{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0138",
  "title": "Sharp Zero Regions and Hurwitz Criteria for BMV Trace Polynomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For positive-definite Hermitian matrices A,B, we study the zeros of P_m(z)=Tr((A+zB)^m). Loewner bounds lB<=A<=uB give a sharp dimension-independent zero region: the union of two explicit closed disks. Every point of this region is attained by a commuting two-dimensional pair, while a nonreal boundary zero forces an endpoint block decomposition and commutativity. Optimizing the associated angle gives the optimal universal Hurwitz guarantee u/l<tan^2(pi/4+pi/(2m)) for m>=3. For arbitrary two-dimensional pairs, including noncommuting ones, a quadratic factorization yields simplicity, an exact stability criterion in terms of Tr(AB)/(Tr A Tr B), and a count of right-half-plane zeros. We also classify real zeros in every dimension. These are explicit zero-location results motivated by an open-ended AIM problem, not a new proof of the BMV coefficient theorem or a classification for every fixed higher-dimensional pair.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA",
    "math.SP"
  ],
  "keywords": [
    "Bessis-Moussa-Villani polynomials",
    "zero localization",
    "positive-definite matrices",
    "Hermitian matrix pencils",
    "Hurwitz stability",
    "Loewner order",
    "AIM-ANALYSIS-0138",
    "math.CA",
    "math.SP"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0138/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0138/paper.pdf?v=8d5f642fd024",
  "doi": "10.5281/zenodo.23024787",
  "zenodo_record_url": "https://zenodo.org/records/23024787",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proofs of the stated zero-location theorems, not a new proof of the BMV coefficient theorem or closure of the entire open-ended AIM programme. The universal zero locus ranges over pairs with given Loewner bounds; fixed higher-dimensional noncommuting pairs are not exhaustively classified. The trace-ratio criterion and maximal right-half-plane count are dimension-two statements. Classical Schur and power-sum methods and the inherited commuting example are credited. Self-audited and unrefereed; no independent certification, formal verification or absolute priority. Folklore or unlocated prior art is not excluded.",
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      "sha256": "8d5f642fd024e366599211f83b3844a38b72fb323a1adea4ff9f2f7c899fb715"
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    "source.zip": {
      "sha256": "715f5f18dc74e035ce1014086840fc0699326776a230b71fd5dcdc05473e3d3e"
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    "verification_report.md": {
      "sha256": "9d4686643b576c9f69e1f18c73ccbda5b020b87cd898b1ec35bda75bdfaf6096"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}