On the gradient of quasi-homogeneous polynomials

Let be the real or the complex field, and let f : ⁿ → be a quasi-homogeneous polynomial with weight w := (w1, w2, . . . , wₙ) and degree d. Assume that ∇f(0) = 0. Lojasiewicz’s well known gradient inequality states that there exists an open neighbourhood U of the origin in ⁿ and two positive constants c and ρ < 1 such that for any x ∈ U we have $ ||∇f(x)k|| ≥ c|f(x)|^\rho$. We prove that if the set $ \tilde{K}_\infty (f) $ of points where the Fedoryuk condition fails to hold is finite, then the gradient inequality holds true with $ \rho=1-\text{min}_j {w_j}/d $. It is also shown that if n = 2, then $ \tilde{K}_\infty (f) $ is either empty or reduced to {0}