Bounds for invariants of numerical semigroups and Wilf’s conjecture

Abstract

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