A Gradient Inequality at Infinity for Tame Functions

Recibido: 17 de enero de 2005 Aceptado: 28 de abril de 2005 Let f be a C1 function defined over Rn and definable in a given o-minimal structureM expanding the real field. We prove here a gradient-like inequality at infinity in a neighborhood of an asymptotic critical value c. When f is C2 we use this inequality to discuss the trivialization by the gradient flow of f in a neighborhood of a regular asymptotic critical level. Key words: Lojasiewicz inequality, asymptotic critical values, bifurcation values, gra-dient trajectories, o-minimal structures