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Let f be a C function defined over R 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 C we use this inequality to discuss the trivialization by the gradient flow of f in a neighborhood of a regular asymptotic critical level.
Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number such that |x|·|∇f| and are separated at infinity. If c is a regular value and , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.
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