Horizontal gradient of polynomials
This paper deals with trajectories of sub-Riemannian (also called horizontal) gradient of polynomials. Given \(p\) analytic vector fields \(X_1, \dots, X_{p}\) on \({\mathbb R}^n\), the horizontal gradient of a function \(f \in C^\infty ({\mathbb R}^n, {\mathbb R})\) is defined as \[ \nabla^h f = \sum_{i=1}^{p} (X_i f) X_i. \] A well-known consequence of the gradient Łojasiewicz inequality is that the length of the bounded trajectories \({\mathbf x}(t)\) of the (Riemannian) gradient \(\nabla f\) of an analytic function \(f:U \subset {\mathbb R}^n \to {\mathbb R}\) have a limit point and are uniformly bounded by a constant. In the sub-Riemannian case this is no longer true. In fact, the authors show an example where the trajectories of the horizontal gradient are not uniformly bounded and another one with trajectories accumulating on a closed curve. However, for the class of sub-Riemannian metrics coming from splitting distributions (which contains those of Heisenberg and Martinet), the trajectories of the horizontal gradient of a generic polynomial \(f\) (that is, for \(f\) in a Zariski open and dense subset of the space of polynomials of degree at most \(d\) in \(n\) variables) have the properties already mentioned for the Riemannian case. For this class of sub-Riemannian metrics the distribution is generated by \(p=n-1\) polynomial vector fields of the form: \[ X_i = \frac{\partial}{\partial x_i} + P_i \frac{\partial}{\partial x_n}, \; i= 1, \dots, n-1 \] where the \(P_i\) are polynomials in the variables \((x_1, \dots, x_n)\).
scientific article; zbMATH DE number 5641406
scientific article