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

TL;DR

This paper proves a specific case of the Abhyankar-Sathaye conjecture by characterizing when a polynomial in three variables is a coordinate over k[z], and explores automorphisms of k[x,y,z], linking Nagata automorphism to non-tame automorphisms.

Contribution

It establishes an equivalence condition for k[z]-coordinates in three variables and connects automorphisms of k[x,y,z] with the Nagata automorphism, advancing understanding of polynomial automorphisms.

Findings

Characterization of k[z]-coordinates in three variables

Solution to a special case of the Abhyankar-Sathaye conjecture

Link between Nagata automorphism and non-tame automorphisms

Abstract

We prove that for a polynomial $f\in k[x,y,z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x,y,z]/(f)\cong k^{[2]}$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a\in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f\in k[x,y,z]$ 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 $R=k[z]/(z^2)$.