{"paper":{"title":"A sharp degree bound in the real Jacobian conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Polynomial maps from the real plane to itself with one component of degree 6 and non-vanishing Jacobian determinant are injective.","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"B. 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; any gap in those cited results would propagate.","pith_extraction_headline":"Polynomial maps from the real plane to itself with one component of degree 6 and non-vanishing Jacobian determinant are injective."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.12302/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-20T14:01:25.307542Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T10:06:10.260533Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:41:58.292080Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T10:39:28.687942Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"a0ea19ba3715882542ff330e49c0632a917884920bbd6752226c8ad9e7daed3a"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}