no. 4
Permutations encoding the local shape of level curves of real polynomials via generic projections
[Permutations qui codent la forme locale des courbes de niveau des polynômes réels via des projections génériques]
Sorea, Miruna-Ştefana1, 2
1 RCMA Lucian Blaga University Str. Doctor Ion Rațiu 5-7 Sibiu 550012 (Romania)
2 SISSA Scuola Internazionale Superiore di Studi Avanzati Geometry and Mathematical Physics via Bonomea, 265 - 34136 Trieste (Italy)
Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1661-1703.

La non-convexité d’une composante connexe lisse et compacte d’une courbe algébrique réelle plane peut être mesurée par un objet combinatoire appelé l’arbre de Poincaré–Reeb associé à la courbe et à une direction de projection. Dans cet article, nous montrons que si la projection choisie évite les bitangentes et les tangences d’inflexions aux courbes de niveaux suffisamment petits d’une fonction polynomiale réelle bivariée, proche d’un minimum local strict à l’origine, alors asymptotiquement l’arbre de Poincaré–Reeb devient un arbre binaire complet et ses sommets sont munis d’une relation d’ordre total. Une telle direction de projection est appelée générique. Nous prouvons que pour toute famille asymptotique de courbes de niveau, il y a un nombre fini d’intervalles sur la droite réelle projective en dehors desquelles toutes les directions sont génériques par rapport à toutes les courbes de la famille. Si le choix de la direction de projection est générique, on peut coder la forme locale des courbes en utilisant des permutations alternées, que nous appelons serpents. Les serpents offrent une description efficace de la géométrie et de la topologie locale, bien adaptée à d’autres calculs.

The non-convexity of a smooth compact connected component of a real algebraic plane curve can be measured by a combinatorial object: the Poincaré–Reeb tree associated to the curve and to a direction of projection. Here we show that if the chosen projection avoids the bitangents and the inflections to the small enough level curves of a real bivariate polynomial function near a strict local minimum at the origin, then the asymptotic Poincaré–Reeb tree becomes complete binary and its vertices become totally ordered. Such a projection direction is called generic. We prove that for any such asymptotic family of level curves, there are finitely many intervals on the real projective line outside of which all the directions are generic with respect to all the curves in the family. If the projection is generic, then the local shape of the curves can be encoded in terms of alternating permutations, called snakes. Snakes offer an effective description of the local geometry and topology, well-suited for computations.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3479
Classification : 14P25, 14P05, 14H20, 14B05, 05C05, 14Q05, 26C, 58K, 14Q05
Keywords: generic projection, real algebraic curve, Poincaré–Reeb tree, permutation, snake, polar curve, bitangent, inflection, dual curve, strict local minimum
Mot clés : projection générique, courbe algébrique réelle, l’arbre de Poincaré–Reeb, permutation, serpent, courbe polaire, bitangente, inflexion, courbe duale, minimum local strict
Sorea, Miruna-Ştefana 1, 2

1 RCMA Lucian Blaga University Str. Doctor Ion Rațiu 5-7 Sibiu 550012 (Romania)
2 SISSA Scuola Internazionale Superiore di Studi Avanzati Geometry and Mathematical Physics via Bonomea, 265 - 34136 Trieste (Italy)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Sorea, Miruna-Ştefana. Permutations encoding the local shape of level curves of real polynomials via generic projections. Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1661-1703. doi : 10.5802/aif.3479. https://aif.centre-mersenne.org/articles/10.5802/aif.3479/

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