On some properties of elementary derivations in dimension six

Given a UFD R containing the rationals, we study elementary derivations of the polynomial ring in three variables over R. A consequence of Theorem 2.1, is that the kernel of every elementary monomial derivation of k[6] (k is a field of characteristic zero) is generated over k by at most six elements. In particular, seven is the lowest dimension in which we can construct a counterexample to Hilbert fourteenthʹs problem of Robertʹs type (see, Roberts, J. Algebra 132 (1990) 461–473).