The Nagata automorphism is shifted linearizable

Abstract

A polynomial automorphism F is called shifted linearizable if there exists a linear map L such that LF is linearizable. We prove that the Nagata automorphism N:=(X−YΔ−ZΔ2,Y+ZΔ,Z) where Δ=XZ+Y2 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=b2, then L(a,b,c)N is linearizable if and only if bc≠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.