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Singularity which has no \(M\)-smoothing. (English) Zbl 0978.14048
Bierstone, Edward (ed.) et al., The Arnoldfest. Proceedings of a conference in honour of V. I. Arnold for his 60th birthday, Toronto, Canada, June 15-21, 1997. Providence, RI: American Mathematical Society. Fields Inst. Commun. 24, 273-309 (1999).
The authors study a local version of the Harnack inequality. The principal local Harnack bound reads as follows: If the germ of a real analytic curve \(C\) has at least one real branch, the number of ovals of the smooothing of \(C\) is bounded from above by \(g=1/2(\mu-r+1)\), where \(\mu\) is the Milnor number and \(r\) the number of all complex branches of \(C\). One of the open questions posed by V. I. Arnol’d [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part 1, 57-69 (1983; Zbl 0519.58019)] asks whether this local Harnack bound is sharp. The authors show a counterexample using so-called Sirler cusp singularities. They improve and correct an earlier result of J.-J. Risler [Invent. Math. 89, 119-137 (1987; Zbl 0672.14020)] suggesting a refinement of the local Harnack inequality for singularities with several real branches.
For the entire collection see [Zbl 0929.00102].
Reviewer: Z.Hajto (Kraków)

MSC:
14P25 Topology of real algebraic varieties
14B05 Singularities in algebraic geometry
57R95 Realizing cycles by submanifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
14H20 Singularities of curves, local rings
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