On the gradient of quasi-homogeneous polynomials (Q2836944)

Let \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{K}=\mathbb{C}\) and \(f:(\mathbb{K} ^{n},0)\rightarrow (\mathbb{K},0)\) be an analytic function germ. Assume that \(\nabla f(0)=0\), i.e., \(f\) is a singularity (not necessary isolated). The Łojasiewicz gradient exponent is the best (the smallest) exponent \(\rho \) such that there exists a constant \(c>0\) for which the inequality \(\|\nabla f(x)\| \geq c\|f(x)\|^{\rho }\) holds in a neighbourhood of the origin in \(\mathbb{K}^{n}.\)NEWLINENEWLINEThe main theorem is: Let \(f:\mathbb{K}^{n}\rightarrow \mathbb{K}\) be a quasi-homogeneous polynomial with weight \(w=(w_{1},\dots ,w_{n})\in (\mathbb{N}-\{0\})^{n}\) and degree \(d>1\) \((d\in \mathbb{N}).\) If the set \(\widetilde{K}_{\infty }(f)\) is finite, then \(\rho \left( f\right) \leq 1-\frac{\min (w_{j})}{d}.\) By definition NEWLINE\[NEWLINE \widetilde{K}_{\infty }(f)=\{\lambda \in \mathbb{K}:\exists x^{k}\rightarrow \infty ,\;f(x^{k})\rightarrow \lambda \text{ and }\|\nabla f(x^{k})\| \rightarrow 0\}, NEWLINE\]NEWLINE the set of points where the Fedoryuk condition fails to hold.