On the special parabolic points and the topology of the parabolic curve of certain smooth surfaces in 3
[Sur les points paraboliques spéciaux et sur la topologie de la courbe parabolique des certaines surfaces lisses en 3 ]
Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 473-478.

Dans ce travail on donne une classe des polynômes réels de degré n⩾3 à deux variables tels que les courbes paraboliques de leurs graphes contiennent au moins (n−1)(n−2)/2 composantes connexes difféomorphes au cercle et exactement n(n−2) points paraboliques spéciaux. Ces polynômes sont de la forme f=l1ln où les li, i=1,…,n, sont des fonctions réelles affines génériques sur le plan.

In this Note we give a class of real polynomials of degree n⩾3 in two variables such that the parabolic curves of theirs graphs have at least (n−1)(n−2)/2 connected components diffeomorphic to the circle and exactly n(n−2) special parabolic points. These polynomials are of the form f=l1ln where li, i=1,…,n, are generic real affine functions on the plane.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02287-2
Ortiz-Rodríguez, Adriana 1

1 Équipe géométrie et dynamique, Université Paris 7, UFR de Math., case 7012, 175, rue du Chevaleret, 75013 Paris, France
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Ortiz-Rodríguez, Adriana. On the special parabolic points and the topology of the parabolic curve of certain smooth surfaces in $ \mathbb{R}^{3}$. Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 473-478. doi : 10.1016/S1631-073X(02)02287-2. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02287-2/

[1] F. Aicardi, Communication personnelle, Août 2001

[2] V.I. Arnold, Communication personnelle, Avril 2001

[3] do Carmo, M.P. Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976

[4] Harnack, A. Über Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann., Volume 10 (1876), pp. 189-199

[5] Landis, E.E. Tangential singularities, Funct. Anal. Appl., Volume 15 (1981), pp. 103-114

[6] Panov, D.A. Special points on surfaces in the three-dimensional projective space, Funct. Anal. Appl., Volume 34 (2000) no. 4, pp. 276-287

[7] Platonova, O.A. Singularities of the mutual disposition of a surface and a line, Russian Math. Surveys, Volume 36 (1981) no. 1, pp. 248-249

[8] Platonova, O.A. Projections des surfaces lisses, J. Soviet Math., Volume 10 (1984) no. 6, pp. 135-149

[9] Benedetti, R.; Risler, J.J. Real Algebraic and Semi-Algebraic Sets, Actualités Math., Hermann, 1990

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