On Zariski's multiplicity problem at infinity

TL;DR

This paper investigates the invariance of the degree of complex algebraic sets at infinity under bi-Lipschitz homeomorphisms, establishing conditions under which degree remains constant, thus contributing to the understanding of Zariski's multiplicity problem at infinity.

Contribution

It proves that the degree is a bi-Lipschitz invariant at infinity under certain conditions and relates various equivalences at infinity to degree preservation.

Findings

Degree is invariant under bi-Lipschitz homeomorphisms with small Lipschitz constants.

Bi-Lipschitz equisingular families at infinity have constant degree.

Polynomials with weakly rugose or bi-Lipschitz equivalences at infinity share the same degree.

Abstract

We address a metric version of Zariski's multiplicity conjecture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, we prove that the degree is a bi-Lipschitz invariant at infinity when the bi-Lipschitz homeomorphism has Lipschitz constants close to 1. In particular, we have that a family of complex algebraic sets bi-Lipschitz equisingular at infinity has constant degree. Moreover, we prove that if two polynomials are weakly rugose equivalent at infinity, then they have the same degree. In particular, we obtain that if two polynomials are rugose equivalent at infinity or bi-Lipschitz contact equivalent at infinity or bi-Lipschitz right-left equivalent at infinity, then they have the same degree.