{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0125",
  "title": "Smooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every prime p, integer n >= 2, and degree d >= n+1, we construct a geometrically smooth hypersurface X in projective n-space over F_p of degree d such that #X(F_p) is not congruent to 1 modulo p. The construction stays over the specified prime field in every characteristic and degree. A finite-field moment polynomial supplies nonzero coefficients with the required point count. In odd characteristic not dividing the degree, a sharper individual-degree bound permits simultaneous avoidance of the singular parameter. When the characteristic divides the degree, positive exponent compositions make a triangular family smooth for all nonzero coefficients, apart from one explicitly handled boundary case. A separate uniform formula treats characteristic two in odd degree. The coefficient selection is a finite deterministic procedure; no efficient complexity bound is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "keywords": [
    "finite fields",
    "smooth hypersurfaces",
    "Chevalley-Warning theorem",
    "point-count congruences",
    "monomial deformations",
    "AIM-ALGEBRAIC_GEOMETRY-0125",
    "math.AG",
    "math.NT",
    "open mathematics",
    "mathematical proof"
  ],
  "manuscript_version_date": "2026-09-06",
  "publication_date": "2026-09-06",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-06",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0125/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0125/paper.pdf?v=97c4f88da7fd",
  "doi": "10.5281/zenodo.22514087",
  "zenodo_record_url": "https://zenodo.org/records/22514087",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
    "paper.pdf": {
      "sha256": "97c4f88da7fd9e70bb50e4679209ba2c5d5483be2648a31b54e274da0e3d0fb4"
    },
    "source.zip": {
      "sha256": "8a74df8b764a199cee3821597d0f95fdb69b46bc9d864332ab8c06150a60bebd"
    },
    "verification_report.md": {
      "sha256": "3af22b5495852b4f2a738b2833a87d464de3fe11f8b2016c7effaf467b49d9c0"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}