{
  "abstract": "In this paper, we improve the upper bound for the tensor rank of the $n \\times n$ determinant tensor to $\\mathcal{B}_n-2^{n-1}+n+1$ by finding a new tensor decomposition of the tensor, where $\\mathcal{B}_n$ denotes the $n$-th Bell number. The main idea is to replace a part of the tensor decomposition suggested by Houston--Goucher--Johnston with a tensor that is essentially the same as the $n \\times n$ permanent tensor, and to use Glynn's formula. This upper bound gives an improved upper bound for the Waring rank of the $n \\times n$ determinant polynomial for $n \\geq 17$ as well.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18394",
  "author_total": 3,
  "authors": [
    {
      "name": "Jeong-Hoon Ju"
    },
    {
      "name": "Hayoung Choi"
    },
    {
      "name": "Taehyeong Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9cy451zss9x8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18394",
  "fulltext_url": null,
  "id": "e9cy451zss9x8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18394",
  "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.AG"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7221801734",
  "pdf_url": "https://arxiv.org/pdf/2610.18394",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9cy451zss9x8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18394",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Improved Upper Bounds for the Rank of the Determinant Tensor via Glynn's Formula",
  "title_text": "Improved Upper Bounds for the Rank of the Determinant Tensor via Glynn's Formula",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/e9cy451zss9x8",
  "venue": "arXiv",
  "versions": []
}