# A note on kz-automorphisms in two variables - Mathematics > Commutative Algebra

Abstract: We prove that for a polynomial $f\in kx,y,z$ equivalent are: 1$f$ is a$kz$-coordinate of $kzx,y$, and 2 $kx,y,z-f\cong k^{2}$ and$fx,y,a$ is a coordinate in $kx,y$ for some $a\in k$. This solves a specialcase of the Abhyankar-Sathaye conjecture. As a consequence we see that acoordinate $f\in kx,y,z$ which is also a $kz$-coordinate, is a$kz$-coordinate. We discuss a method for constructing automorphisms of$kx,y,z$, and observe that the Nagata automorphism occurs naturally as thefirst non-trivial automorphism obtained by this method - essentially linkingNagata with a non-tame $R$-automorphism of $Rx$, where $R=kz-z^2$.

Author: Eric Edo, Arno van den Essen, Stefan Maubach

Source: https://arxiv.org/