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A long-standing problem is to determine the largest integer $k$ such that the condition $\\deg p\\le k$ guarantees the global injectivity of $F$. Although several partial results have been obtained over the past $30$ years, the sharp degree bound has remained unknown. In this paper, we prove that $F$ is injective whenever $\\deg p=6$. On the other hand, we construct a non-injective polynomial map with nowhere vanishing Jacobian determinant for which $\\deg p=7$. 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Or\\'efice-Okamoto, F. Braun, F. Fernandes, J. Gwo\\'zdziewicz","submitted_at":"2026-05-12T15:53:52Z","abstract_excerpt":"Let $F=(p,q):\\mathbb R^2\\to \\mathbb R^2$ be a polynomial map with nowhere zero Jacobian determinant. A long-standing problem is to determine the largest integer $k$ such that the condition $\\deg p\\le k$ guarantees the global injectivity of $F$. Although several partial results have been obtained over the past $30$ years, the sharp degree bound has remained unknown. In this paper, we prove that $F$ is injective whenever $\\deg p=6$. On the other hand, we construct a non-injective polynomial map with nowhere vanishing Jacobian determinant for which $\\deg p=7$. Combined with the previously known i"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if F=(p,q):R²→R² is a polynomial map such that the degree of p is 6 and whose Jacobian determinant is nowhere zero, then F is injective.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The argument relies on a collection of earlier results for lower degrees and on standard facts about real polynomial rings and the topology of the plane; any gap in those cited results would propagate.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Polynomial maps R²→R² with one component of degree 6 and nowhere-zero Jacobian are injective.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Polynomial maps from the real plane to itself with one component of degree 6 and non-vanishing Jacobian determinant are injective.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e04564b7850a5070d85bf5702f6a7a411576f8ac51e47d5c4c57ac0b93146f22"},"source":{"id":"2605.12302","kind":"arxiv","version":2},"verdict":{"id":"b44de881-6434-40bd-b3b7-040469321a39","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T04:19:47.990075Z","strongest_claim":"if F=(p,q):R²→R² is a polynomial map such that the degree of p is 6 and whose Jacobian determinant is nowhere zero, then F is injective.","one_line_summary":"Polynomial maps R²→R² with one component of degree 6 and nowhere-zero Jacobian are injective.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The argument relies on a collection of earlier results for lower degrees and on standard facts about real polynomial rings and the topology of the plane; 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