On the limit set at infinity of a gradient trajectory of a semialgebraic function (Q863919)

Let \(f\) be a real analytic function defined over an open domain \(\Omega\) of \(\mathbb R^n\), endowed with the induced Euclidean structure. Let \(\Gamma\) be a maximal non-stationary trajectory of the gradient vector field \(\nabla f\). Then the author addresses the question of studying the behaviour of the trajectory of \(\Gamma\) nearby infinity (this question is motivated by the fact that such a trajectory may be never contained in any compact set of \(\mathbb R^n\)). Let \(y_0\in\Gamma\), let \(\Gamma^-(y_0)=\{x\in\Gamma: f(x)\leqslant f(y_0)\}\) and let \(\Gamma^+(y_0)=\{x\in\Gamma: f(x)\geqslant f(y_0)\}\). Any such semi-trajectory is denoted by \(\Gamma^-\) or by \(\Gamma^+\), respectively. Let us assume that \(\Gamma^+\) leaves any compact set of \(\mathbb R^n\). Then the author considers the problem of determining if the behaviour of \(\Gamma^+\), when going to infinity, is similar to that of the affine situation described by the gradient conjecture [see the article of \textit{K. Kurdyka, T. Mostowski} and \textit{A. Parusinski}, Ann. Math. (2) 152, No. 3, 763--792 (2000; Zbl 1053.37008)], that is, if the following limit exists: \[ \lim_{x\to \infty, x\in\Gamma^+}\frac{x}{| x|}. \] Let \(\widetilde \Gamma^+\) be the radial projection of \(\Gamma^+\) on the unit sphere \(\mathbb S^{n-1}\) centered at the origin of \(\mathbb R^n\). Then, the main result states that the length of \(\widetilde \Gamma^+\) is finite. As indicated by the author, the proof of this result follows the steps of Kurdyka, Mostowski and Parusinski (loc. cit.), but also uses some new specific ingredients, in particular a Łojasiewicz inequality at infinity near a value \(c\in\mathbb R\) called the Kurdyka-Łojasiewicz inequality at infinity at \(c\) and its related exponent (see Proposition 3.2 of the article of \textit{D. D'Acunto} and \textit{V. Grandjean} [Ann. Pol. Math. 87, 39--49 (2005; Zbl 1092.32004)]). The author also proves in the last section of the article a sufficient condition to trivialise the function \(f\) over a regular asymptotic critical value at infinity.