{
  "schema_version": 1,
  "problem_number": "AMR-036-0017",
  "title": "Polynomial-Exponential Counterexamples to a Derivative-Zero Half-Plane Classification",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let P be the compact-open closure of the polynomials whose zeros lie in the closed lower half-plane. The entire function exp(iz)+z+2i has every zero of every derivative in that half-plane, but lies neither in P nor in the exceptional families C exp(az) and C(exp(ibz)-a), with complex C,a and real b. This gives a negative answer to the precise alternative in Eremenko's version of a problem of B. Ya. Levin and in the frozen UnsolvedMath record AMR-036-0017. For every positive integer d, an explicit affine holomorphic d-parameter family gives the stronger property that every derivative is zero-free in the closed upper half-plane, while remaining outside P and the stated exceptional families. These exceptions cannot be covered by countably many holomorphic families of uniformly bounded finite dimension. The proof uses polynomial dominance and a logarithmic-derivative condition. Buhovski's earlier exponential example is credited. This is not a classification of all admissible entire functions. The preprint is AI-assisted, self-audited and unrefereed, with no independent review, formal verification or absolute priority claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CV"
  ],
  "keywords": [
    "entire functions",
    "zeros of derivatives",
    "half-plane",
    "polynomial approximation",
    "counterexample",
    "Levin problem"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-036-0017/",
  "pdf_url": "https://eulersolve.org/papers/amr-036-0017/paper.pdf?v=cc3c03f6647c",
  "doi": "10.5281/zenodo.23289844",
  "zenodo_record_url": "https://zenodo.org/records/23289844",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The precise proposed alternatives are refuted, with counterexample families of every finite parameter dimension. The full class of admissible entire functions is not classified. Buhovski and the classical closure facts are credited. Self-audited and unrefereed; no independent review, formal verification or certified priority.",
  "files": {
    "paper.pdf": {
      "sha256": "cc3c03f6647c3c24defabbb26f31d04946108f552e88e98b9886eb397beadd01"
    },
    "source.zip": {
      "sha256": "39c8dc7556bc5d058e27a364915191e91d4b9591bb8676bc42bb28a11b9684c6"
    },
    "verification_report.md": {
      "sha256": "7144a2f9c06aeab51ef91e293099d38a033443989bb7efa1ab8c8c1371cff127"
    }
  },
  "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."
}