Coordinates of R[x,y]: Constructions and classifications

TL;DR

This paper classifies specific polynomial coordinates over a PID and constructs non-tame automorphisms of polynomial rings, revealing complex automorphism structures.

Contribution

It provides a complete classification of certain polynomial coordinates and constructs non-tame automorphisms with particular subgroup properties.

Findings

Classified all coordinates of a specific form over PIDs.

Constructed non-tame automorphisms in three variables.

Showed the subgroup generated by these automorphisms contains all tame automorphisms.

Abstract

Let R be a PID. We construct and classify all coordinates of R[x,y] of the form p_2y+Q_2(p_1x+Q_1(y)) with p_1 and p_2 in qt(R) and Q_1 and Q_2 in qt(R)[y]. From this construction (with R=K[z]) we obtain non tame automorphisms s of K[x,y,z] (where K is a field of characteristic 0) such that the sub-group generated by s and the affine automorphisms contains all tame automorphisms.