{
  "abstract": "For positive integers $q$ and $m$ with $m\\geq 2q+3$, consider the symmetric numerical semigroup $S_{m,q}=\\langle m,m+1,qm+2q+2,qm+2q+3,\\ldots,qm+m-1\\rangle$. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answer this question affirmatively for every $q\\geq 1$, including an independent proof of the previously known case $q=1$. Let $P_{m,q}$ be the semigroup polynomial, and set $t=m-2q-3$, $L=2(q+1)(m+1)-1$, and $D=°P_{m,q}=2qm+2q+2$. For $m\\geq 2q+4$, let $ρ_1,\\ldots,ρ_D$ be the roots of $P_{m,q}$, counted with multiplicity. An explicit computation of a single coefficient of the formal logarithm gives $\\left|\\sum_{j=1}^{D}ρ_j^{-L}\\right|=L\\left|[x^L]\\log P_{m,q}(x)\\right|\\geq Lt-1>D$. The strict inequality forces $P_{m,q}$ to have a root off the unit circle. Hence $S_{m,q}$ is noncyclotomic whenever $m\\geq 2q+4$. The boundary case $m=2q+3$ has embedding dimension $3$ and is cyclotomic. Therefore $S_{m,q}$ is cyclotomic if and only if $m=2q+3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29484",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhi-Lin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1vm8cw09zd1qc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29484",
  "fulltext_url": null,
  "id": "1vm8cw09zd1qc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29484",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214388455",
  "pdf_url": "https://arxiv.org/pdf/2609.29484",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1vm8cw09zd1qc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29484",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29484"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups",
  "title_text": "A power-sum obstruction to cyclotomicity in a family of symmetric numerical semigroups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/1vm8cw09zd1qc",
  "venue": "arXiv",
  "versions": []
}