{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1130",
  "content_hash": null,
  "slug": "polynomial-bijection-q2-q-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted answer. Poonen gives a conditional polynomial injection, while Bresciani proves conditional nonexistence of a polynomial bijection under a weak Bombieri-Lang conjecture. The dated search found no unconditional resolution. Give an explicit polynomial \\(F\\in\\mathbb Q[X,Y]\\) and prove that every rational \\(t\\) has exactly one ordered rational preimage.",
  "relevance": "Defines the dated frontier and the exact remainder that research on this problem must resolve.",
  "relevance_source": "recorded",
  "body": "UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted answer. Poonen gives a conditional polynomial injection, while Bresciani proves conditional nonexistence of a polynomial bijection under a weak Bombieri-Lang conjecture. The dated search found no unconditional resolution.\n\nA complete resolution must satisfy this condition: Give an explicit polynomial \\(F\\in\\mathbb Q[X,Y]\\) and prove that every rational \\(t\\) has exactly one ordered rational preimage.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q",
      "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/21003/polynomial-bijection-from-mathbb-q-times-mathbb-q-to-mathbb-q",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "relations": [
    {
      "slug": "R1129",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "polynomial-bijection-q2-q",
      "title": "polynomial bijection q2 q",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}