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Horizontal Gradient of Polynomial Functions for the Standard Engel Structure on ℝ4

Journal of Dynamical and Control Systems volume 22, pages 15–34 (2016)Cite this article

Abstract

We investigate the set V f of horizontal critical points of a polynomial function f for the standard Engel structure defined by the 1-forms ω 3 = d x 3 − x 1 d x 2 and ω 4 = d x 4 − x 3 d x 2, endowed with the sub-Riemannian metric \(g_{\text {SR}}=d{x_{1}^{2}}+d{x^{2}_{2}}\). For a generic polynomial, we show that the set Γ f of points in V f , where V f is not transverse to the Engel distribution, does not have a connected component which is contained in a fiber of f. Then, we prove that each trajectory of the horizontal gradient of f approaching the set V f has a limit.

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Acknowledgments

This work was supported by the Fields Institute, ANR grant STAAVF (France), LIA Formath Vietnam Project, VAST and Vietnam National Foundation for Science and Technology Development (NAFOSTED) grant 101.04-2014.23.

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Authors and Affiliations

  1. Institute of Mathematics, VAST 18 Hoang Quoc Viet Road, Cau Giay District, 10307, Hanoi, Vietnam

    Si Tiep Dinh

  2. Laboratoire de Mathématiques (LAMA) UMR-5127 CNRS, Campus Scientifique, Université de Savoie, 73376, Le Bourget-du-Lac cedex, France

    Krzysztof Kurdyka

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  1. Si Tiep Dinh
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Correspondence to Krzysztof Kurdyka.

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Dinh, S.T., Kurdyka, K. Horizontal Gradient of Polynomial Functions for the Standard Engel Structure on ℝ4 . J Dyn Control Syst 22, 15–34 (2016). https://doi.org/10.1007/s10883-014-9242-7

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Mathematics Subject Classifications (2010)

  • 14P10
  • 53C17
  • 58A30
  • 58Kxx