{
  "abstract": "We show that the local rings of quasianalytic Denjoy-Carleman classes need not be Noetherian. More precisely, we construct quasianalytic Denjoy-Carleman classes $C^M$ whose local rings $C^M(\\mathbb{R}^d,0)$ are not Noetherian for any $d\\geq 2$. Conversely, we exhibit quasianalytic classes whose local rings in dimension two are regular and Noetherian. Our approach is based on the problem of exact division: if $f=φg$, where $f\\in C^M$, $φ$ is real-analytic, and $g\\in C^\\infty$, under which conditions does it follow that $g\\in C^M$? We introduce a quantitative scale-growth condition on the defining weight sequence $M$ and prove that exact division holds in two variables whenever this condition is sufficiently strong relative to the rate at which the nonreal branches of the complex zero set of $φ$ approach the real plane. We also construct quasianalytic weight sequences with prescribed growth thresholds and establish corresponding counterexamples to exact division. These yield nonclosed principal ideals and, consequently, non-Noetherian local rings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40265",
  "author_total": 2,
  "authors": [
    {
      "name": "André Belotto da Silva"
    },
    {
      "name": "Avner Kiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7e74mpjqw4ehh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40265",
  "fulltext_url": null,
  "id": "7e74mpjqw4ehh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40265",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40265",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7e74mpjqw4ehh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40265",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40265"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Division and noetherianity in quasianalytic Denjoy-Carleman classes",
  "title_text": "Division and noetherianity in quasianalytic Denjoy-Carleman classes",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7e74mpjqw4ehh",
  "venue": "arXiv",
  "versions": []
}