The Nagata automorphism is shifted linearizable

2008/04/30 by Stefan Maubach, Maubach, Stefan, Pierre-Marie Poloni +1
Mathematics · Medicine · #14L17 #14R10 #14R20 #32M17 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #math.AC #math.AG #math.CV #msc:14L17 #msc:14R10 #msc:14R20 #msc:32M17

paper · pdf · doi:10.48550/arxiv.0804.4870

14 pages

arxiv created 2008/04/30 · openalex publication_date 2008/04/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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-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\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.

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