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LOCAL Cr-RIGHT EQUIVALENCE OF Cr+1 FUNCTIONS

Part of: Theory of singularities and catastrophe theory Local theory

Published online by Cambridge University Press:  10 June 2016

PIOTR MIGUS
PIOTR MIGUS*
Affiliation:
Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland e-mail: migus@math.uni.lodz.pl
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Abstract

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Let f,g:(ℝn, 0) → (ℝ, 0) be Cr+1 functions, r ∈ ℕ. We will show that if ∇f(0)=0 and there exist a neighbourhood U of 0 ∈ ℝn and a constant C > 0 such that

$$\begin{equation*} \left|\partial^m(g-f)(x)\right| ≤ C \left|\nabla f(x)\right|^{r+2-|m|} \quad \textrm{ for } x\in U, \end{equation*} $$
and for any m ∈ ℕ0n such that |m| ≤ r, then there exists a Cr diffeomorphism ϕ:(ℝn, 0) → (ℝn, 0) such that f = g ° ϕ in a neighbourhood of 0 ∈ ℝn.

MSC classification

Primary: 58K40: Classification; finite determinacy of map germs
Secondary: 14B05: Singularities
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 265 - 272
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

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