Homogeneous locally nilpotent derivations having slices and embeddings of affine spaces

Let m and n be positive integers such that n⩾m and let B be a polynomial ring in m+n+1 variables over a field k of characteristic 0. We give a bijective correspondence between the equivalence classes of embeddings Am→An and the equivalence classes of sequences of mutually commuting locally nilpotent derivations δi (1⩽i⩽m) on B in some form, which are homogeneous with respect to a Z-grading on B and have slices. The intersection A of the kernels of δi for 1⩽i⩽m inherits the Z-grading on B. We show that A is a polynomial ring with homogeneous coordinates if and only if the corresponding embedding is rectifiable.