On the complex \L ojasiewicz inequality with parameter

TL;DR

This paper establishes a parameter-independent Łojasiewicz inequality for a family of holomorphic functions by proving a continuity property of currents, enhancing understanding of the inequality's stability under parameter variations.

Contribution

It introduces a continuity property of currents that leads to a uniform Łojasiewicz inequality with an effective exponent, independent of parameters.

Findings

Proved a continuity property of currents for holomorphic functions.

Derived a Łojasiewicz inequality with a parameter-independent exponent.

Enhanced the stability analysis of holomorphic function families.

Abstract

We prove a continuity property in the sense of currents of a continuous family of holomorphic functions which allows us to obtain a \L ojasiewicz inequality with an effective exponent independent of the parameter.