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

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 \(C^3\).