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We will show that if $\\nabla f(0)=0$ and there exist a neigbourhood $U$ of $0\\in \\mathbb{R}^n$ and a constant $C>0$ such that $$ \\left|\\partial^m(g-f)(x)\\right|\\leq C \\left|\\nabla f(x)\\right|^{r+2-|m|}, \\quad x\\in U, $$ for any $m\\in \\mathbb{N}_0^n$ such that $|m|\\leq r$, then there exists a $C^r$ diffeomorphism $\\varphi:(\\mathbb{R}^n,0)\\rightarrow (\\mathbb{R}^n,0)$ such that $f=g\\circ \\varphi$ in a neighbourhood of $0$."},"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":"1506.02589","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-06-08T17:23:29Z","cross_cats_sorted":[],"title_canon_sha256":"f5a23a4b3b935eb5c2fd4c3ef7c94f5af01a8569a9034230e3ecf42c9b17296d","abstract_canon_sha256":"7e672f607ed2ea8f45061380cf04e71c79dcd04024d55b2bc8cf215f332ea6f6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:45.896627Z","signature_b64":"iHvD7szwLf+tigP7zDBrJ/IrX7KlwO0Yz9j2XTgWAVGNfRM92XStbIeJ+c04LYTJykT/mBEpsvfmWWdVwM+cDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f81f0e959c07062d429ed40c14c998505f759e4dffe8f9d482e34dcc20ed3d00","last_reissued_at":"2026-05-18T01:41:45.896074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:45.896074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local $C^r$-right equivalence of $C^{r+1}$ functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Piotr Migus","submitted_at":"2015-06-08T17:23:29Z","abstract_excerpt":"Let $f,g:(\\mathbb{R}^n,0)\\rightarrow (\\mathbb{R},0)$ be $C^{r+1}$ functions, $r\\in \\mathbb{N}$. We will show that if $\\nabla f(0)=0$ and there exist a neigbourhood $U$ of $0\\in \\mathbb{R}^n$ and a constant $C>0$ such that $$ \\left|\\partial^m(g-f)(x)\\right|\\leq C \\left|\\nabla f(x)\\right|^{r+2-|m|}, \\quad x\\in U, $$ for any $m\\in \\mathbb{N}_0^n$ such that $|m|\\leq r$, then there exists a $C^r$ diffeomorphism $\\varphi:(\\mathbb{R}^n,0)\\rightarrow (\\mathbb{R}^n,0)$ such that $f=g\\circ \\varphi$ in a neighbourhood of $0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02589","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1506.02589","created_at":"2026-05-18T01:41:45.896161+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.02589v2","created_at":"2026-05-18T01:41:45.896161+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.02589","created_at":"2026-05-18T01:41:45.896161+00:00"},{"alias_kind":"pith_short_12","alias_value":"7APQ5FM4A4DC","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_16","alias_value":"7APQ5FM4A4DC2QU6","created_at":"2026-05-18T12:29:07.941421+00:00"},{"alias_kind":"pith_short_8","alias_value":"7APQ5FM4","created_at":"2026-05-18T12:29:07.941421+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB","json":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB.json","graph_json":"https://pith.science/api/pith-number/7APQ5FM4A4DC2QU62QGBJSMYKB/graph.json","events_json":"https://pith.science/api/pith-number/7APQ5FM4A4DC2QU62QGBJSMYKB/events.json","paper":"https://pith.science/paper/7APQ5FM4"},"agent_actions":{"view_html":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB","download_json":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB.json","view_paper":"https://pith.science/paper/7APQ5FM4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.02589&json=true","fetch_graph":"https://pith.science/api/pith-number/7APQ5FM4A4DC2QU62QGBJSMYKB/graph.json","fetch_events":"https://pith.science/api/pith-number/7APQ5FM4A4DC2QU62QGBJSMYKB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB/action/storage_attestation","attest_author":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB/action/author_attestation","sign_citation":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB/action/citation_signature","submit_replication":"https://pith.science/pith/7APQ5FM4A4DC2QU62QGBJSMYKB/action/replication_record"}},"created_at":"2026-05-18T01:41:45.896161+00:00","updated_at":"2026-05-18T01:41:45.896161+00:00"}