The Łojasiewicz exponent of an analytic function at an isolated zero (Q1125555)

Summary: 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 \(\alpha,\beta,\theta\) for which we have the following estimates: \[ \bigl|f(x)\bigr |\geq c|x|^\alpha,\;\bigl|\text{grad} f(x)\bigr |\geq c|x|^\beta,\;\bigl|\text{grad} f(x)\bigr |\geq c\bigl|f(x)\bigr|^\theta \] for \(x\) near zero with \(c>0\). We prove that \(\alpha=\beta+1\), \(\theta=\beta/\alpha\). Moreover \(\beta=N +a/b\) where \(0\leq a<b\leq N^{n-1}\). If \(f\) is a polynomial then \(|f(x)|\geq c|x|^{(\deg f-1)^n+1}\) in a small neighborhood of zero.