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

Abstract:

The Abhyankar-Sathaye Problem asks whether any biregular embedding $\varphi :\mathbb {C}^k\hookrightarrow \mathbb {C}^n$ can be rectified, that is, whether there exists an automorphism $\alpha \in \operatorname {Aut} \mathbb {C}^n$ such that $\alpha \circ \varphi$ is a linear embedding. Here we study this problem for the embeddings $\varphi :\mathbb {C}^3\hookrightarrow \mathbb {C}^4$ whose image $X=\varphi (\mathbb {C}^3)$ is given in $\mathbb {C}^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in \mathbb {C}[x,y]\backslash \{0\}$ and $g\in \mathbb {C}[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $\mathbb {C}^{[4]}=\mathbb {C}[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $\mathbb {C}^4$). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings $\mathbb {C}^2\hookrightarrow \mathbb {C}^3$. Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial $p$ as above, a criterion for when $X=p^{-1}(0)\simeq \mathbb {C}^3$.