Automorphisms of a polynomial ring which admit reductions of type I

2007/08/16 by Shigeru Kuroda, Kuroda, Shigeru · 1 citation
Computer Science · Mathematics · #13N15 #14R10 #14R20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13N15 #msc:14R10 #msc:14R20

paper · pdf · doi:10.48550/arxiv.0708.2120

Question 4.1 of the first version was answered

openalex publication_date 2007/08/16 · arxiv created 2007/09/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Recently, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I--IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen--Makar-Limanov--Willems gave a family of such automorphisms. In this paper, we present a new construction of such automorphisms using locally nilpotent derivations. As a consequence, we discover that there exists an automorphism admitting a reduction of type I which satisfies some degree condition for each possible value.

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• Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism

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