# Multiplicity and degree as bi-Lipschitz invariants for complex sets

## TL;DR

This paper investigates how multiplicity and degree, key invariants of complex sets, remain unchanged under outer bi-Lipschitz transformations, establishing their equivalence and invariance for curves and surfaces.

## Contribution

It demonstrates the invariance of multiplicity and degree under bi-Lipschitz homeomorphisms, linking local and global invariants and analyzing their behavior for complex curves and surfaces.

## Key findings

- Invariance of multiplicity is equivalent to invariance of degree.
- Proves invariance of tangent cone and relative multiplicities at infinity.
- Topology of homogeneous surface germs determines their multiplicity.

## Abstract

We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topology of a homogeneous surface germ determines its multiplicity.