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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 $\\deg p=7$. Combined with the previously known i","authors_text":"B. Or\\'efice-Okamoto, F. Braun, F. Fernandes, J. Gwo\\'zdziewicz","cross_cats":["math.AC"],"headline":"Polynomial maps from the real plane to itself with one component of degree 6 and non-vanishing Jacobian determinant are injective.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-05-12T15:53:52Z","title":"A sharp degree bound in the real Jacobian conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.12302","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-13T04:19:47.990075Z","id":"b44de881-6434-40bd-b3b7-040469321a39","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"Polynomial maps from the real plane to itself with one component of degree 6 and non-vanishing Jacobian determinant are injective.","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.","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."}},"verdict_id":"b44de881-6434-40bd-b3b7-040469321a39"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ca05bad11fd446d11a79327dec29ece5a2c08b839964038f1f06a81c546e3355","target":"record","created_at":"2026-05-26T01:03:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"358f64cac1326a1286a26df8d7262033ba549a54596b9ab0a0d23b5b8202b533","cross_cats_sorted":["math.AC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-05-12T15:53:52Z","title_canon_sha256":"8a06a13aecc739f639e2b350d357d43ce6fa8e5994d25834c3a19b43e44d750e"},"schema_version":"1.0","source":{"id":"2605.12302","kind":"arxiv","version":2}},"canonical_sha256":"9609eb6c61149c84f569f7963d5db06ee380d5ed9f3ab119b897f5b88411efe7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9609eb6c61149c84f569f7963d5db06ee380d5ed9f3ab119b897f5b88411efe7","first_computed_at":"2026-05-26T01:03:33.668432Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T01:03:33.668432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nE/Kmvww/xQjRo3Xztz3cATNmFJ1cM//2yJcUeYL2Y90aDqs4+baZYbtF1bGZSKKH+FTaxbcEMQzHxzcILiRBw==","signature_status":"signed_v1","signed_at":"2026-05-26T01:03:33.669159Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12302","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ca05bad11fd446d11a79327dec29ece5a2c08b839964038f1f06a81c546e3355","sha256:5ebbc1f4a2283e6e4eb3dc06158c0c6944abb4d83cb634876bac4dfb7d9cf021"],"state_sha256":"c65301cfb483b382c7337a3243a9da06c2db449124cdf0042f8c71eaed34993d"}