Locally nilpotent derivations and their rings of constants.

Abstract

Given a UFD R containing Q , we study R-elementary derivations of B = R[Y1,..., Ym], i.e., R-derivations satisfying D( Yi) ∈ R for all i; in the particular case of m = 3, we will show that if R is a polynomial ring in n variables over a field k (of characteristic zero), and a1, a3, a3 ∈ R are three monomials, then the kernel of the derivation i=13ai6 /6Yi of B is generated over R by at most three linear elements in the Yi's. This gives a partial answer to a question of A. van den Essen ([27]) about the existence of elementary derivations in dimension six whose kernels are not finitely generated. A set of generators is given for the kernel of R-elementary fixed point free derivations of B. Also, some interesting examples of elementary derivations in dimensions six and seven are provided as well as a criterion for a derivation of R[2] (i.e., a polynomial ring in two variables over R) to be R-elementary. Given a field k of characteristic zero, it is well-known that the kernel of any linear derivation of k[X 1,..., Xn] (that is, a k-derivation which maps each Xi to a linear form in X1,..., Xn) is a finitely generated k-algebra (see [28]). All known proofs of this result are non-constructive in the sense that we do not know a generating set for the kernel. Nowicki conjectured in [25] that the kernel of the derivation d = i=1nXi6 /6Yi of k[X1,..., Xn, Y1,..., Yn] is generated over k by the elements uij = XiYj - XjYi for 1 ≤ i ≤ j ≤ n. Using the theory of Groebner bases, we prove this conjecture in the more general case of the derivation D = i=1nXti i6/6Yi where each ti is a nonnegative integer. Note that in the case of the derivation D, the finite generation of the kernel is no longer evident. Note also that the generators of ker D are linear in the Yi's over k[X1,..., Xn]; we will show that this is not always the case for elementary derivations by giving an example of an elementary derivation in dimension seven whose kernel is finitely generated but cannot be generated by linear forms.