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 − 2 Y Δ − Z Δ 2 , Y + Z Δ , Z ) where Δ = X Z + 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 a c = b 2 , then L ( a , b , c ) N is linearizable if and only if b c ≠ 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.