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 $N:= (X-Y\Delta-Z\Delta^2,Y+Z\Delta,Z)$ 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 $bc \neq 1$. 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.