S=langle g_1,ldots ,g_e rangle is almost-symmetric.