Journal of the Mathematical Society of Japan

Equivalence relations for two variable real analytic function germs


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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

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

First available in Project Euclid: 24 January 2013

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Zentralblatt MATH identifier

Primary: 32S15: Equisingularity (topological and analytic) [See also 14E15]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 57R45: Singularities of differentiable mappings

blow-analytic equivalence tree model Puiseux pairs C1 equivalence bi-Lipschitz equivalence


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.

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