On gradient at infinity of semialgebraic functions

• Autorzy: Didier D'Acunto , Vincent Grandjean
• Języki publikacji: EN

Abstrakty

EN

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 $ϱ_c ≤ 1$ 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.

Kategorie tematyczne

• 34Cxx: Qualitative theory
• 32Sxx: Singularities
• 32Bxx: Local analytic geometry
• 14P10: Semialgebraic sets and related spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2005
• Tom: 87
• Numer: 1
• Strony: 39-49

Daty

wydano

2005

Twórcy

autor

Didier D'Acunto

• Dipartimento di Matematica, Università degli Studi di Pisa, Via Filippo Buonarotti 2, 56127 Pisa, Italy

autor

Vincent Grandjean

• Department of Computer Science, University of Bath, Bath BA2 7AY, England, UK

Identyfikatory

DOI

10.4064/ap87-0-4