{
  "abstract": "We construct an explicit Hermitian polynomial in six selfadjoint variables, $f=-1+ω+ω^*+M\\sum_{j=1}^{15}(2-u_j-u_j^*)$, with integer coefficients, degree $72$, and exactly $33$ monomials. Here $ω,u_1,\\ldots,u_{15}$ are specified words, $*$ reverses words, and $M$ is a specified positive integer. Its normalized trace is at least $3/4$ on selfadjoint matrix contraction tuples of every dimension, but equals $-1$ at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Thus $f$ is a counterexample to the algebraic formulation of Connes' embedding conjecture. We also construct a Hermitian quartic in $37$ selfadjoint variables, with coefficients in $\\mathbb{Z}[i]=\\mathbb{Z}+i\\mathbb{Z}$ and $387$ monomials, attaining the same trace bounds without norm restrictions on selfadjoint matrix inputs. Degree four is minimal in this unrestricted setting. Finally, an encoding in two selfadjoint variables yields a counterexample on the contraction domain with integer coefficients and degree at most $48$; two variables are minimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01536",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiaqi Wang"
    },
    {
      "name": "Lihong Zhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cna43dw13x5p8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01536",
  "fulltext_url": null,
  "id": "cna43dw13x5p8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01536",
  "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.OA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7215918409",
  "pdf_url": "https://arxiv.org/pdf/2610.01536",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cna43dw13x5p8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.01536",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01536"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "An Explicit Polynomial Counterexample to Connes' Embedding Conjecture",
  "title_text": "An Explicit Polynomial Counterexample to Connes' Embedding Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/cna43dw13x5p8",
  "venue": "arXiv",
  "versions": []
}