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$% D. D’Acunto and V. Grandjean, A gradient inequality at infinity for tame functions, Rev. Mat. Complut. 18 (2005), no. 2, 493–501.%$

A Gradient Inequality at Infinity for Tame Functions

Didier DACUNTO and Vincent GRANDJEAN
Dipartimento di Matematica
Universitàá degli Studi di Pisa
Via Filippo Buonarotti 2
56127 Pisa Italia
Department of Computer Science
University of Bath
BATH BA2 7AY England (UK)

Received: January 17, 2005
Accepted: April 28, 2005

Let f  be a C1  function defined over Rn  and definable in a given o-minimal structure M 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  2
C  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: Łojasiewicz inequality, asymptotic critical values, bifurcation values, gradient trajectories, o-minimal structures.
2000 Mathematics Subject Classification: Primary 03C64, 34A26; Secondary 34C08.