Homogeneization of locally nilpotent derivations and an application to k[X,Y,Z]

Let k be a field of characteristic zero and let B be a graded k-algebra. We obtain information on a given derivation D:B→B by studying the behavior of the associated homogeneous derivation grD. As an application, we give a complete classification of locally nilpotent derivations D:k[X,Y,Z]→k[X,Y,Z] satisfying D2X=D2Y=0; in particular, it is proved that every k-derivation D satisfying D2X=D2Y=D2Z=0 is essentially a partial derivative.