## On blow-analytic equivalence of embedded curve singularities.(English)Zbl 0899.32002

Fukuda, T. (ed.) et al., Real analytic and algebraic singularities. Harlow: Longman. Pitman Res. Notes Math. Ser. 381, 30-37 (1998).
A homeomorphism $$h:({\mathbb{R}}^2,0) \to ({\mathbb{R}}^2,0)$$ is blow-analytic if there exists a modification $$\pi: (X,E)\to ({\mathbb{R}}^2,0)$$ such that the composite $$h\circ \pi$$ is analytic and if the corresponding condition for $$h^{-1}$$ holds.
By a result of T. Fukui [Compos. Math. 105, No. 1, 95-108 (1997; Zbl 0873.32008)] the multiplicity of a germ of a curve is a blow-analytic invariant. This is no longer the case if one considers only the underlying topological spaces and ignores analytic structures. The main result of this paper is that the germ $$C$$ of any unibranch real plane curve singularity can be transformed into the germ of a smooth line by a blow-analytic homeomorphism. The proof is based on the fact that two embedded resolutions are equivalent if their weighted graphs are the same modulo 2.
For the entire collection see [Zbl 0882.00014].

### MSC:

 32C07 Real-analytic sets, complex Nash functions 32S15 Equisingularity (topological and analytic)

Zbl 0873.32008