{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R413",
  "content_hash": null,
  "slug": "iqf101-attempt-literature-audit",
  "type": "attempt",
  "title": "The literature supplies general methods for prescribed coefficients",
  "summary": "Rabin supplies the exact test, and prescribed-coefficient papers place the family in a well-studied class. This audit found no published value for the fixed maximum.",
  "relevance": "For Most irreducible constant slices of a sparse quintic over F_101, record iqf101-attempt-literature-audit (“The literature supplies general methods for prescribed coefficients”) documents a concrete method, search boundary, or failed route. The record states: Rabin supplies the exact test, and prescribed-coefficient papers place the family in a well-studied class.",
  "relevance_source": "recorded",
  "body": "Rabin's 1980 paper gives the finite-field factorization and irreducibility machinery used in the exhaustive artifact. Granger gives a deterministic method for exact expressions when the first \\(\\ell<p\\) coefficients of a monic degree-\\(n\\) polynomial are prescribed and \\(\\gcd(n,p)=1\\). Here a slice fixes the first four coefficients after the leading term, namely \\(0,0,a,b\\), with \\(\\ell=4\\), \\(p=101\\), and \\(n=5\\). Summing over the constant term produces \\(N(a,b)\\).\n\nGao, Kuttner, and Wang develop a general group-algebra generating-function framework for prescribed leading and ending coefficients. Their framework confirms that exact coefficient counts belong to an established research area. A focused search for this sparse degree-five family over \\(\\mathbb F_{101}\\), the value 29, and the invariant class 62 found no published table or theorem stating this fixed maximum. Publication novelty remains unverified.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published irreducibility tests and prescribed-coefficient enumeration literature checked for the fixed F_101 quintic slice maximum through 2026-07-25",
    "bounds": {
      "field_order": {
        "min": 101,
        "max": 101
      },
      "degree": {
        "min": 5,
        "max": 5
      },
      "search_year": {
        "min": 2026,
        "max": 2026
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1016/j.ffa.2019.01.001",
      "locator": "Robert Granger, On the Enumeration of Irreducible Polynomials over GF(q) with Prescribed Coefficients, Finite Fields and Their Applications 57 (2019), 156-229; Michael O. Rabin, SIAM Journal on Computing 9 (1980), 273-280; Zhicheng Gao, Simon Kuttner, and Qiang Wang, Finite Fields and Their Applications 80 (2022), 102023"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.ffa.2019.01.001",
    "locator": "Robert Granger, On the Enumeration of Irreducible Polynomials over GF(q) with Prescribed Coefficients, Finite Fields and Their Applications 57 (2019), 156-229; Michael O. Rabin, SIAM Journal on Computing 9 (1980), 273-280; Zhicheng Gao, Simon Kuttner, and Qiang Wang, Finite Fields and Their Applications 80 (2022), 102023"
  },
  "relations": [
    {
      "slug": "R414",
      "title": "The exact maximum slice count is 29",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "irreducible-quintic-f101-slices",
      "title": "irreducible quintic f101 slices",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}