The Łojasiewicz numbers and plane curve singularities
Volume 87 / 2005
Annales Polonici Mathematici 87 (2005), 127-150
MSC: 32S05, 14H20.
DOI: 10.4064/ap87-0-11
Abstract
For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent $\mathcal{L}_{0}(f)$ defined to be the smallest $\theta>0$ such that $\vert {\rm grad} f(z)\vert \geq c \vert z\vert ^{\theta}$ near $0\in\mathbb{C}^{2}$ for some $c>0$. We investigate the set of all numbers $\mathcal{L}_{0}(f)$ where $f$ runs over all holomorphic functions with an isolated critical point at $0\in\mathbb{C}^{2}$.
