{
  "abstract": "Let $\\mathbb{R}\\langle X,Y\\rangle$ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone $Θ$ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved that the two notions coincide in degree at most four; from degree six onwards they differ. We give a complete, explicit description of both cones on the six-dimensional space $Z$ of symmetric homogeneous sextics in two symmetric letters whose abelianization vanishes identically, and show that on $Z$ they coincide: for $f\\in Z$, trace-positivity is equivalent to membership in $Θ_3$, to non-negativity of the trace on pairs of $2\\times2$ symmetric matrices, and to being cyclically equivalent to a sum of four Hermitian squares of cubics with vanishing abelianization. Membership in $Θ$ reduces to a $4\\times4$ positive semidefinite Gram matrix whose diagonal blocks are fixed by the content-$(4,2)$ and $(2,4)$ parts of $f$; each of those parts ranges over a two-dimensional cone with two linear facets, and the content-$(3,3)$ part is then confined to a rectangle whose sides are the geometric means $\\sqrt{p_+q_+}$ and $\\sqrt{p_-q_-}$ of the eigenvalues of those blocks. Each facet is certified by one of two families of pairs of $2\\times2$ symmetric matrices. The abelianization hypothesis is natural: every bivariate degree-six example we are aware of -- a trace-positive polynomial outside $Θ$ -- arises by lifting a non-negative, non-SOS polynomial in two variables to a cyclically sorted polynomial; and that lifting cannot operate when the abelianization vanishes. Symmetry of $f$ is essential: dropping it adds one antisymmetric direction, on which the trace vanishes identically, yet whose non-zero elements all lie outside $Θ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12490",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel Keren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/41gxecvvc718w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12490",
  "fulltext_url": null,
  "id": "41gxecvvc718w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12490",
  "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.CO"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7221513731",
  "pdf_url": "https://arxiv.org/pdf/2610.12490",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/41gxecvvc718w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12490",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12490"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators",
  "title_text": "Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/41gxecvvc718w",
  "venue": "arXiv",
  "versions": []
}