Comptes Rendus
Algebraic Geometry
On the number of connected components of the parabolic curve
Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 287-289.

We construct a polynomial of degree d in two variables whose Hessian curve has (d4)2 connected components using Viro patchworking. In particular, this implies the existence of a smooth real algebraic surface of degree d in RP3 whose parabolic curve is smooth and has d(d4)2 connected components.

À l'aide du patchwork de Viro, nous construisons un polyôme de degré d en deux variables dont la courbe Hessienne a (d4)2 composantes connexes. Cela implique en particulier l'existence d'une surface algébrique réelle de degré d dans RP3 dont la courbe parabolique, lisse, a d(d4)2 composantes connexes.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2010.01.028

Benoît Bertand 1; Erwan Brugallé 2

1 Institut mathématique de Toulouse, I.U.T. de Tarbes, 1 rue Lautréamont, BP 1624, 65016 Tarbes, France
2 Université Pierre et Marie Curie, Institut Mathématiques de Jussieu, 175 rue du Chevaleret, 75 013 Paris, France
@article{CRMATH_2010__348_5-6_287_0,
     author = {Beno{\^\i}t Bertand and Erwan Brugall\'e},
     title = {On the number of connected components of the parabolic curve},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {287--289},
     publisher = {Elsevier},
     volume = {348},
     number = {5-6},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.028},
     language = {en},
}
TY  - JOUR
AU  - Benoît Bertand
AU  - Erwan Brugallé
TI  - On the number of connected components of the parabolic curve
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 287
EP  - 289
VL  - 348
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2010.01.028
LA  - en
ID  - CRMATH_2010__348_5-6_287_0
ER  - 
%0 Journal Article
%A Benoît Bertand
%A Erwan Brugallé
%T On the number of connected components of the parabolic curve
%J Comptes Rendus. Mathématique
%D 2010
%P 287-289
%V 348
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2010.01.028
%G en
%F CRMATH_2010__348_5-6_287_0
Benoît Bertand; Erwan Brugallé. On the number of connected components of the parabolic curve. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 287-289. doi : 10.1016/j.crma.2010.01.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.028/

[1] V.I. Arnold Arnold's Problems, Springer Verlag, Berlin, 2004

[2] I.M. Gelfand; M.M. Kapranov; A.V. Zelevinsky Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994

[3] F. Klein Eine neue Relation zwischen den Singularitäten einer algebraischen Curve, Math. Ann., Volume 10 (1876) no. 2, pp. 199-209

[4] G. Mikhalkin Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology, Volume 43 (2004), pp. 1035-1060

[5] A. Ortiz-Rodriguez Quelques aspects sur la géométrie des surfaces algébriques réelles, Bull. Sci. Math., Volume 127 (2003), pp. 149-177

[6] A. Ortiz-Rofriguez; F. Sottile Real Hessian curves, Boletín de la Sociedad Mathemática Mexicana, Volume 13 (2007), pp. 157-166

[7] J.J. Risler Construction d'hypersurfaces réelles (d'après Viro), Séminaire Bourbaki, Volume 763 (1992) (in French)

[8] F. Ronga Klein's paper on real flexes vindicated (W. Pawlucki; B. Jakubczyk; J. Stasica, eds.), Singularities Symposium – Lojasiewicz 70, vol. 44, Banach Center Publications, 1998

[9] F. Schuh An equation of reality for real and imaginary plane curves with higher singularities, Proc. Section of Sciences of the Royal Academy of Amsterdam, Volume 6 (1903–1904), pp. 764-773

[10] O.Ya. Viro Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187-200

[11] O.Ya. Viro Some integral calculus based on Euler characteristic, Lecture Notes in Math., vol. 1346, Springer Verlag, 1988, pp. 127-138

[12] O.Ya. Viro Real plane algebraic curves: constructions with controlled topology, Leningrad Math. J., Volume 1 (1989) no. 5, pp. 1059-1134

Cited by Sources:

Comments - Policy