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

This paper considers real forms of closed algebraic ${\mathbb{C}}^{\ast }$-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 ${\mathbb{R}}^2$.