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Nearly Gorenstein numerical semigroups with five generators have bounded type

Alessio Moscariello, Francesco Strazzanti

TL;DR

This work proves that the type of nearly Gorenstein numerical semigroups with five generators is bounded. By introducing NG-vectors and RF-matrices, the authors bound the two subsets of pseudo-Frobenius numbers, showing $| ext{PF}_2(S)|\le 6$ and a refined bound $| ext{PF}_1(S)|\le 31$ in the non-almost-symmetric 5-generated case, which together yield $t(S)\le 40$ (with $t(S)\le 39$ in a narrower scenario). A sharper bound occurs when the NG-vector has three distinct entries, giving $t(S)\le 5$ in that subcase. Extending the discussion to general embedding dimension, they argue that while the 5-generator case is bounded, higher dimensions can produce unbounded type, and they outline RF-matrix-based constructions illustrating this, along with connections to Backelin examples and numerical duplication.

Abstract

We prove that the type of nearly Gorenstein numerical semigroups minimally generated by $5$ integers is bounded. In particular, if such a semigroup is not almost symmetric, then its type is at most $40$. Finally, we make some considerations in higher embedding dimension.

Nearly Gorenstein numerical semigroups with five generators have bounded type

TL;DR

This work proves that the type of nearly Gorenstein numerical semigroups with five generators is bounded. By introducing NG-vectors and RF-matrices, the authors bound the two subsets of pseudo-Frobenius numbers, showing and a refined bound in the non-almost-symmetric 5-generated case, which together yield (with in a narrower scenario). A sharper bound occurs when the NG-vector has three distinct entries, giving in that subcase. Extending the discussion to general embedding dimension, they argue that while the 5-generator case is bounded, higher dimensions can produce unbounded type, and they outline RF-matrix-based constructions illustrating this, along with connections to Backelin examples and numerical duplication.

Abstract

We prove that the type of nearly Gorenstein numerical semigroups minimally generated by integers is bounded. In particular, if such a semigroup is not almost symmetric, then its type is at most . Finally, we make some considerations in higher embedding dimension.
Paper Structure (7 sections, 13 theorems, 26 equations)

This paper contains 7 sections, 13 theorems, 26 equations.

Table of Contents

  1. Preliminaries
  2. RF-matrices
  3. On the type of 5-generated nearly Gorenstein semigroups
  4. A bound for $|\operatorname{PF}_2(S)|$
  5. Bounds for $|\operatorname{PF}_1(S)|$
  6. Bounds for the type
  7. On the type of numerical semigroups

Key Result

Proposition 1.1

MS The following statements hold: In particular, an almost symmetric numerical semigroup is nearly Gorenstein.

Theorems & Definitions (34)

  • Proposition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • Corollary 1.5
  • Definition 1.6
  • Definition 1.7
  • Lemma 1.8
  • Remark 1.9
  • ...and 24 more