The Łojasiewicz exponent of an analytic function at an isolated zero

Abstract

Let f be a real analytic function defined in a neighborhood of $0 \in {\Bbb R}^n$ such that $f^{-1}(0)=\{0\}$ . We describe the smallest possible exponents !, #, / for which we have the following estimates: $|f(x)|\geq c|x|^{\alpha}$ , $|{\rm grad}\,f(x)|\geq c|x|^{\beta}$ , $|{\rm grad}\,f(x)|\geq c|f(x)|^{\theta}$ for x near zero with $c > 0$ . We prove that $\alpha=\beta+1$, $\theta=\beta/\alpha$ . Moreover $\beta=N+a/b$ where $0 h a < b h N^{n-1}$ . If f is a polynomial then $|f(x)|\geq c|x|^{(\deg f-1)^n+1}$ in a small neighborhood of zero.