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

Abstract

Let $X \subset \mathbf{\mathbb{R}}^n$ be a compact semialgebraic set and let $f : X \to \mathbf{\mathbb{R}}$ be a nonzero Nash function. We give a Solernó and D'Acunto–Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the Łojasiewicz gradient inequality $|\nabla f(x)| \ge C|f(x)|^\varrho$ for $x \in X$, $|f(x)| < \varepsilon$ for some constants $C,\varepsilon > 0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x)) = 0$, $x \in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash function singularities.