  Equivariant compactifications of reductive groups can be described by
combinatorial data. On the other hand, equivariant compactifications of the
additive group G^n_a are more complicated in at least two respects. First, they
often admit moduli. Second, even simple varieties (like projective spaces)
admit many different structures as equivariant compactifications of the
additive group. We give a dictionary relating Artinian local rings, certain
systems of partial differential equations, and equivariant compactifications of
G^n_a. As an application, we classify the structures on varieties like
projective spaces, ruled surfaces, and certain threefolds. We consider these
results as a first step in a systematic study of G^n_a-equivariant birational
geometry.
