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Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\\geq 1$ and give some partial results for $r=2$.\n  Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d \\ge 2$, we give the inverse polynomial map under the condition that $JH^3=0$. We shall show that ${\\operatorname{deg}}(F^{-1})\\leq d^k$ if $k \\le 2$ and $JH^{k+1}=0$, but also give an example with $d = 2$ and $JH^4=0$ such that ${\\operatorname{deg}}(F^{-1})>"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1302.5864","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.AG","submitted_at":"2013-02-24T03:07:57Z","cross_cats_sorted":[],"title_canon_sha256":"8cbe9fdb2803832f417633abacb0f01a591a2878481ad08882cbeab63938cf3b","abstract_canon_sha256":"ff0f952d333c7dcd8fa216aa9d3138828411bbc15324d9d6da8b77fda7152ce0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:03.361059Z","signature_b64":"5jKhLZBfADtmvoTL9vvzVKw+bkMLhsqy+k1P7KVpBDhlz21LdEtFa+BQBBLzG+G4zHSZiYRvU2UnHaWQHFDDDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb4584aead92698963a4352e3b3317f485c6d952bab7ab9173b4f6ba90b38b15","last_reissued_at":"2026-05-18T02:49:03.360630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:03.360630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Remarks on the Jacobian Conjecture and Dru{\\.z}kowski mappings","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Dan Yan, Michiel de Bondt","submitted_at":"2013-02-24T03:07:57Z","abstract_excerpt":"In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. 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