{
  "abstract": "We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. We show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that H is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several key results. 1) Rigidty: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result is a ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Huneke-Ulrich structure: the toric ideal can be realized as the ideal of entries of an alternating matrix multiplied by a vector of variables, and this yields the licci property. 3) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 4) Herzog--Watanabe question: we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24880",
  "author_total": 6,
  "authors": [
    {
      "name": "Kazufumi Eto"
    },
    {
      "name": "Naoyuki Matsuoka"
    },
    {
      "name": "Alessio Moscariello"
    },
    {
      "name": "Takahiro Numata"
    },
    {
      "name": "Alessio Sammartano"
    },
    {
      "name": "Kei-ichi Watanabe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55y7150fyj88p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24880",
  "fulltext_url": null,
  "id": "55y7150fyj88p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24880",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214023884",
  "pdf_url": "https://arxiv.org/pdf/2609.24880",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55y7150fyj88p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2609.24880",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24880"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The structure of almost symmetric almost complete intersection numerical semigroups",
  "title_text": "The structure of almost symmetric almost complete intersection numerical semigroups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/55y7150fyj88p",
  "venue": "arXiv",
  "versions": []
}