Images of locally nilpotent derivations of polynomial algebras in three variables

Let k be a field of characteristic zero. We study the problem of whether or not the image of a locally nilpotent derivation of the polynomial algebra k[x,y,z] is a Mathieu-Zhao subspace. The problem arose from the Jacobian conjecture. We show that the problem has an affirmative answer for rank-two l...

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Bibliographic Details
Published inJournal of algebra Vol. 569; pp. 401 - 415
Main Authors Sun, Xiaosong, Liu, Dayan
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.03.2021
Subjects
Jacobian conjecture
Local slice construction
Locally nilpotent derivation
Mathieu-Zhao subspace
Locally nilpotent derivation
Local slice construction
14R10
14R15
Mathieu-Zhao subspace
13N15
Jacobian conjecture
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Summary:Let k be a field of characteristic zero. We study the problem of whether or not the image of a locally nilpotent derivation of the polynomial algebra k[x,y,z] is a Mathieu-Zhao subspace. The problem arose from the Jacobian conjecture. We show that the problem has an affirmative answer for rank-two locally nilpotent derivations. We also give some new results on local slice constructions, and by use of which, we show that the problem above has an affirmative answer for rank-three homogeneous locally nilpotent derivations.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2020.10.025