Milnor number of an invariant singularity: generalization of Chulkov's inequality

TL;DR

This paper establishes a lower bound for the Milnor number of invariant function germs under finite abelian group actions, demonstrating tightness and stability conditions for functions with multiple variables.

Contribution

It generalizes Chulkov's inequality to invariant singularities and characterizes those reaching the bound as equivariantly stable Morse-like singularities.

Findings

Lower bound for Milnor number established

Bound is tight for functions with many variables

Invariant stable Morse-like singularities identified

Abstract

We prove a lower bound for the Milnor number of function germ invariant with respect to a finite abelian group action. It is shown that this bound is tight for functions of arbitrarily many variables. We also prove the function germs that reach this lower bound are equivariantly stable, i.e. invariant analogues of Morse singularities.