Modified analytic trivialization via weighted blowing up
Toshizumi Fukui · Journal of the Mathematical Society of Japan · 1992
We consider the classification of real function-germs.It is well-known that there are modulus near non-simple germ for the differentiable equivalence.For topological equivalence it does not cause modulus, but seems to be too weak to provide a workable theory.T.-C.Kuo introduced the notion of the modified analytic trivialization (MAT) for a family of function-germs in [4], and general- ized it naturally $[5, 6]$ .(We give the definition of it in more general form in \S 1, and also call it MAT.)The associated equivalence relation preserves computability, but is slightly weaker than bianalyticity and much stronger than homeomorphism.He showed a finite classification theorem for isolated singular- ities in $[5, 6]$ .The next problem to be considered would be to describe MAT constant strata explicitly or what kind of singularities form a modified analytic equivalence class ?Several authors have studied this problem, see $e.g$ .$[4,2,7]$ .In this paper we show a generalization of Kuo's theorem in [4], establishing MAT for a class of singularities in $R^{n}$ .As a consequence, we obtain that the $Briangon-Speder$ family, for example, admits a MAT.In [3], S. Koike showed that the $Briangon-Speder$ family does not preserve " tangency of arcs."Thus MAT does not preserve " tangency of arcs," as S. Koike conjectured before.