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


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.

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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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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).

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Mathematics Subject Classifications (2010)

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