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 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. (C) 2020 Elsevier Inc. All rights reserved.