logo AFST
Topologie et dénombrement des courbes algébriques réelles
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 383-422.

We describe the topology of singular real algebraic curves in a smooth surface. We enumerate and bound the number of topological types of singular algebraic curves in the real projective plane as a function of the degree.

Nous décrivons la topologie des courbes algébriques réelles singulières dans une surface lisse. Nous énumérons et bornons en fonction du degré le nombre de types topologiques de courbes algébriques singulières du plan projectif réel.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1698
Keywords: Courbe algébrique réelle, éclatement, classe combinatoire, décomposition de Cunningham d’un graphe, grammaire algébrique, opérade, combinatoire analytique
Christopher-Lloyd Simon 1

1 ENS de Lyon, 15 parvis René Descartes, 69342 Lyon Cédex 07, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AFST_2022_6_31_2_383_0,
     author = {Christopher-Lloyd Simon},
     title = {Topologie et d\'enombrement des courbes alg\'ebriques r\'eelles},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {383--422},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 31},
     number = {2},
     year = {2022},
     doi = {10.5802/afst.1698},
     zbl = {07549944},
     language = {fr},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1698/}
}
TY  - JOUR
TI  - Topologie et dénombrement des courbes algébriques réelles
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2022
DA  - 2022///
SP  - 383
EP  - 422
VL  - 6e s{\'e}rie, 31
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1698/
UR  - https://zbmath.org/?q=an%3A07549944
UR  - https://doi.org/10.5802/afst.1698
DO  - 10.5802/afst.1698
LA  - fr
ID  - AFST_2022_6_31_2_383_0
ER  - 
%0 Journal Article
%T Topologie et dénombrement des courbes algébriques réelles
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2022
%P 383-422
%V 6e s{\'e}rie, 31
%N 2
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1698
%R 10.5802/afst.1698
%G fr
%F AFST_2022_6_31_2_383_0
Christopher-Lloyd Simon. Topologie et dénombrement des courbes algébriques réelles. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 2, pp. 383-422. doi : 10.5802/afst.1698. https://afst.centre-mersenne.org/articles/10.5802/afst.1698/

[1] Cyril Banderier; Michael Drmota Formulae and asymptotics for coefficients of algebraic functions, Comb. Probab. Comput., Volume 24 (2015), pp. 1-53 | DOI | MR | Zbl

[2] Jacek Bochnak; Michel Coste; Marie-Françoise Roy Real Algebraic Geometry, Springer, 1992

[3] André Bouchet Reducing prime graphs and recognizing circle graphs, Combinatorica, Volume 7 (1987), pp. 243-254 | DOI | MR | Zbl

[4] André Bouchet Circle Graph Obstructions, J. Comb. Theory, Ser. B, Volume 60 (1994) no. 1, pp. 107-144 | DOI | MR | Zbl

[5] Henri Cartan Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. Fr., Volume 85 (1957), pp. 77-99 | DOI | Numdam | Zbl

[6] Henri Cartan Sur les fonctions de plusieurs variables complexes : les espaces analytiques, Congrès International des Mathématiciens (Edinburgh, 1958), 1960, pp. 33-52 | Zbl

[7] Sergei Chmutov; Sergeĭ Duzhin; Jonathan Mostovoy Introduction to Vassiliev Knot Invariants, Cambridge University Press, 2012 | DOI

[8] William Cunningham Decomposition of directed graphs, SIAM J. Algebraic Discrete Methods, Volume 3 (1982), pp. 214-228 | DOI | MR | Zbl

[9] Jean Dieudonné Cours de géométrie algébrique 1, Presses Universitaires de France, 1974

[10] Philippe Flajolet; Robert Sedgewick Analytic Combinatorics, Cambridge University Press, 2009 | DOI

[11] Benoit Fresse Homotopy of operads and Grothendieck-Teichmuller groups, Mathematical Surveys and Monographs, 217, American Mathematical Society, 2017

[12] Étienne Ghys A singular mathematical promenade, ENS Editions, 2017

[13] Étienne Ghys; Christopher-Lloyd Simon On the topology of a real analytic curve in the neighborhood of a singular point, Some aspects of the theory of dynamical systems : a tribute to Jean-Christophe Yoccoz (Astérisque), Volume 415, Société Mathématique de France, 2020, pp. 1-33 | Zbl

[14] Emeric Gioan; Christophe Paul Split decomposition and graph-labelled trees : characterizations and fully dynamic algorithms for totally decomposable graphs, Discrete Appl. Math., Volume 160 (2012) no. 6, pp. 708-733 | DOI | MR | Zbl

[15] Hans Grauert On Levi’s problem and the imbedding of real-analytic manifolds, Ann. Math., Volume 68 (1958), pp. 460-472 | DOI | MR | Zbl

[16] Viatcheslav Kharlamov; Stepan Orevkov Asymptotic growth of the number of classes of real plane algebraic curves as the degree grows, J. Math. Sci., New York, Volume 113 (2003) no. 5, pp. 666-674

[17] Viatcheslav Kharlamov; Stepan Orevkov The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves, J. Comb. Theory, Ser. A, Volume 105 (2004) no. 1, pp. 127-142 | DOI | MR | Zbl

[18] János Kollár Nash’s work in algebraic geometry, Bull. Am. Math. Soc., Volume 54 (2017) no. 2, pp. 307-324 | DOI | MR | Zbl

[19] Sergei Lando; Alexander Zvonkin Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004 | DOI

[20] Neil Sloane The On-Line Encyclopedia of Integer Sequences (https://oeis.org)

[21] William T. Tutte A census of slicings, Can. J. Math., Volume 14 (1962), pp. 708-722 | DOI | MR | Zbl

[22] Hassier Whitney Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc., Volume 36 (1934), pp. 63-89 | DOI | MR

Cited by Sources: