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On minimal presentations of numerical monoids

Alessio Moscariello, Alessio Sammartano

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.

On minimal presentations 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.
Paper Structure (7 sections, 22 theorems, 28 equations)

This paper contains 7 sections, 22 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Algebraic toolbox
  3. Betti numbers
  4. Monoid algebras
  5. Regions where the bound is sharp
  6. Regions where the bound is not sharp
  7. Results for the type

Key Result

Theorem 1.2

Let $e,m \in \mathbb{N}$ be such that $3 \leq e < m$.

Theorems & Definitions (43)

  • Theorem 1.2
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 33 more