Strongly residual coordinates over A[x]

TL;DR

This paper investigates when strongly residual coordinates over A[x] are actual coordinates, establishing new classes of such coordinates and applying results to Venereau-type polynomials, advancing understanding of polynomial automorphisms.

Contribution

It introduces conditions under which strongly residual coordinates are genuine coordinates, extending known results to higher dimensions and specific polynomial classes.

Findings

All strongly residual coordinates are coordinates for n=2.

A large class generated by elementaries upon inverting x are coordinates for any n.

Venereau-type polynomials are proven to be 1-stable coordinates.

Abstract

For a domain A of characteristic zero, a polynomial f over A[x] is called a strongly residual coordinate if f becomes a coordinate (over A) upon going modulo x, and f becomes a coordinate upon inverting x. We study the question of when a strongly residual coordinate is a coordinate, a question closely related to the Dolgachev-Weisfeiler conjecture. It is known that all strongly residual coordinates are coordinates for n=2 . We show that a large class of strongly residual coordinates that are generated by elementaries upon inverting x are in fact coordinates for arbitrary n, with a stronger result in the n=3 case. As an application, we show that all Venereau-type polynomials are 1-stable coordinates.