Description:
We prove that for a polynomial f e 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)~k2 and f(x,y,a) is a coordinate in k[x,y] for some a e k. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate f e 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).