{
  "abstract": "Let $H=\\langle a_1,\\ldots,a_n\\rangle$ be a numerical semigroup. A conjecture on numerical semigroup rings predicts that the defining ideal $I_H$ has a determinantal presentation if and only if the set of pseudo-Frobenius numbers of $H$ has form $\\{h+α,h+2α,\\ldots,h+(n-1)α\\}$ for $h\\ge 0, α>0$. We prove this conjecture under the condition that $h$ has a unique factorization in $H$. We also give characterization this unique-factorization condition. As an application, we consider the family of affine-orbit numerical semigroups determined by $m=cA_{n-1}+1$, $1\\le c\\le a$. We prove that the relevant element $h$ has a unique factorization and the defining ideal is generated by the 2-minors of a $2\\times n$-matrix. In particular, the Eagon--Northcott complex of the matrix mentioned above gives the graded minimal free resolution of $k[H]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39218",
  "author_total": 1,
  "authors": [
    {
      "name": "Do Van Kien"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/47fjemfdtvvzv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39218",
  "fulltext_url": null,
  "id": "47fjemfdtvvzv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39218",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7215127412",
  "pdf_url": "https://arxiv.org/pdf/2609.39218",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/47fjemfdtvvzv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39218",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39218"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Arithmetic pseudo-Frobenius numbers and determinantal numerical semigroup rings",
  "title_text": "Arithmetic pseudo-Frobenius numbers and determinantal numerical semigroup rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/47fjemfdtvvzv",
  "venue": "arXiv",
  "versions": []
}