  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.
