Abhyankar-Sathaye embedding problem in dimension three

Abstract

Abhyankar-Moh and Suzuki proved that if an irreducible polynomial $f \in \mathbf{C}[x_{1}, x_{2}]$ in two complex variables $x_{1}$ and $x_{2}$ defines the affine plane curve $C = (f = 0) \subset \mathbf{A}^{2}$, which is isomorphic to the affine line: $C \cong \mathbf{A}^{1}$, then $f$ is a variable of $\mathbf{C}[x_{1}, x_{2}]$, i.e., there exists a polynomial $g \in\mathbf{C}[x_{1}, x_{2}]$ such that $\mathbf{C}[f, g] = \mathbf{C}[x_{1}, x_{2}]$ (cf. [A-M75], [Su74]). In this article, we prove under some additional assumptions that the similar result holds in the three-dimensional case, namely, if an irreducible polynomial $f \in \mathbf{C}[x_{1}, x_{2}, x_{3}]$ in three complex variables $x_{1}$, $x_{2}$ and $x_{3}$ defines the hypersurface $S = (f = 0) \subset \mathbf{A}^{3}$, which is isomorphic to the affine plane: $S \cong \mathbf{A}^{2}$, then $f$ is a variable of $\mathbf{C}[x_{1}, x_{2}, x_{3}]$, i.e., there are polynomials $g, h \in \mathbf{C}[x_{1}, x_{2}, x_{3}]$ such that $\mathbf{C}[f, g, h] = \mathbf{C}[x_{1}, x_{2}, x_{3}]$. Moreover, we shall determine the detailed form of such a polynomial $f \in\mathbf{C}[x_{1}, x_{2}, x_{3}]$ for the special case.