On polynomial automorphisms commuting with a simple derivation

2024/12/12 by Pierre-Louis Montagard, Iván Pan, Montagard, Pierre-Louis +3 · 1 citation
Mathematics · #13N15 #14R10 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2412.09519

openalex publication_date 2024/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D be a simple derivation of the polynomial ring \mathbbk[x₁,…,x_n], where \mathbbk is an algebraically closed field of characteristic zero, and denote by Aut(D)\subsetAut(\mathbbk[x₁,…,x_n]) the subgroup of \mathbbk-automorphisms commuting with D. We show that the connected component of Aut(D) passing through the identity is a unipotent algebraic group of dimension at most n-2, this bound being sharp. Moreover, Aut(D) is an algebraic group if and only if it is a connected ind-group. Given a simple derivation D, we characterize when Aut(D) contains a normal subgroup of translations. As an application of our techniques we show that if n=3, then either Aut(D) is a discrete group or it is isomorphic to the additive group acting by translations, and give some insight on the case n=4.

Cited by

• On the Isotropy Groups of Non-Invertible Simple Derivations

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