{
  "abstract": "Let $R$ be a Noetherian unique factorization domain containing $\\mathbb Q$, with $\\dim R\\leq3$. We prove that $R[[t]]$ is a unique factorization domain if and only if two conditions hold: every two-generated parameter ideal $q$ in every height-two localization satisfies $\\overline{q^2}\\subseteq q$, and every height-three localization has depth three. No excellence or completion hypothesis is imposed. The main local result is that $A[[t]]$ is not a UFD whenever $A$ is a local $\\mathbb Q$-UFD of dimension at least three and depth two. We also prove that the UFD property of $R[[t]]$ passes to every coefficient localization in dimension at most three, give an all-dimensional criterion when the completions of the maximal localizations satisfy Serre's condition $(S_2)$, and obtain an arbitrary-characteristic criterion for excellent UFDs of dimension at most three.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24551",
  "author_total": 1,
  "authors": [
    {
      "name": "Elad Paran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2d93ecy0anxdb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24551",
  "fulltext_url": null,
  "id": "2d93ecy0anxdb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24551",
  "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": "W7223055853",
  "pdf_url": "https://arxiv.org/pdf/2610.24551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2d93ecy0anxdb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Unique Factorization of Power Series Rings over Q-UFDs of Dimension at Most 3",
  "title_text": "Unique Factorization of Power Series Rings over Q-UFDs of Dimension at Most 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/2d93ecy0anxdb",
  "venue": "arXiv",
  "versions": []
}