On the kernels of some higher derivations in polynomial rings

Let A = R[x(1), .... , x(n)] be the polynomial ring in n variables over an integral domain R with unit, let D be a rational higher R-derivation on A and let (D) over bar be the extension of D to the quotient field of A. We prove that, if the transcendental degree of the kernel of D over R is not less than n - 1, then the quotient field of the kernel of D equals the kernel of (D) over bar. Moreover, when n = 2, we give a necessary and sufficient condition for an R-subalgebra of A to be expressed as the kernel of a rational higher R-derivation on A. (C) 2011 Elsevier B.V. All rights reserved.