A set on which the local Lojasiewicz exponent is attained

TL;DR

This paper proves that for a holomorphic map near zero, the Lojasiewicz exponent is achieved on the set where the product of its component functions vanishes, providing insights into the behavior of such mappings.

Contribution

It demonstrates that the local Lojasiewicz exponent for holomorphic mappings is attained on the zero set of the product of component functions.

Findings

The Lojasiewicz exponent is attained on the zero set of the product of component functions.

The result applies to holomorphic mappings near the origin in complex space.

Provides a geometric characterization of the Lojasiewicz exponent in this context.

Abstract

Let U be a neighbourhood of 0 \in C^n.We show that for a holomorphic mapping F = (f_1,...,f_m) : U -> C^m, F(0) = 0, the Lojasiewicz exponent of F at 0 is attained on the set {z \in U : f_1(z)...f_m(z) = 0}.