Permutations encoding the local shape of level curves of real polynomials via generic projections
Annales de l'Institut Fourier, Online first, 43 p.

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.

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.

Received:
Revised:
Accepted:
Online First:
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
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)
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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, Online first, 43 p.

[1] Altituderando, Comment lire une carte IGN de randonné ? (https://www.altituderando.com/Comment-lire-une-carte-IGN-de-randonnee)

[2] André, Désiré Développements de secx et de tangx, C. R. Math. Acad. Sci. Paris, Volume 88 (1879), pp. 965-967 | Zbl

[3] André, Désiré Sur les permutations alternées, Resal J. (3), Volume 7 (1881), pp. 167-184 | Zbl

[4] Arnold, Vladimir I. Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers of Coxeter groups, Usp. Mat. Nauk, Volume 47 (1992) no. 1(283), p. 3-45, 240 | DOI | Zbl

[5] Arnold, Vladimir I. Nombres d’Euler, de Bernoulli et de Springer pour les groupes de Coxeter et les espaces de morsification; le calcul des serpents, Leçons de mathématiques d’aujourd’hui (Le Sel et le Fer), Volume 4, Cassini, 2000, pp. 61-98 | Zbl

[6] Arwann, (http://arwann.com/?page_id=3869)

[7] Bank, Bernd; Giusti, Marc; Heintz, Joos; Safey El Din, Mohab; Schost, Éric On the geometry of polar varieties, Appl. Algebra Eng. Commun. Comput., Volume 21 (2010) no. 1, pp. 33-83 | DOI | Zbl

[8] Bassino, Frédérique; Bouvel, Mathilde; Féray, Valentin; Gerin, Lucas; Pierrot, Adeline The Brownian limit of separable permutations, Ann. Probab., Volume 46 (2018) no. 4, pp. 2134-2189 | DOI | Zbl

[9] Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics, 10, Springer, 2006 | Zbl

[10] Bodin, Arnaud Newton polygons and families of polynomials, Manuscr. Math., Volume 113 (2004) no. 3, pp. 371-382 | DOI | Zbl

[11] Bodin, Arnaud Jump of Milnor numbers, Bull. Braz. Math. Soc. (N.S.), Volume 38 (2007) no. 3, pp. 389-396 | DOI | Zbl

[12] Bose, Prosenjit; Buss, Jonathan Frederick; Lubiw, Anna Pattern matching for permutations, Inf. Process. Lett., Volume 65 (1998) no. 5, pp. 277-283 | DOI | Zbl

[13] Brieskorn, Egbert; Knörrer, Horst Plane algebraic curves, Birkhäuser, 1986 (translated from the German by John Stillwell) | DOI | Zbl

[14] Brugallé, Erwan A.; López de Medrano, Lucia M. Inflection points of real and tropical plane curves, J. Singul., Volume 4 (2012), pp. 74-103 | Zbl

[15] Castellini, Roberto The topology of A’Campo deformations of singularities: an approach through the lotus, Ph. D. Thesis, Université de Lille 1, Lille, France (2015) (https://www.theses.fr/2015LIL10062)

[16] Cayley, Arthur The collected mathematical papers. Volume 4, Cambridge Library Collection - Mathematics, Cambridge University Press, 2009 (Reprint of the 1891 original) | DOI | Zbl

[17] Cheng, Jinsan; Lazard, Sylvain; Peñaranda, Luis; Pouget, Marc; Rouillier, Fabrice; Tsigaridas, Elias On the topology of real algebraic plane curves, Math. Comput. Sci., Volume 4 (2010) no. 1, pp. 113-137 | DOI | Zbl

[18] Coste, Michel An introduction to semialgebraic geometry (2002), 78 pages (Notes published by RAAG network school, vol. 145, https://perso.univ-rennes1.fr/michel.coste/polyens/SAG.pdf)

[19] Coste, Michel; de la Puente, María Jesús Atypical values at infinity of a polynomial function on the real plane: an erratum, and an algorithmic criterion, J. Pure Appl. Algebra, Volume 162 (2001) no. 1, pp. 23-35 | DOI | Zbl

[20] Degtyarev, Alex; Itenberg, Ilia; Zvonilov, Victor Real trigonal curves and real elliptic surfaces of type I, J. Reine Angew. Math., Volume 686 (2014), pp. 221-246 | DOI | Zbl

[21] Fischer, Gerd Plane algebraic curves, Student Mathematical Library, 15, American Mathematical Society, 2001 (translated from the 1994 German original by Leslie Kay) | DOI | Zbl

[22] Flores, Arturo Giles; Teissier, Bernard Local polar varieties in the geometric study of singularities, Ann. Fac. Sci. Toulouse, Math., Volume 27 (2018) no. 4, pp. 679-775 | DOI | Zbl

[23] García Barroso, Evelia R. Invariants des singularités de courbes planes et courbure des fibres de Milnor, Ph. D. Thesis, Universidad de La Laguna, San Crist’obal, Spain (1996) (http://ergarcia.webs.ull.es/tesis.pdf)

[24] García Barroso, Evelia R. Un théorème de décomposition pour les polaires génériques d’une courbe plane, C. R. Math. Acad. Sci. Paris, Volume 326 (1998) no. 1, pp. 59-62 | DOI | Zbl

[25] García Barroso, Evelia R. Sur les courbes polaires d’une courbe plane réduite, Proc. Lond. Math. Soc., Volume 81 (2000) no. 1, pp. 1-28 | DOI | Zbl

[26] García Barroso, Evelia R.; González Pérez, Pedro D.; Popescu-Pampu, Patrick The valuative tree is the projective limit of Eggers–Wall trees, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, Volume 113 (2019) no. 4, pp. 4051-4105 | DOI | Zbl

[27] García Barroso, Evelia R.; González Pérez, Pedro D.; Popescu-Pampu, Patrick The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses, Handbook of geometry and topology of singularities. I (Cisneros-Molina, José Luis, ed.), Springer, 2020, pp. 1-150 | DOI | MR | Zbl

[28] Gelfand, Izrail M.; Kapranov, Mikhail M.; Zelevinsky, Andrei V. Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser, 1994 | DOI | Zbl

[29] Ghys, Étienne Intersecting curves (variation on an observation of Maxim Kontsevich), Am. Math. Mon., Volume 120 (2013) no. 3, pp. 232-242 | DOI | Zbl

[30] Ghys, Étienne A singular mathematical promenade, ENS Éditions, 2017 | Zbl

[31] Kaihnsa, Nidhi; Kummer, Mario; Plaumann, Daniel; Namin, Mahsa Sayyary; Sturmfels, Bernd Sixty-four curves of degree six, Exp. Math., Volume 28 (2019) no. 2, pp. 132-150 | DOI | Zbl

[32] Kitaev, Sergey Patterns in permutations and words, Monographs in Theoretical Computer Science. An EATCS Series, Springer, 2011 (With a foreword by Jeffrey B. Remmel) | DOI | Zbl

[33] Knuth, Donald E. The art of computer programming. Vol. 1. Fasc. 1. The MMIX – a RISC computer for the new millennium, Addison-Wesley Publishing Group, 2005 | Zbl

[34] Lê, Dũng-Tráng; Michel, Françoise; Weber, Claude Sur le comportement des polaires associées aux germes de courbes planes, Compos. Math., Volume 72 (1989) no. 1, pp. 87-113 | Zbl

[35] Maugendre, Hélène Topologie comparée d’une courbe polaire et de sa courbe discriminante, Rev. Mat. Complut., Volume 12 (1999) no. 2, pp. 439-450 | DOI | Zbl

[36] Piene, Ragni Polar varieties revisited, Computer algebra and polynomials (Gutierrez, Jaime, ed.) (Lecture Notes in Computer Science), Volume 8942, Springer, 2015, pp. 139-150 | DOI | Zbl

[37] Plücker, Julius Sur les points singuliers des courbes, J. Math. Pures Appl., Volume 2 (1837), pp. 11-15

[38] Poncelet, Jean-Victor Théorie des polaires réciproques, Annales de Mathématiques pures et appliqués, Volume 8 (1817), pp. 201-232

[39] Popescu-Pampu, Patrick Le cerf-volant d’une constellation, Enseign. Math., Volume 57 (2011) no. 3-4, pp. 303-347 | DOI | Zbl

[40] Safey El Din, Mohab; Schost, Éric Polar varieties and computation of one point in each connected component of a smooth algebraic set, Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, Philadelphia, PA, USA, August 3–6, 2003 (2003), pp. 224-231 | DOI | Zbl

[41] Sorea, Miruna-Ştefana The shapes of level curves of real polynomials near strict local minima, Ph. D. Thesis, Université de Lille, France (2018) (https://hal.archives-ouvertes.fr/tel-01909028v1)

[42] Sorea, Miruna-Ştefana Constructing Separable Arnold Snakes of Morse Polynomials, Port. Math. (N.S.), Volume 77 (2020) no. 2, pp. 219-260 | DOI | Zbl

[43] Sorea, Miruna-Ştefana Measuring the local non-convexity of real algebraic curves, J. Symb. Comput., Volume 109 (2022), pp. 482-509 | DOI | Zbl

[44] Stanley, Richard P. A survey of alternating permutations, Combinatorics and graphs. Selected papers based on the presentations at the 20th anniversary conference of IPM on combinatorics, Tehran, Iran, May 15–21, 2009. Dedicated to Reza Khosrovshahi on the occasion of his 70th birthday (Brualdi, Richard A., ed.) (Contemporary Mathematics), Volume 531, American Mathematical Society, 2010, pp. 165-196 | DOI | Zbl

[45] Teissier, Bernard Variétés polaires. I. Invariants polaires des singularités d’hypersurfaces, Invent. Math., Volume 40 (1977) no. 3, pp. 267-292 | DOI | Zbl

[46] Teissier, Bernard Quelques points de l’histoire des variétés polaires, de Poncelet à nos jours, Séminaire d’Analyse, 1987–1988 (Clermont-Ferrand, 1987–1988), Univ. Clermont-Ferrand II, Clermont-Ferrand, 1990, 4 | Zbl

[47] Wall, Charles T. C. Singular points of plane curves, London Mathematical Society Student Texts, 63, Cambridge University Press, 2004 | DOI | Zbl

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