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.

The factorial conjecture was proposed by van den Essen et al. [‘On the image conjecture’, J. Algebra 340(1) (2011), 211–224] to study the image conjecture, which arose from the Jacobian conjecture. We show that the factorial conjecture holds for all homogeneous polynomials in two variables. We also give a variation of the result and use it to show that the image of any linear locally nilpotent derivation of $\mathbb{C}[x,y,z]$ is a Mathieu–Zhao subspace.

We strengthen certain results concerning actions of $\left( \mathbb{C},\,+ \right)$ on ${{\mathbb{C}}^{3}}$ and embeddings of ${{\mathbb{C}}^{2}}$ in ${{\mathbb{C}}^{3}}$, and show that these results are in fact valid over any field of characteristic zero.