The ?ojasiewicz exponent of an analytic function at an isolated zero

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 α, β, θ 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 \( 0 ≤ a < b ≤ N^{n-1} \). If f is a polynomial then \( |f(x)|\geq c|x|^{(\deg f-1)^n+1} \) in a small neighborhood of zero.