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

TL;DR

This paper investigates Łojasiewicz inequalities for non-degenerate holomorphic and mixed functions, providing explicit estimates of Łojasiewicz exponents and extending these results to mixed functions under certain non-degeneracy conditions.

Contribution

It offers explicit Łojasiewicz exponent estimations for non-degenerate holomorphic functions and extends these results to mixed functions, including weighted homogeneous polynomials.

Findings

Explicit estimation of Łojasiewicz exponents for holomorphic functions with isolated singularities.

Improved estimation for weighted homogeneous polynomials under non-degeneracy conditions.

Generalization of Łojasiewicz inequalities to strongly non-degenerate 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.