{"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"}