Simple birational extensions of the polynomial ring $\C^{[3]}$

Abstract

The Abhyankar-Sathaye Problem asks whether any biregular embedding of affine spaces $A^m_k\to A^n_k$ can be rectified, that is, is equivalent to a linear embedding up to an automorphism of the target space. Here we study this problem for the embeddings $C^3 \to C^4$ whose image $X$ is given in $C^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in C[x,y],$ $f\neq 0$ and $g\in C[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $C[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $C^4$). This is an analog of a theorem due to Sathaye which concerns the case of embeddings $C^2\to C^3$. Besides, we generalize a theorem of Miyanishi giving, for a polynomial $p$ as above, a criterion for as when $X$ is isomorphic to $C^3$.