Open problems on relations of numerical semigroups

TL;DR

This paper reviews open problems related to the algebraic and combinatorial properties of numerical semigroups, focusing on minimal presentations, defining ideals, and resolutions of their associated rings.

Contribution

It compiles and discusses both longstanding and recent open problems in the study of numerical semigroups and their algebraic structures.

Findings

Identification of key open problems in minimal presentations

Discussion of unresolved questions about defining ideals

Highlighting recent developments and challenges

Abstract

We collect some open problems about minimal presentations of numerical semigroups and, more generally, about defining ideals and free resolutions of their semigroup rings and associated graded rings. We emphasize both long-standing problems and more recent questions and developments.