Effective Łojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs

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 ϱ ∈ [0,1) in the Łojasiewicz gradient inequality |∇𝑓(𝑥)| ≥ 𝐶|𝑓(𝑥)|^ϱ 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 function singularities.