{
  "abstract": "The unfair $0$--$1$ polynomial conjecture asks whether a factorization \\[C(x)=A(x)B(x),\\] with $A$ and $B$ monic and having nonnegative real coefficients, and $C$ a polynomial with all the coefficients 0 and 1, must already be a factorization into $0$--$1$ polynomials. Let $k$ be odd and $0<a<1$. We study the possibility that \\[1+a x^2+x^k\\] divides a $0$--$1$ polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case $k=5$. We prove that no such factorization exists for any odd $k\\ge 341$. The intermediate cases (5 < k < 341) are treated in the companion paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19173",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexander Dvorsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2d1mmn9w28dd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19173",
  "fulltext_url": null,
  "id": "c2d1mmn9w28dd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19173",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7203884969",
  "pdf_url": "https://arxiv.org/pdf/2608.19173",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2d1mmn9w28dd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.19173",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19173"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Unfair 0-1 Polynomial Problem and High-Degree Trinomials",
  "title_text": "The Unfair 0-1 Polynomial Problem and High-Degree Trinomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/c2d1mmn9w28dd",
  "venue": "arXiv",
  "versions": []
}