On the uniqueness of polynomial embeddings of the real 1-sphere in the plane

TL;DR

This paper proves that, up to coordinate change, there is a unique polynomial embedding of the real 1-sphere in the plane, building on recent classification results of algebraic embeddings.

Contribution

It establishes the uniqueness of polynomial embeddings of the real 1-sphere in the plane, based on prior classification of algebraic embeddings.

Findings

Only one polynomial embedding of the real 1-sphere exists up to coordinate change.

The result relies on recent classification of algebraic $ ext{C}^*$-embeddings.

Confirms the rigidity of polynomial embeddings of the real 1-sphere in the plane.

Abstract

This paper considers real forms of closed algebraic $\mathbb{C}^*$-embeddings in $\mathbb{C}^2$. The classification of such embeddings was recently completed by Cassou-Nogues, Koras, Palka and Russell. Based on their classification, this paper shows that, up to an algebraic change of coordinates, there is only one polynomial embedding of the real 1-sphere $\mathbb{S}^1$ in the affine plane $\mathbb{R}^2$.