The Łojasiewicz exponent of an analytic function at an isolated zero

Abstract

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