An explicit bound for the Łojasiewicz exponent of real polynomials

Abstract

Let f : Rⁿ → R be a polynomial function of degree d with f(0) = 0. The classical Łojasiewiz inequality states that there exist c > 0 and α > 0 such that |f(x)| ≥ cd(x, f^–1(0))^α in a neighbourhod of the origin 0 $\in$ Rⁿ, where d(x, f^–1 (0)) denotes the distance from x to the set f^–1(0). We prove that the smallest such exponent α is not greater than R(n, d) with R(n, d) := max{d(3d – 4)^n–1, 2d(3d – 3)^n–2}.