A criterion for topological equivalence of two variable complex analytic function germs

TL;DR

This paper establishes a precise criterion for when two complex analytic function germs in two variables are topologically equivalent, based on their zero set components and intersection properties.

Contribution

It provides a complete topological classification criterion for complex analytic function germs in two variables using geometric invariants.

Findings

Topological equivalence characterized by zero set component correspondence

Preservation of multiplicities, Puiseux pairs, and intersection numbers is necessary and sufficient

Offers a practical method for classifying complex analytic function germs

Abstract

We show that two analytic function germs $(\C^2,0) \to (\C,0)$ are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pairs of distinct components.