Numerical semigroups with large embedding dimension satisfy Wilf's conjecture

TL;DR

This paper proves Wilf's conjecture for certain numerical semigroups with large embedding dimension, specifically when 2 times the embedding dimension exceeds the multiplicity, and also for cases with small multiplicity or generated by generalized arithmetic sequences.

Contribution

It extends the class of numerical semigroups for which Wilf's conjecture is verified, including those with large embedding dimension and specific generating sets.

Findings

Wilf's conjecture holds when 2ν ≥ m.

The conjecture is valid for semigroups with m ≤ 8.

Semigroups generated by generalized arithmetic sequences satisfy the conjecture.

Abstract

We give an affirmative answer to Wilf's conjecture for numerical semigroups satisfying 2 \nu \geq m, where \nu and m are respectively the embedding dimension and the multiplicity of a semigroup. The conjecture is also proved when m \leq 8 and when the semigroup is generated by a generalized arithmetic sequence.