{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1132",
  "content_hash": null,
  "slug": "polynomial-image-z2-nonnegative-integers-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem. Give an explicit \\(F\\in\\mathbb Q[X,Y]\\) and prove the equality \\(F(\\mathbb Z^2)=\\mathbb Z_{\\ge0}\\), including integrality and nonnegativity on every integer pair.",
  "relevance": "For polynomial image z2 nonnegative integers, pins the dated research frontier: UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem. Give an explicit \\(F\\in\\mathbb Q[X,Y]\\) and prove the equality.",
  "relevance_source": "recorded",
  "body": "UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem.\n\nA complete resolution must satisfy this condition: Give an explicit \\(F\\in\\mathbb Q[X,Y]\\) and prove the equality \\(F(\\mathbb Z^2)=\\mathbb Z_{\\ge0}\\), including integrality and nonnegativity on every integer pair.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers",
      "locator": "See dataset.references[0] for the exact external source and locator."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1131",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "polynomial-image-z2-nonnegative-integers",
      "title": "polynomial image z2 nonnegative integers",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}