On minimal presentations of numerical monoids

TL;DR

This paper investigates the maximum size of minimal presentations of numerical monoids with fixed parameters, using algebraic tools to solve many cases and reveal challenges in others.

Contribution

It introduces a novel algebraic approach employing Hilbert functions and free resolutions to analyze minimal presentations of numerical monoids.

Findings

Determined maximum cardinalities for many numerical monoids cases.

Identified algebraic challenges in unresolved cases.

Derived results for the type of numerical monoids.

Abstract

We consider the classical problem of determining the largest possible cardinality of a minimal presentation of a numerical monoid with given embedding dimension and multiplicity. Very few values of this cardinality are known. In addressing this problem, we apply tools from Hilbert functions and free resolutions of artinian standard graded algebras. This approach allows us to solve the problem in many cases and, at the same time, identify subtle difficulties in the remaining cases. As a by-product of our analysis, we deduce results for the corresponding problem for the type of a numerical monoid.