On gradient at infinity of semialgebraic functions

Didier D'Acunto; Vincent Grandjean

Annales Polonici Mathematici, Tome 85 (2005), p. 39-49 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $ϱ_{c} \leq 1$ such that |x|·|∇f| and ${| f (x) - c |}^{ϱ_{c}}$ are separated at infinity. If c is a regular value and $ϱ_{c} < 1$ , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.

Publié le : 2005-01-01

Zbl 1092.32004

EUDML-ID : urn:eudml:doc:280497

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Didier D'Acunto; Vincent Grandjean. On gradient at infinity of semialgebraic functions. Annales Polonici Mathematici, Tome 85 (2005) pp. 39-49. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-4/