{
  "abstract": "Let $T$ be a nonzero real symmetric third-order tensor. Denote by $r(T)$ its CP rank, by $s(T)$ its symmetric rank, by $d(T)$ its essential dimension, and put $e(T)=r(T)-d(T)$. We establish two complementary upper bounds for the Comon rank gap $$Δ(T):= s(T)-r(T)\\leq\\min\\{C_{ec}(e(T)),\\ C_{dc}(d(T))\\}.$$ The excess-controlled function $C_{ec}(e)$ is zero for $0\\leq e\\leq3$ and equals $9e-3$ for $e\\geq4$. In particular, the equality $C_{ec}(e(T))=0$ for $0\\leq e\\leq3$ extends Friedland's flattening-rank criterion for real symmetric third-order tensors by two additional rank layers. The dimension-controlled function $C_{dc}(d)$ is zero for $d=0$, equals $(d^2-4d+10)/3$ for $1\\leq d\\leq18$, and equals $d^2/4+d/4+3/2$ for $d\\geq19$. The excess estimate follows from the alternating equations and decomposable-bivector support geometry; a low-wedge descent gives the vanishing range $e\\leq3$. The dimension estimate combines a transverse polar-completion bound with a shifted estimate for the maximum real Waring rank. Thus the two bounds measure different features of the same tensor and remain effective in complementary regimes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14002",
  "author_total": 1,
  "authors": [
    {
      "name": "Jianze Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ewferkzxt1v7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14002",
  "fulltext_url": null,
  "id": "9ewferkzxt1v7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14002",
  "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.AG"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7221782514",
  "pdf_url": "https://arxiv.org/pdf/2610.14002",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ewferkzxt1v7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14002",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14002"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors",
  "title_text": "Upper Bounds on the Comon Rank Gap of Real Symmetric Third-Order Tensors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/9ewferkzxt1v7",
  "venue": "arXiv",
  "versions": []
}