Simple birational extensions of the polynomial algebra ℂ ^[3]

The Abhyankar-Sathaye Problem asks whether any biregular embedding φ : ℂ ^k → ℂ ⁿ can be rectified, that is, whether there exists an auto-morphism α ∈ Aut ℂ ⁿ such that α o φ is a linear embedding, Here we study this problem for the embeddings φ : ℂ ³ → ℂ ⁴ whose image X = φ(ℂ ³) is given in ℂ ⁴ 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 ℂ ⁴). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings ℂ ² → 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) ≃ ℂ ³.