Multiplicity of singularities is not a bi-Lipschitz invariant

TL;DR

This paper disproves the conjecture that the multiplicity of a singularity remains invariant under bi-Lipschitz transformations by providing counterexamples with differing multiplicities.

Contribution

It introduces counterexamples showing that multiplicity is not preserved under bi-Lipschitz equivalence, challenging a longstanding conjecture in singularity theory.

Findings

Constructed bi-Lipschitz equivalent singularities with different multiplicities

Disproved the conjecture that multiplicity is a bi-Lipschitz invariant

Provided new insights into the structure of algebraic singularities

Abstract

It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture, constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.