Homogeneization of locally nilpotent derivations and an application to [formula omitted

Abstract

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 gr D . As an application, we give a complete classification of locally nilpotent derivations D : k [ X , Y , Z ] → k [ X , Y , Z ] satisfying D 2 X = D 2 Y = 0 ; in particular, it is proved that every k -derivation D satisfying D 2 X = D 2 Y = D 2 Z = 0 is essentially a partial derivative.