On gradient at infinity of semialgebraic functions

Volume 87 / 2005

Didier D'Acunto, Vincent Grandjean Annales Polonici Mathematici 87 (2005), 39-49 MSC: Primary 32Bxx, 34Cxx; Secondary 32Sxx, 14P10. DOI: 10.4064/ap87-0-4


Let $f: \mathbb{R}^n \to\mathbb{R}$ be a $C^2$ semialgebraic function and let $c$ be an asymptotic critical value of $f$. We prove that there exists a smallest rational number $\varrho_c\leq 1$ such that $|x|\cdot |\nabla f|$ and $|f(x)-c|^{\varrho_c}$ are separated at infinity. If $c$ is a regular value and $\varrho_c< 1$, then $f$ is a locally trivial fibration over $c$, and the trivialisation is realised by the flow of the gradient field of $f$.


  • Didier D'AcuntoDipartimento di Matematica
    Università degli Studi di Pisa
    Via Filippo Buonarotti 2
    56127 Pisa, Italy
  • Vincent GrandjeanDepartment of Computer Science
    University of Bath
    Bath BA2 7AY, England, UK

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