Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Volume 87 / 2005

Didier D'Acunto, Krzysztof Kurdyka Annales Polonici Mathematici 87 (2005), 51-61 MSC: Primary 32Bxx, 34Cxx, Secondary 32Sxx, 14P10. DOI: 10.4064/ap87-0-5


Let $f:\mathbb{R}^n\to\mathbb{R}$ be a polynomial function of degree $d$ with $f(0)=0$ and $\nabla f(0)=0$. Łojasiewicz's gradient inequality states that there exist $C>0$ and $\varrho\in (0,1)$ such that $|\nabla f|\geq C|f|^{\varrho}$ in a neighbourhood of the origin. We prove that the smallest such exponent $\varrho$ is not greater than $1- R(n,d)^{-1}$ with $R(n,d)= d(3d-3)^{n-1}$.


  • Didier D'AcuntoDipartimento di Matematica
    Università degli Studi di Pisa
    Via Filippo Buonarroti, 2
    56127 Pisa, Italy
  • Krzysztof KurdykaLaboratoire de Mathématiques LAMA
    UMR 5127 CNRS, Université de Savoie
    73376 Le Bourget-du-Lac Cedex, France

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