Extensions birationnelles simples de l'anneau de polynêmes C^[3]

The Abhyankar-Sathaye problem asks whether any biregular embedding φ:C^kCⁿ can be rectified, that is, whether there exists an automorphism α∈AutCⁿ such that αφ is a linear embedding. In the spirit of [5], here we study this problem for the embeddings φ:C³C⁴ whose image X=φ(C³) is given in C⁴ by an equation p=f(x,y)u+g(x,y,z)=0, where f∈C[x,y]{0} and g∈C[x,y,z]. Under certain additional assumptions we show that, indeed, the polynomial p is a variable of the polynomial ring C^[4]=C[x,y,z,u] (i.e., a coordinate of a polynomial automorphism of C⁴). For that, we first study the acyclicity of X in a more general setting. Then we give several equivalent conditions for X=p^-1(0)≃C³ generalizing in particular a theorem of Miyanishi [4, Thm. 2].