Effective {\L}ojasiewicz gradient inequality and finite determinacy of non-isolated Nash function singularities

Abstract

Let $X\subset \mathbb{R}^n$ be a compact semialgebraic set and let $f:X\to \mathbb{R}$ be a nonzero Nash function. We give a Solern\'o and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the {\L}ojasiewicz 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 functions singularities