On pairs of commuting derivations of the polynomial ring in one or two variables

It is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector. We prove an analogous statement for derivations of k[x] and k[x, y] over any field k of zero characteristic. In particular, if D(1) and D(2) are commuting derivations of k[x, y] and they are linearly independent over k, then either (i) they have a common polynomial eigenfunction: i.e., a nonconstant polynomial f is an element of k[x, y] such that D(1)(f) = lambda f and D(2)(f) = mu f for some A, mu is an element of k[x, y], or (ii) they are Jacobian derivations [GRAPHICS] defined by some u, v is an element of k[x, y] for which D(u)(v) is a nonzero constant. (C) 2010 Elsevier Inc. All rights reserved.