## Journal of the Mathematical Society of Japan

### Equivalence relations for two variable real analytic function germs

#### Abstract

For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to other natural equivalence relations. Our main theorem states that $C^1$ equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the $C^1$ equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence, in terms of the real tree model.

We also give several examples of bi-Lipschitz equivalent germs that are not blow-analytically equivalent.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 1 (2013), 237-276.

Dates
First available in Project Euclid: 24 January 2013

https://projecteuclid.org/euclid.jmsj/1359036454

Digital Object Identifier
doi:10.2969/jmsj/06510237

Mathematical Reviews number (MathSciNet)
MR3034404

Zentralblatt MATH identifier
1266.32009

#### Citation

KOIKE, Satoshi; PARUSIŃSKI, Adam. Equivalence relations for two variable real analytic function germs. J. Math. Soc. Japan 65 (2013), no. 1, 237--276. doi:10.2969/jmsj/06510237. https://projecteuclid.org/euclid.jmsj/1359036454

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