$C^r$-right equivalence of analytic functions

TL;DR

This paper proves that under certain conditions involving the derivatives and ideal membership, two analytic functions are $C^r$-right equivalent, extending understanding of function equivalence in analytic geometry.

Contribution

It establishes a new criterion for $C^r$-right equivalence of analytic functions based on ideal membership and derivative conditions.

Findings

If $ abla f(0)=0$, then $f$ and $g$ are $C^r$-right equivalent under the given conditions.

The result links ideal membership $(f)^{r+2}$ to the equivalence of functions.

Provides a theoretical foundation for analyzing equivalence of analytic functions in singularity theory.

Abstract

Let $f,g:(\mathbb{R}^n,0)\rightarrow (\mathbb{R},0)$ 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 $r\in \mathbb{N}$.