Locally nilpotent derivations of polynomial rings.

Abstract

Let k be a field of characteristic 0. We classify locally nilpotent derivations D : k[ X, Y, Z] → k[X, Y, Z] satisfying D2X = D2 Y = 0; in particular, it is proved that every k-derivation D satisfying D 2X = D2Y = D2Z = 0 is essentially a partial derivative. Then we study three classes of k-derivations D : k [X1,..., Xn] → k [X 1,...Xn], namely the elementary , constructible and nice derivations. We note the simple fact that elementary ⇒ constructible ⇒ nice, and we investigate under which conditions the converses hold. We find that if n = 3 then all three notions are the same; in dimension 4, if D is irreducible then constructible is equivalent to elementary; in dimension 5, there is an example which is constructible, irreducible but not elementary. Another result states that rank D ≤ n -- 2 holds for all constructible derivations and all n ≥ 3.