The Nagata automorphism is shifted linearizable

TL;DR

This paper proves the Nagata automorphism is shifted linearizable, meaning it becomes linearizable after composition with a suitable linear map, and explores broader implications for automorphism groups and derivations.

Contribution

It introduces the concept of shifted linearizability and proves the Nagata automorphism possesses this property, expanding understanding of automorphism linearization.

Findings

Nagata automorphism is shifted linearizable under certain conditions.

Any exponent of a homogeneous locally finite derivation is shifted linearizable.

Conjecture that linearizable automorphisms generate the automorphism group.

Abstract

A polynomial automorphism $F$ is called {\em shifted linearizable} if there exists a linear map $L$ such that $LF$ is linearizable. We prove that the Nagata automorphism $N:=(X-Y\Delta -Z\Delta^2,Y+Z\Delta, Z)$ where $\Delta=XZ+Y^2$ is shifted linearizable. More precisely, defining $L_{(a,b,c)}$ as the diagonal linear map having $a,b,c$ on its diagonal, we prove that if $ac=b^2$, then $L_{(a,b,c)}N$ is linearizable if and only if $bc\not = 1$. We do this as part of a significantly larger theory: for example, any exponent of a homogeneous locally finite derivation is shifted linearizable. We pose the conjecture that the group generated by the linearizable automorphisms may generate the group of automorphisms, and explain why this is a natural question.