Equivalence relations for two variable real analytic function germs

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

2008-01-17

arxiv.org/abs/0801.2650v1

Mathematics

Algebraic Geometry

30 pages, 7 figures

Scientific paper

For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to the 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.

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