Pecker, Daniel On the local Harnack’s theorem. (Sur le théorème local de Harnack.) (French) Zbl 0934.14037 C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 5, 573-576 (1998). 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. Reviewer: I.Itenberg (Rennes) Cited in 1 Document MSC: 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14P25 Topology of real algebraic varieties 14H20 Singularities of curves, local rings Keywords:germ of a plane real curve; Milnor number Citations:Zbl 0672.14020 PDFBibTeX XMLCite \textit{D. Pecker}, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 5, 573--576 (1998; Zbl 0934.14037) Full Text: DOI