×

On the local Harnack’s theorem. (Sur le théorème local de Harnack.) (French) Zbl 0934.14037

The note is devoted to the following result. Let \(f\) be an irreducible germ of a plane real curve, \(\mu\) its Milnor number, and \(B\) a Milnor ball for \(f\). Let \(\alpha\) and \(c\) be nonnegative integers such that \(\alpha+c\leq\mu/2\). Then there exists a deformation of \(f\) having \(\alpha\) ovals and \(c\) nondegenerate isolated double points in \(B\cup\mathbb{R}^2\), and without other singular points in \(B\). In the particular case \(\alpha=\mu/2\), the statement was proved by J.-J. Risler [Invent. Math. 89, 119-137 (1987; Zbl 0672.14020)]. The proof of the result formulated above is obtained varying the parametrization of the germ.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14P25 Topology of real algebraic varieties
14H20 Singularities of curves, local rings

Citations:

Zbl 0672.14020
PDFBibTeX XMLCite
Full Text: DOI