{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1131",
  "content_hash": null,
  "slug": "polynomial-image-z2-nonnegative-integers-attempt-resolution-route",
  "type": "attempt",
  "title": "Resolve the stated acceptance condition",
  "summary": "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 A two-variable polynomial whose integer image is exactly the nonnegative integers, record polynomial-image-z2-nonnegative-integers-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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_source": "recorded",
  "body": "Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1132",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "polynomial-image-z2-nonnegative-integers",
      "title": "polynomial image z2 nonnegative integers",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}