{
  "abstract": "Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables $x_1,...,x_g,x_1^*,...,x_g^*$, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial $f$ is the sequence of hypersurfaces $\\mathscr{Z}_n(f)=\\{X\\in M_n(\\mathbb{C})^g:\\det f(X) = 0\\}$. When $f$ is Hermitian, the positivity domain of $f$ is the sequence of semialgebraic sets $\\mathcal{D}_n(f)=\\{X\\in M_n(\\mathbb{C})^g:f(X)\\text{ is positive semidefinite}\\}$. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through $Γ$-convexity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14053",
  "author_total": 3,
  "authors": [
    {
      "name": "Abdellah Islam"
    },
    {
      "name": "Jurij Volčič"
    },
    {
      "name": "Hugo J. Woerdeman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/et7y88295r99g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14053",
  "fulltext_url": null,
  "id": "et7y88295r99g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14053",
  "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",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7221798687",
  "pdf_url": "https://arxiv.org/pdf/2610.14053",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/et7y88295r99g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14053",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14053"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hereditary NC Polynomials & Positivity Domains",
  "title_text": "Hereditary NC Polynomials & Positivity Domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/et7y88295r99g",
  "venue": "arXiv",
  "versions": []
}