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Numerical semigroups from rational matrices IV: computation of the matricial dimensions of numerical semigroups with small Frobenius number or genus

Theo Chinn, Junshu Feng, Stephan Ramon Garcia, Peiting Jiang

TL;DR

This work advances the study of numerical semigroups by introducing a module-theoretic framework that translates exponent semigroups into linear-algebraic data, yielding constructive criteria for realizing $S$ as $\mathcal{S}(A)$ and providing general bounds $\operatorname{dim_{mat}} S\le m(S)$. It couples this framework with a companion-matrix method (CMM), an integer linear-programming approach, to produce small-dimension representing matrices and to realize many semigroups with $F(S)\le 10$ or $g(S)\le 6$—cases previously out of reach. The paper also derives precise results for semigroups of low multiplicity and demonstrates how intersections can tighten dimension bounds. Collectively, the methodology enables a near-complete computation of matricial dimensions within the stated bounds and offers practical tools for extending these computations, aided by HPC for the most challenging instances.

Abstract

We introduce a module-theoretic approach and a linear-programming method to compute the matricial dimension of numerical semigroups. We use these to compute the matricial dimension of every numerical semigroup with Frobenius number at most $10$ or genus at most $6$. Many of these evaluations were beyond the scope of previous techniques.

Numerical semigroups from rational matrices IV: computation of the matricial dimensions of numerical semigroups with small Frobenius number or genus

TL;DR

This work advances the study of numerical semigroups by introducing a module-theoretic framework that translates exponent semigroups into linear-algebraic data, yielding constructive criteria for realizing as and providing general bounds . It couples this framework with a companion-matrix method (CMM), an integer linear-programming approach, to produce small-dimension representing matrices and to realize many semigroups with or —cases previously out of reach. The paper also derives precise results for semigroups of low multiplicity and demonstrates how intersections can tighten dimension bounds. Collectively, the methodology enables a near-complete computation of matricial dimensions within the stated bounds and offers practical tools for extending these computations, aided by HPC for the most challenging instances.

Abstract

We introduce a module-theoretic approach and a linear-programming method to compute the matricial dimension of numerical semigroups. We use these to compute the matricial dimension of every numerical semigroup with Frobenius number at most or genus at most . Many of these evaluations were beyond the scope of previous techniques.
Paper Structure (6 sections, 8 theorems, 23 equations)

This paper contains 6 sections, 8 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Module-theoretic approach
  3. Numerical semigroups of low multiplicity
  4. The companion-matrix method
  5. Matricial dimension by Frobenius number and genus
  6. Open questions

Key Result

Proposition 1

Let $\mathcal{V}$ be a $d$-dimensional $\mathbb{Q}$-vector space containing $\mathcal{M} \subseteq \mathcal{V}$, a $\mathbb{Z}$-module of rank $d$. Let $L \in \operatorname{End}(\mathcal{V})$ and let $\beta$ denote an integral basis for $\mathcal{M}$. Then $A = {}_{\beta}[L]_{\beta} \in \mathsf{M}_d

Theorems & Definitions (23)

  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Example 3
  • Remark 4
  • Theorem 5
  • proof
  • Example 6
  • Example 7
  • ...and 13 more