Effective Łojasiewicz gradient inequality and finite determinacy of non-isolated Nash function singularities

Abstract

Let X ⊂ R n be a compact semialgebraic set and let f : X → R be a nonzero Nash function. We give a Solernó and D'Acunto-Kurdyka type estimation of the exponent ̺ ∈ [0, 1) in the Łojasiewicz gradient inequality |∇f (x)| ≥ C|f (x)| ̺ for x ∈ X, |f (x)| < ε for some constants C, ε > 0, in terms of the degree of a polynomial P such that P (x, f (x)) = 0, x ∈ X. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash functions singularities.