Lipschitz normally embedded set and tangent cones at infinity

TL;DR

This paper establishes that complex analytic sets with a unique tangent cone at infinity are algebraic, and for Lipschitz normally embedded algebraic sets, their degree equals that of their tangent cone at infinity.

Contribution

It proves the algebraicity of sets with a unique tangent cone at infinity and relates the degree of Lipschitz normally embedded sets to their tangent cones.

Findings

Analytic sets with a unique tangent cone at infinity are algebraic.

Degree of Lipschitz normally embedded algebraic sets equals the degree of their tangent cone at infinity.

Abstract

We prove that any analytic set in $\C^n$ with a unique tangent cone at infinity is an algebraic set. We prove that the degree of a complex algebraic set in $\C^n$, which is Lipschitz normally embedded at infinity, is equal to the degree of its tangent cone at infinity.