Łojasiewicz inequalities in a certain class of smooth functions
Volume 128 / 2024
Banach Center Publications 128 (2024), 69-93
MSC: Primary 58K55; Secondary 14P25, 32C99
DOI: 10.4064/bc128-5
Abstract
Let $f$ be a germ of a smooth function at the origin in $\mathbb {R}^n.$ We show that if $f$ is Kouchnirenko’s nondegenerate and satisfies the so called Kamimoto–Nose condition then it admits the Łojasiewicz inequalities. We compute the Łojasiewicz 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 Łojasiewicz exponents can be expressed very simply in terms of its Newton polyhedron.
