The commuting derivations conjecture

Abstract

This paper proves the Commuting Derivations Conjecture in dimension three: if D 1 and D 2 are two locally nilpotent derivations which are linearly independent and satisfy [ D 1, D 2]=0 then the intersection of the kernels, A D 1 ∩ A D 2 equals C[f] where f is a coordinate. As a consequence, it is shown that p( X) Y+ Q( X, Z, T) is a coordinate if and only if Q( a, Z, T) is a coordinate for every zero a of p( X). Next to that, it is shown that if the Commuting Derivations Conjecture in dimension n, and the Cancellation Problem and Abhyankar–Sataye Conjecture in dimension n−1, all have an affirmative answer, then we can similarly describe all coordinates of the form p( X) Y+ q( X, Z 1,…, Z n−1 ). Also, conjectures about possible generalisations of the concept of “coordinate” for elements of general rings are made.