On polynomials in three variables annihilated by two locally nilpotent derivations

Abstract

Let B be a polynomial ring in three variables over an algebraically closed field k of characteristic zero. We are interested in irreducible polynomials f ∈ B satisfying the following condition: there exist nonzero locally nilpotent derivations D 1 , D 2 : B → B such that ker ( D 1 ) ≠ ker ( D 2 ) and D 1 ( f ) = 0 = D 2 ( f ) . The main result asserts that a nonconstant polynomial f ∈ B satisfies the above requirement if and only if its “generic fiber” k ( f ) ⊗ k [ f ] B is isomorphic, as an algebra over the field K = k ( f ) , to K [ X , Y , Z ] / ( X Y − φ ( Z ) ) for some nonconstant φ ( Z ) ∈ K [ Z ] .