Real Algebraic and Analytic Geometry
Submission: 2005, April 28.
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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