Journal of Dynamical and Control Systems

, Volume 22, Issue 1, pp 15–34 | Cite as

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

  • Si Tiep Dinh
  • Krzysztof KurdykaEmail author


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 3x 1 d x 2 and ω 4 = d x 4x 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.

Mathematics Subject Classifications (2010)

14P10 53C17 58A30 58Kxx 



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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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Laboratoire de Mathématiques (LAMA) UMR-5127 CNRS, Campus Scientifique, Université de SavoieLe Bourget-du-Lac cedexFrance

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