Gradient horizontal de fonctions polynomiales

Dinh, Si Tiep; Kurdyka, Krzysztof; Orro, Patrice

Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1999-2042.

Nous étudions les trajectoires du gradient sous-riemannien (appellé horizontal) de fonctions polynômes. Dans ce cadre l’inégalité de Łojasiewicz n’est pas valide et une trajectoire du gradient horizontal peut être de longueur infinie, et peut même s’accumuler sur une courbe fermée. Nous montrons que ces comportement sont exceptionnels ; et que, pour une fonction générique les trajectoires de son gradient horizontal ont des propriétés similaires au cas du gradient riemannien. Pour obtenir la finitude des longueurs des trajectoires, nous changeons la métrique sous-riemanienne de façon convenable. Nous considérons une classe de distributions dites scindées, incluant celles d’Heisenberg et de Martinet. Pour un polynôme générique $f$ l’ensemble $V_f$ des points critiques horizontaux de $f$ est un ensemble algébrique lisse de dimension $1$ ou est vide et la restriction ${f|}_{V_f}$ est une fonction de Morse. Nous montrons aussi que pour un polynôme générique $f$, chaque trajectoire du gradient horizontal (qui approche $V_f$) possède une limite comme dans le cas riemannien étudié par S. Łojasiewicz.

We study trajectories of sub-Riemannian (also called horizontal) gradient of polynomials. In this setting Łojasiewicz’s gradient inequality does not hold and a trajectory of a horizontal gradient may be of infinite length, moreover it may accumulate on a closed curve. We show that these phenomena are exceptional; for a generic polynomial function the behavior of the trajectories of horizontal gradients are similar to the behavior of the trajectories of a Riemannian gradient. To obtain the finiteness of the length of trajectories we change suitably the sub-Riemannian metric. We consider a class of splitting distributions which contains those of Heisenberg and Martinet. For a generic polynomial $f$ the set $V_f$ of horizontal critical points, is a smooth algebraic set of dimension $1$ or the empty set, moreover ${f|}_{V_f}$ is a Morse function. We show that for a generic polynomial function any trajectory of the horizontal gradient (which approaches $V_f$) has a limit, as in the Riemannian case studied by S. Łojasiewicz.

Classification : 14P10, 53C17, 58Kxx, 58A30, 58K14, 93F14
Mot clés : semi-algébrique, sous-riemannien, généricité, gradient, inégalité de Łojasiewicz
Keywords: Sub-Riemannian, gradient, inegality

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