{\L}ojasiewicz exponents of non-degenerate holomorohic and mixed functions

Abstract

We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz exponent in a slightly weaker form than the assertion in Fukui.For a weighted homogeneous polynomial, we give a better estimation in the form which is conjectured by Brzostowski, Krasinski and Oleksik under under some condition (the \L ojasiewicz non-degeneracy). We also introduce \L ojasiewicz inequality for strongly non-degenerate mixed functions and generalize this estimation for mixed functions.