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Open problems on relations of numerical semigroups

Alessio Moscariello, Alessio Sammartano

TL;DR

This work surveys open problems connecting numerical semigroups with their defining relations and free resolutions, recasting questions in terms of Betti numbers of the associated rings $A_\Gamma$, $R_\Gamma$, and $G_\Gamma$. It organizes the landscape by key invariants—embedding dimension $\mathrm{edim}(\Gamma)$, multiplicity $\mathrm{mult}(\Gamma)$, width $\mathrm{width}(\Gamma)$, symmetry (and almost symmetricness), and the Frobenius number—showing how these control minimal presentations ($\rho(\Gamma)$) and higher syzygies, with main threads including Ci/Cyclotomic characterizations, Wilf-type bounds, and uniform boundedness phenomena. The paper collects precise results for small $\mathrm{edim}$, outlines sharp open problems (e.g., Bresinsky’s problem for symmetric semigroups, Rossi’s Hilbert-function question for CI semigroups), and presents influential conjectures (ERV, width-based bounds) to guide future research. By embedding numerical semigroup questions into commutative-algebraic language of free resolutions, it provides a unified framework linking combinatorial data to algebraic invariants and highlights the depth and breadth of unresolved questions in this area.

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.

Open problems on relations of numerical semigroups

TL;DR

This work surveys open problems connecting numerical semigroups with their defining relations and free resolutions, recasting questions in terms of Betti numbers of the associated rings , , and . It organizes the landscape by key invariants—embedding dimension , multiplicity , width , symmetry (and almost symmetricness), and the Frobenius number—showing how these control minimal presentations () and higher syzygies, with main threads including Ci/Cyclotomic characterizations, Wilf-type bounds, and uniform boundedness phenomena. The paper collects precise results for small , outlines sharp open problems (e.g., Bresinsky’s problem for symmetric semigroups, Rossi’s Hilbert-function question for CI semigroups), and presents influential conjectures (ERV, width-based bounds) to guide future research. By embedding numerical semigroup questions into commutative-algebraic language of free resolutions, it provides a unified framework linking combinatorial data to algebraic invariants and highlights the depth and breadth of unresolved questions in this area.

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.
Paper Structure (8 sections, 17 theorems, 8 equations)

This paper contains 8 sections, 17 theorems, 8 equations.

Table of Contents

  1. Generalities
  2. Embedding dimension and multiplicity
  3. Width
  4. Symmetric semigroups
  5. Almost symmetric semigroups
  6. Uniform boundedness
  7. Complete intersections
  8. Frobenius number

Key Result

Theorem 1

Let $\Gamma$ be a numerical semigroup with $\mathrm{edim}(\Gamma)= 3$. If $\Gamma$ is symmetric, then $b_1(R_\Gamma) =2$ and $b_2(R_\Gamma) =1$. If $\Gamma$ is not symmetric, then $b_1(R_\Gamma) =3$ and $b_2(R_\Gamma) =2$.

Theorems & Definitions (17)

  • Theorem 1: Herzog
  • Theorem 2: RosalesGarciaSanchez98
  • Theorem 3: MoscarielloSammartano
  • Theorem 4: MoscarielloSammartano
  • Theorem 5: ERV
  • Theorem 6: MoscarielloSammartano
  • Theorem 7: MoscarielloSammartano
  • Theorem 8: CavigliaMoscarielloSammartano
  • Theorem 9: CavigliaMoscarielloSammartano
  • Theorem 10: LTV
  • ...and 7 more