Un analogue local du théorème de Harnack. (A local analogue of Harnack's theorem) (Q1119984)

The result of Harnack referred to in the title of this paper, says that any smooth, real algebraic plane curve of degree d, has at most \((d-1)(d- 2)+1=g+1\) connected components, and for every d there exists a curve with that many connected components. The author shows in this paper, that any irreducible real plane curve singularity, at the origin, with Milnor number \(\mu\), may be deformed into a smooth curve with \(1/2\;\mu\) compact and 1 non-compact, connected components, in the neighborhood of the origin [see also the result of \textit{N. A'Campo}, in Math. Ann. 213, 1-32 (1975; Zbl 0316.14011)]. Since an easy argument shows that for a real plane curve singularity f, with r irreducible components, and Milnor number \(\mu\), the Milnor fibre \(f_{\epsilon}^{-1}(0)\cap B\) has at most \((\mu-r+1)\) compact components (ovals), and exactly r noncompact ones, the result above is, in the irreducible case, the best possible. - In the general situation, the author obtains a best possible result only when f has r distinct tangents. The proofs consist of a succession of intelligent applications of Harnack's construction.