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On strong $$C^ 0$$-equivalence of real analytic functions. (English) Zbl 0788.32024
The author introduces the notion of strong $$C^ 0$$-equivalence for real analytic germs $$(\mathbb{R}^ n,0) \to(\mathbb{R},0)$$ which is the usual $$C^ 0$$- equivalence preserving the tangency of analytic arcs at the origin $$0 \in \mathbb{R}^ n$$. Then he considers two $$\mu$$-constant one-parameter families of complex polynomial germs $x^{15}+xy^ 7+z^ 5+ty^ 6z,\;x^ 8+y^{16}+z^{16}+x^ 3yz^ 3+tx^ 5z^ 2,$ constructed by J. Briançon and J.-P. Speder [C. R. Acad. Sci, Paris, Ser. A 280, 365-367 (1975; Zbl 0331.32010)] and M. Oka [Contemp. Math. 90, 199- 210 (1989; Zbl 0682.32011)] respectively. Both families have a weak simultaneous resolution but have no strong simultaneous resolution. However, the author proves that in the real case these families are topologically trivial but not strongly $$C^ 0$$-trivial.

##### MSC:
 32S15 Equisingularity (topological and analytic) 26E05 Real-analytic functions 57R45 Singularities of differentiable mappings in differential topology
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