Bounds for invariants of numerical semigroups and Wilf's Conjecture

TL;DR

This paper establishes new bounds for invariants of numerical semigroups, including the Frobenius number, and proves Wilf's conjecture in specific cases, advancing understanding of the structure of these semigroups.

Contribution

It provides bounds for the Frobenius number and type of numerical semigroups, and proves Wilf's conjecture for almost-symmetric semigroups.

Findings

Proved that F+1 ≤ q e n for numerical semigroups.

Established that F+1 ≤ e n^2, providing tighter bounds.

Simplified proof of Wilf's conjecture for almost-symmetric semigroups.

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 $q= \left \lceil \frac{F+1}{g_1} \right \rceil$, 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.