{
  "abstract": "Let $(R,\\mathfrak{m})$ be a one-dimensional Cohen-Macaulay local ring. In this paper, we find the reduction number $r_{z}(\\mathfrak{m}^d)$ of $\\mathfrak{m}^d$ with respect to a witness $z\\in \\mathfrak{m}^d \\setminus \\mathfrak{m}^{d+1}$ to the generalized Loewy length $\\text{g}\\ell\\ell(R)$ for several infinite families of hypersurfaces $\\left\\{(R,\\mathfrak{m}) \\right\\}$. For every principal reduction $w$ of $\\mathfrak{m}^d$, we have $\\text{g}\\ell\\ell(R) \\leq d(r_{w}(\\mathfrak{m}^d)+1)$. We give examples of families $\\left\\{ (R,\\mathfrak{m}) \\right\\}$ such that $d(r_{z}(\\mathfrak{m}^d)+1)-\\text{g}\\ell\\ell(R)=0$ and $d(r_{z}(\\mathfrak{m}^d)+1)-\\text{g}\\ell\\ell(R)=1$. In particular, we prove that for every prime $p$, there are local hypersurfaces $R$ over $\\mathbb{F}_p$ such that $d(r_{z}(\\mathfrak{m}^d)+1)-\\text{g}\\ell\\ell(R)=0$ and $d(r_{z}(\\mathfrak{m}^d)+1)-\\text{g}\\ell\\ell(R)=1.$ We also prove that the difference $d(r_{z}(\\mathfrak{m}^d)+1)-\\text{g}\\ell\\ell(R)$ can vary independently of $\\text{g}\\ell\\ell(R)-e(R)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26003",
  "author_total": 3,
  "authors": [
    {
      "name": "Richard Bartels"
    },
    {
      "name": "Sarah Dajani"
    },
    {
      "name": "Gabriel Koomson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71dfkjw44xbet.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26003",
  "fulltext_url": null,
  "id": "71dfkjw44xbet",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26003",
  "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": "W7204435318",
  "pdf_url": "https://arxiv.org/pdf/2608.26003",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71dfkjw44xbet.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.26003",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26003"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Reduction numbers for witnesses to the generalized Loewy length",
  "title_text": "Reduction numbers for witnesses to the generalized Loewy length",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/71dfkjw44xbet",
  "venue": "arXiv",
  "versions": []
}