Actions of Ga on A3 defined by homogeneous derivations

The first example of an algebraic action of G(a) on affine 3-space having maximal rank 3 is produced. Its fixed points consist of a single line in A(3), and G(a) is realized as an algebraic subgroup of Aut(k)(A(3)) whose non-trivial elements are of degree 41. The corresponding derivation is homogeneous and irreducible of degree 4. Since triangulable actions are never of maximal rank, this action is non-triangulable. This action is embedded, for each n greater than or equal to 3, into a G(a)-action on A(n), in such a way that the resulting action has rank n, thus showing that algebraic G(a)-actions on A(n) having maximal rank exist for each n greater than or equal to 3. Also considered is the general case of a homogeneous locally nilpotent derivation on k[3]. The main tool here is the exponent of a polynomial relative to the derivation. By describing such derivations of type (2, d + 1), where d is the degree of the derivation, it is shown that actions induced by homogeneous derivations of degree less than four have rank at most 2. The rank 3 example mentioned above appears as a special case of Theorem 4.2. (C) 1998 Elsevier Science B.V.