On the kernels of some higher derivations in polynomial rings

Let A=Rx₁,…,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 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 . 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.