A note on [formula omitted]-automorphisms in two variables

Abstract

We prove that for a polynomial f ∈ k [ x , y , z ] , the following are equivalent: (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 proves 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 the Nagata automorphism with a non-tame R -automorphism of R [ x ] , where R = k [ z ] / ( z 2 ) .