Images of locally nilpotent derivations acting on ideals of polynomial algebras (Q6548040)

In connection with Jacobian Conjecture \textit{W. Zhao} [J. Algebra 350, No. 1, 245--272 (2012; Zbl 1255.16018)] introduced the notion of Mathieu-Zhao (MZ) subspace which is a generalization of the notion of ideal. A \(k\)-subspace \(M\) of a commutative \(k\)-algebra \(S\) is called a MZ subspace if for \(f ,g \in S\) and \(f^{m} \in M\) for \(m \geq 1\) we have \(gf^{m} \in M\) for \(m \gg 0\) (\(k\) -- is a field of characteristic zero). W. Zhao proposed the following LND conjecture: Let \(D\) be a locally nilpotent derivation of \(k[x_{1} ,\ldots ,x_{n}] .\) Then, for any ideal \(I \subset k[x_{1} ,\ldots ,x_{n}] ,\) \(D(I)\) is an MZ subspace of \(k[x_{1} ,\ldots ,x_{n}] .\)\par The authors show that the LND conjecture holds in dimension \(n =2\) (for \(n =1\) it is true) for principal ideals and some others classes of ideals.