{
  "schema_version": 1,
  "problem_number": "OWR-15219-013",
  "title": "A Symmetric Counterexample to Brändén's Majorization Conjectures for Mixed Characteristic Polynomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an extended abstract in Oberwolfach Rep. 14 (2017), no. 1, 771–862 (pp. 825–826), Brändén considered the set D_{m,d} of stable homogeneous polynomials P of degree d in m variables with ∂P/∂x_i(1) = d/m for all i, and the zero vector λ(P) of the mixed characteristic polynomial χ_P(t) = ∏_i (1 − ∂/∂x_i)P evaluated at x = t·1. He conjectured that λ(P) is majorized by λ(P_∞), where P_∞ = (x_1 + ⋯ + x_m)^d/m^d (Conjecture 2), and, for symmetric P, that λ(Ψ_r P) is majorized by λ(Ψ_s P) for 0 ≤ r ≤ s along a flow Ψ_s tending to P_∞ (Conjecture 5). We show that both conjectures fail for the symmetric polynomial P = (x_1 + x_2)(x_1 + x_3)(x_2 + x_3)/8 in D_{3,3}, by exact proofs that need no numerical computation. Here χ_P = (t − 1)(4t² − 8t + 1)/4, and λ_1(P) + λ_2(P) = 2 + √3/2 exceeds the same sum for P_∞ because 9√3 < 16. Along the flow, Ψ_s P is again a product of three linear forms, χ_{Ψ_s P} is an explicit cubic with positive discriminant, and λ_1 + λ_2 of Ψ_s P is smaller than that of P for every s > 0; so Conjecture 5 fails with r = 0 and every s > 0. The flow is printed as exp(s(I − T)), a sign under which the source's Lemma 3 fails; we prove that lemma for exp(s(T − I)), and show that Conjecture 5 fails under either sign. Conjecture 2 itself is not new: it was reportedly refuted in a 2017 KTH degree project (Samuel Zackrisson, DiVA diva2:1106332; full text not read), according to its abstract by way of Wanless's counterexamples to the Holens–Đoković permanent conjecture, and according to a search-engine excerpt also by an explicit example. Our example satisfies the Holens–Đoković inequalities, so it does not arise from Wanless's counterexamples; we claim no priority for it. We found no refutation of Conjecture 5 in the sources searched; since the thesis could not be read, we cannot exclude that it also treats Conjecture 5. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.CV",
    "math.CA"
  ],
  "keywords": [
    "mixed characteristic polynomials",
    "stable polynomials",
    "doubly stochastic polynomials",
    "majorization",
    "zeros of polynomials",
    "van der Waerden conjecture",
    "Kadison-Singer problem",
    "permanents",
    "Holens-Đoković conjecture",
    "counterexample",
    "Brändén's conjectures",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-15219-013",
    "math.CO",
    "math.CV",
    "math.CA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15219-013/",
  "pdf_url": "https://eulersolve.org/papers/owr-15219-013/paper.pdf?v=4ad4fc96106f",
  "doi": "10.5281/zenodo.23498312",
  "zenodo_record_url": "https://zenodo.org/records/23498312",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Conjecture 2 was reportedly refuted earlier in a 2017 KTH degree project (not read); the Conjecture 5 refutation was not found in the sources searched.",
  "files": {
    "paper.pdf": {
      "sha256": "4ad4fc96106f787d3885723da3995c5e52bbe70935297e2d325a6687c1296506"
    },
    "source.zip": {
      "sha256": "670cfe55fca398edd433b6d9f78921a06d9aaeb43307937f48c9743b80ea90e8"
    },
    "verification_report.md": {
      "sha256": "895ffcf86852dd81a0d83428aa2bc16e59dcfdebd2e601d254d539846c9fa58c"
    }
  },
  "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."
}