zbMATH — the first resource for mathematics

Un analogue local du théorème de Harnack. (A local analogue of Harnack’s theorem). (French) Zbl 0672.14020
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 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.
Reviewer: O.A.Laudal

14H20 Singularities of curves, local rings
14N05 Projective techniques in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
Full Text: DOI EuDML
[1] [A] A’Campo, N.: Le groupe de monodromic du déploiement des singularités de courbes planes, I. Math. Ann. 213, 1-32 (1975) · Zbl 0316.14011
[2] [B-C-R] Bochnak, J., Coste, M., Roy, M.-F.: Géométri algébrique réelle, livre à paraître. Berlin-Heidelberg-New York: Springer 1987
[3] [B-C-T] Benedetti, R., Risler, J.-J., Tognoli, A.: Topologie des ensembles algébriques réels, livre à paraître. Paris: Hermann 1987
[4] [C] Chevallier, B.: Sur les courbes maximales de Harnack, note aux. C.R. Acad. Sci., Paris300, (no 4) 109-114 (1985) · Zbl 0587.14036
[5] [M] Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud.61, Princeton (1968) · Zbl 0184.48405
[6] [R1] Risler, J.-J.: Le théorème des zéros en géométrie algébrique et analytique réelle. Bull. Soc. Math. France104, 113-127 (1976)
[7] [R2] Risler, J.-J.: Sur le 16ieme problème de Hilbert: un résumé et quelques questions. Publ. Math. Univ. Paris7 (no 9), 11-25 (1981)
[8] [Z] Zariski, O.: Studies in equisingularity I. Am. J. Math.87, 507-586 (1965) · Zbl 0132.41601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.