Wilf's conjecture for numerical semigroups with large second generator

TL;DR

This paper investigates Wilf's conjecture for a specific class of numerical semigroups where the second generator exceeds a certain threshold, demonstrating the conjecture holds under bounded multiplicity conditions.

Contribution

It establishes the validity of Wilf's conjecture for semigroups with large second generators and bounded multiplicity relative to the embedding dimension.

Findings

Wilf's conjecture holds for semigroups with second generator exceeding (c + μ)/3.

Validation of Wilf's conjecture when multiplicity is quadratically bounded by embedding dimension.

Provides conditions under which Wilf's conjecture is proven for a specific class of numerical semigroups.

Abstract

We study Wilf's conjecture for numerical semigroups $S$ such that the second least generator $a_2$ of $S$ satisfies $a_2>\frac{c(S)+\mu(S)}{3}$, where $c(S)$ is the conductor and $\mu(S)$ the multiplicity of $S$. In particular, we show that for these semigroups Wilf's conjecture holds when the multiplicity is bounded by a quadratic function of the embedding dimension.