Trimming the numerical semigroups tree to probe Wilf's conjecture to higher genus

TL;DR

This paper introduces a method to efficiently explore the numerical semigroups tree by using properties called cutting semigroups, which helps in verifying Wilf's conjecture for larger genus values.

Contribution

It proposes the concept of cutting semigroups and a trimmed tree approach, enabling more efficient computational verification of Wilf's conjecture for larger genus.

Findings

No counter-examples to Wilf's conjecture found up to genus 60

Introduction of the cutting semigroup concept for pruning the search tree

Application to find all semigroups with negative Eliahou number up to higher genus

Abstract

This paper aims to contribute to validate, for numerical semigroups of reasonably large genus, the so-called Conjecture of Wilf. There is no counter-example for the conjecture among the over 3*10^{10} numerical semigroups of genus up to 60, as it has been computationally verified by Fromentin and Hivert. The computations use the idea of parsing a semigroups tree, making tests in each node. As a mean to combine parsing of the semigroups tree with known theoretical results, we introduce the concept of cutting semigroup. Assume that there exists a property on numerical semigroups that implies that a semigroup satisfying it necessarily satisfies Wilf's conjecture. Assume also that, besides, this property is hereditary, that is, if a semigroup satisfies it, then all its descendants in the semigroups tree also have the same property. Such properties exist. Interesting ones can be easily deduced from some deep results of Eliahou. A semigroup satisfying such a property is called a cutting semigroup (for that property). When looking for counter-examples to Wilf's conjecture, if a cutting semigroup is found, then it can be cut off, since no counter example lies among its descendants. One can therefore consider the trimmed tree obtained by cutting off all the cutting semigroups. Depending on the properties, these ideas allow in some cases to avoid parsing a very significant part of the tree and can be applied to other related problems. For instance, they can be used to find all numerical semigroups with negative Eliahou number up to a genus bigger than 60, which is the current record. Computations by Fromentin had shown that there are exactly 5 numerical semigroups in these circumstances.