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

• Autorzy: Didier D'Acunto , Krzysztof Kurdyka
• Języki publikacji: EN

Abstrakty

EN

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

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: 51-61

Daty

wydano

2005

Twórcy

autor

Didier D'Acunto

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

autor

Krzysztof Kurdyka

• Laboratoire de Mathématiques LAMA, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France

Identyfikatory

DOI

10.4064/ap87-0-5