{
  "schema_version": 1,
  "problem_number": "AIM-COMPUTATION-0095",
  "title": "A Sharp Obstruction to Uniform Low-Degree Tails over Finite Fields",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a prime power q and integers m >= n >= 2, let D_q(n,m) be the least degree bound on a polynomial g over F_q for which X^m+g(X) has an irreducible factor of degree n. We give the elementary sharp identity max_{m>=n} D_q(n,m)=n-1: equality holds whenever m is congruent to -1 modulo q^n-1. More generally, the residue m congruent to -j forces D_q(n,m)>=n-j for 1<=j<n. In particular, over F_2, no tail of degree at most 6 makes X^254+g(X) have an irreducible factor of degree 8. This disproves the unrestricted-exponent formulation of Question 7 in the AIM algorithmic number theory workshop list. It does not settle Gao's original conjecture, where the exponent is the least power of q at least n, or the separate sparse-polynomial variants.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.NT",
    "math.AC"
  ],
  "keywords": [
    "finite fields",
    "irreducible polynomials",
    "low-degree tails",
    "reciprocal exponents",
    "AIM-COMPUTATION-0095"
  ],
  "manuscript_version_date": "2026-10-04",
  "publication_date": "2026-10-04",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-04",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-computation-0095/",
  "pdf_url": "https://eulersolve.org/papers/aim-computation-0095/paper.pdf?v=7f4cd7946b8d",
  "doi": "10.5281/zenodo.23132141",
  "zenodo_record_url": "https://zenodo.org/records/23132141",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the unrestricted-exponent main assertion of AIM-COMPUTATION-0095, with a sharp uniform tail-degree bound. Gao's original least-q-power exponent conjecture and the separate auxiliary sparse-polynomial questions remain unresolved. Novelty and absolute priority are not certified. AI-assisted, self-audited and unrefereed; no independent specialist review or proof-assistant verification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "7f4cd7946b8daa2d409c9516ea021fad4df4f2f3cca4172e7f3ee3f0d4e75ddd"
    },
    "source.zip": {
      "sha256": "254348680a00a9fa690402603c6fa50d45c32effdc5580ba42bbbfd3528f0c25"
    },
    "verification_report.md": {
      "sha256": "aad8acc23310a4f514e7bdf911ac23250794dd60fd877df1f675477a7984cb69"
    }
  },
  "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."
}