Lojasiewicz inequalities in a certain class of smooth functions

Abstract

Let $f$ be a germ of a smooth function at the orirgin in $\RR^n.$ We show that if $f$ is Kouchnirenko's nondegenerate and satisfies the so called Kamimoto--Nose condition then it admits the \L ojasiewicz inequalities. We compute the \L ojasiewicz exponents for some special cases. In particular, if $f$ is a germ of a smooth convex Kouchnirenko's nondegenerate function and satisfies the Kamimoto--Nose condition, then all its \L ojasiewicz exponents can be expressed very simply in terms of its Newton polyhedron.