A set on which the Lojasieewicz exponent at infinity is attained

TL;DR

This paper demonstrates that for polynomial mappings from complex n-space to m-space, the Lojasiewicz exponent at infinity is achieved on the zero set of the product of the component functions.

Contribution

It establishes that the Lojasiewicz exponent at infinity for polynomial maps is attained specifically on the set where the product of the component functions vanishes.

Findings

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

The result applies to polynomial mappings from C^n to C^m.

Provides a precise characterization of where the exponent is attained.

Abstract

We show that for a polynomial mapping F = (f_1,...,f_m): C^n \to C^m the Lojasiewicz exponent at infinity of F is attained on the set {z \in C^n : f_1(z)...f_m(z) = 0}