On the lnd conjecture

Let k be a field of characteristic zero and $k<^>{[n]}$ the polynomial algebra in n variables over k. The LND conjecture concerning the images of locally nilpotent derivations arose from the Jacobian conjecture. We give a positive answer to the LND conjecture in several cases. More precisely, we prove that the images of rank-one locally nilpotent derivations of $k<^>{[n]}$ acting on principal ideals are MZ-subspaces for any $n\geq 2$ , and that the images of a large class of locally nilpotent derivations of $k<^>{[3]}$ (including all rank-two and homogeneous rank-three locally nilpotent derivations) acting on principal ideals are MZ-subspaces.