{
  "abstract": "Tensors over commutative rings naturally appear in number theory, geometry, and group theory. For example, $2\\times 2\\times 2$ tensors over $\\mathbb{Z}$ form the starting point of Bhargava's celebrated works generalising Gauss's composition law (Bhargava, Ann. Math., 2004). Symmetric tensors over $\\mathbb{Z}$ are central to the classification of Calabi--Yau threefolds (Yau, Commun. Pure Appl. Math., 1978; Wall, Invent. Math., 1966), geometric objects of significance in string theory. Additionally, tensors over finite commutative rings closely correspond to finite nilpotent groups of class 2 (Baer, Trans. Am. Math. Soc., 1938). In this work, we study isomorphism testing of tensors over commutative rings from a complexity-theoretic viewpoint, and its applications. Some of our main results are: 1. Let $\\mathrm{R}$ be a commutative ring. We introduce two complexity classes: $\\mathsf{3TI}_\\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three modules over $\\mathrm{R}$, and $\\mathsf{3FTI}_\\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three free modules over $\\mathrm{R}$. 2. We show that some classical problems considered by Grunewald and Segal (ibid.), and the problem of classifying Calabi--Yau threefolds, are $\\mathsf{3FTI}_\\mathbb{Z}$-complete. We also show that many natural problems are complete for $\\mathsf{3TI}_{\\mathbb{Z}/p^e\\mathbb{Z}}$. 3. We show that testing isomorphism of tensors in $\\mathbb{Z}^2\\otimes\\mathbb{Z}^2\\otimes\\mathbb{Z}^2$ is polynomial-time equivalent to the principal ideal problem in algorithmic number theory. 4. We present an $N^{O((\\log N)^{8/9})}$-time algorithm for testing isomorphism of finite nilpotent groups of class $2$ and odd order $N$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00735",
  "author_total": 4,
  "authors": [
    {
      "name": "Joshua A. Grochow"
    },
    {
      "name": "Youming Qiao"
    },
    {
      "name": "Katherine E. Stange"
    },
    {
      "name": "Xiaorui Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46nhch56rdtkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00735",
  "fulltext_url": null,
  "id": "46nhch56rdtkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00735",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00735",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46nhch56rdtkv.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00735",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00735"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings",
  "title_text": "On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:05Z",
  "url": "https://research.draconitic.com/paper/46nhch56rdtkv",
  "venue": "arXiv",
  "versions": []
}