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

Didier D'Acunto; Krzysztof Kurdyka

Annales Polonici Mathematici, Tome 85 (2005), p. 51-61 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

Publié le : 2005-01-01

Zbl 1093.32011

EUDML-ID : urn:eudml:doc:280831

@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-5,
     author = {Didier D'Acunto and Krzysztof Kurdyka},
     title = {Explicit bounds for the \L ojasiewicz exponent in the gradient inequality for polynomials},
     journal = {Annales Polonici Mathematici},
     volume = {85},
     year = {2005},
     pages = {51-61},
     zbl = {1093.32011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-5}
}

Didier D'Acunto; Krzysztof Kurdyka. Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials. Annales Polonici Mathematici, Tome 85 (2005) pp. 51-61. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-5/