A note on k [ z ] - automorphisms in two variables

We prove that for a polynomial f ∈ k[x, y, z] equivalent are: (1)f is a k[z]-coordinate of k[z][x, y], and (2) k[x, y, z]/(f) ∼= k[2] and f(x, y, a) is a coordinate in k[x, y] for some a ∈ k. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate f ∈ k[x, y, z] which is also a k(z)-coordinate, is a k[z]-coordinate. We discuss a method for constructing automorphisms of k[x, y, z], and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method essentially linking Nagata with a non-tame R-automorphism of R[x], where R = k[z]/(z2).