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

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