Alain Haraux
Universitatis Iagellonicae Acta Mathematica, Volume 49, 2011, pp. 45-57
https://doi.org/10.4467/20843828AM.12.003.0455Let K be the real or the complex field, and let f : Kn → K be a quasi-homogeneous polynomial with weight w := (w1;w2;...,wn) and degree d. Assume that rf(0) = 0. Łojasiewicz well known gradient inequality states that there exists an open neighbourhood U of the origin in Kn and two positive constants c and p < 1 such that for any x → U we have rf(x) > cf(x)p: We prove that if the set K - (f) of points where the Fedoryuk condition fails to hold is nite, then the gradient inequality holds true with p = 1-minj wj/d. It is also shown that if n = 2; then K-(f) is either empty or reduced to {0}.