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

Didier D'Acunto, and Krzysztof Kurdyka. "Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials." Annales Polonici Mathematici 87.1 (2005): 51-61. <http://eudml.org/doc/280831>.

@article{DidierDAcunto2005,
abstract = {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 $R(n,d) = d(3d-3)^\{n-1\}$.},
author = {Didier D'Acunto, Krzysztof Kurdyka},
journal = {Annales Polonici Mathematici},
keywords = {real polynomial; Lojasiewicz inequality; Lojasiewicz exponent; valley line},
language = {eng},
number = {1},
pages = {51-61},
title = {Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials},
url = {http://eudml.org/doc/280831},
volume = {87},
year = {2005},
}

TY - JOUR
AU - Didier D'Acunto
AU - Krzysztof Kurdyka
TI - Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials
JO - Annales Polonici Mathematici
PY - 2005
VL - 87
IS - 1
SP - 51
EP - 61
AB - 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 $R(n,d) = d(3d-3)^{n-1}$.
LA - eng
KW - real polynomial; Lojasiewicz inequality; Lojasiewicz exponent; valley line
UR - http://eudml.org/doc/280831
ER -