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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.

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