$C^r$-right equivalence of analytic functions

Abstract

Let $f,g:({\mathbb{R}}^n,0)\to (\mathbb{R},0)$ be analytic functions. We will show that if $\nabla f(0)=0$ and $g-f\in (f{)}^{r+2}$ then $f$ and $g$ are $C^r$-right equivalent, where $(f)$ denote ideal generated by $f$ and $r\in \mathbb{N}$.

9 pages. Main result has been significantly improved

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• Local $C^r$-right equivalence of $C^{r+1}$ functions