The Nagata automorphism is shifted linearizable

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-2Y Delta - Z Delta(2). Y + Z Delta. Z) where Delta = XZ + Y-2 is shifted linearizable. More precisely, defining L-(a,L-b,L-c) as the diagonal linear mal) having a. b. c oil its diagonal, we prove that if ac = b(2), then L-(a,L-b,L-c) N is linearizable if and only if bc not equal 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. (C) 2008 Elsevier Inc. All rights reserved.