Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2005, April 28.

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
there exist C>0 and 0<ρ<1 such that |∇ f(x)| ≥ C|f|^ρ in a neighbourhood of the origin.  We prove that the smallest of
such exponents $\rho$ is not greater than  1- R(n,d)^-1 with  R(n,d)=d(3d-3)^n-1.

Mathematics Subject Classification (2000): 32Bxx, 34Cxx, 32Sxx, 14P10.

Keywords and Phrases: polynomials, Łojasiewicz inequality, Łojasiewicz exponent,  valley lines.

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