Comptes Rendus
Théorie des singularités
Contact unimodal map germs from the plane to the plane
Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 923-930.

In this article, we correct the classification of unimodal map germs from the plane to the plane of Boardman symbol (2,2) given by Dimca and Gibson. Also, we characterize this classification of unimodal map germs in terms of certain invariants. Moreover, on the basis of this characterization we present an algorithm to compute the type of unimodal map germs of the Boardman symbol (2,2) without computing the normal form and give its implementation in the computer algebra system Singular [8].

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DOI : 10.5802/crmath.114
Classification : 58Q05, 14H20
Muhammad Ahsan Binyamin 1 ; Saima Aslam 1 ; Khawar Mehmood 2

1 Department of Mathematics, GC University, Faisalabad, Pakistan
2 Department of Applied Mathematics & Statistics, Institute of Space Technology P.O.Box 2750, Islamabad 44000, Pakistan
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Muhammad Ahsan Binyamin; Saima Aslam; Khawar Mehmood. Contact unimodal map germs from the plane to the plane. Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 923-930. doi : 10.5802/crmath.114. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.114/

[1] Deeba Afzal; Shamsa Kanwal; Gerhard Pfister Tangent space of the orbit of an algebraic group action, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér., Volume 61 (2018) no. 109, pp. 135-146 | MR | Zbl

[2] Vladimir I. Arnolʼd Singularities of systems of rays, Russ. Math. Surv., Volume 38 (1983) no. 2, pp. 87-176 | Zbl

[3] Saima Aslam; Muhammad Ahsan Binyamin; Gerhard Pfister Recognition of unimodal map germs from the plane to the plane by invariants, Int. J. Algebra Comput., Volume 28 (2018) no. 7, pp. 1199-1208 | DOI | MR | Zbl

[4] Muhammad Ahsan Binyamin; Saima Aslam; Hasan Mahmood; K. Mehmood; M. Ishaq Recognition of contact map germs from the plane to the plane by invariants, C. R. Acad. Bulg. Sci., Volume 73 (2020) no. 6, pp. 758-766 | Zbl

[5] Muhammad Ahsan Binyamin; Hasan Mahmood; Shamsa Kanwal On the classification of simple maps from the plane to the plane, J. Algebra Appl., Volume 16 (2017) no. 10, 1750199, 8 pages | MR | Zbl

[6] James W. Bruce Projections and reflections of generic surfaces in 3 , Math. Scand., Volume 54 (1984), pp. 262-278 | DOI | MR

[7] James W. Bruce; Peter J. Giblin Outlines and their duals, Proc. Lond. Math. Soc., Volume 50 (1985), pp. 552-570 | DOI | MR

[8] W. Decker; Gert-Martin Greuel; Gerhard Pfister; H. Schönemann Singular 4-1-0 — A computer algebra system for polynomial computations (http://www.singular.uni-kl.de)

[9] Alexandru Dimca; Christopher G. Gibson Contact unimodular germs from the plane to the plane, Q. J. Math, Volume 34 (1983), pp. 281-295 | DOI | MR | Zbl

[10] Gert-Martin Greuel; Gerhard Pfister A Singular Introduction to Commutative Algebra, Springer, 2007 | Zbl

[11] Yutaro Kabata Recognition of plane-to-plane map-germs, Topology Appl., Volume 202 (2016), pp. 216-238 | Zbl

[12] Joachim H. Rieger Families of maps from the plane to the plane, J. Lond. Math. Soc., Volume 36 (1987) no. 1-2, pp. 351-369 | DOI | MR | Zbl

[13] Joachim H. Rieger A-unimodal map-germs into the plane, Hokkaido Math. J., Volume 33 (2004) no. 1, pp. 47-64 | DOI | MR | Zbl

[14] Hassler Whitney On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane, Ann. Math., Volume 62 (1955), pp. 374-410 | DOI | MR | Zbl

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