{
  "abstract": "Let $K$ be a nonempty compact semialgebraic subset of the real line, and let $S$ be a saturated scalar description of $K$. We prove that every hermitian polynomial with bounded-operator coefficients that is positive semidefinite on $K$ belongs to the operator quadratic module generated by $S$. We also show that if such a polynomial has degree at most $\\tt{d}$, then it admits a certificate of weighted degree at most $\\tt{d}+c_S$, where $c_S$ depends only on $S$, not on the polynomial or the coefficient Hilbert space. In contrast to the scalar case, where natural preorderings admit the degree bound $\\tt{d}$ independently of $K$, we show that no uniform additive constant exists for natural matrix preorderings. We construct quadratic $2\\times2$ matrix polynomials positive semidefinite on unions of two intervals whose minimum certificate degrees in the corresponding natural preorderings diverge as one interval shrinks to a point, while the gap length remains bounded below. Conversely, we establish uniform bounds of the form $\\tt{d}+C$ for families of interval unions contained in a fixed compact interval when interval lengths and gap lengths have uniform positive lower bounds. Our proofs develop two techniques that may be of independent interest. First, a quadratic reduction principle represents arbitrary-degree operator polynomials as compressions of quadratic ones on enlarged Hilbert spaces. A perturbation argument and the closedness of truncated operator quadratic modules then extend uniform degree bounds from the quadratic case to arbitrary degrees with linear degree bounds of slope $1$. Second, a rank-one symmetric matrix pencil transfers positivity on interval unions to $[0,1]$, where the operator Fejér--Riesz theorem yields certificates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02967",
  "author_total": 2,
  "authors": [
    {
      "name": "Abhay Jindal"
    },
    {
      "name": "Aljaž Zalar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3cxspfwsbheef.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02967",
  "fulltext_url": null,
  "id": "3cxspfwsbheef",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02967",
  "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.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02967",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3cxspfwsbheef.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.02967",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02967"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Operator Fejér--Riesz Theorems with Gaps and Degree Bounds",
  "title_text": "Operator Fejér--Riesz Theorems with Gaps and Degree Bounds",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:14Z",
  "url": "https://research.draconitic.com/paper/3cxspfwsbheef",
  "venue": "arXiv",
  "versions": []
}