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

Abstract

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 ∈ k [ x , y ] such that D 1 ( f ) = λ f and D 2 ( f ) = μ f for some λ , μ ∈ k [ x , y ] , or (ii) they are Jacobian derivations D u ( g ) : = ∂ u ∂ x ∂ u ∂ y ∂ g ∂ x ∂ g ∂ y , D v ( g ) : = ∂ v ∂ x ∂ v ∂ y ∂ g ∂ x ∂ g ∂ y for all g ∈ k [ x , y ] defined by some u , v ∈ k [ x , y ] for which D u ( v ) is a nonzero constant.