The commuting derivations conjecture

This paper proves the Commuting Derivations Conjecture in dimension three: if D1 and D2 are two locally nilpotent derivations which are linearly independent and satisfy [D1,D2]=0 then the intersection of the kernels, AD1∩AD2 equals 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,Z1,…,Zn−1). Also, conjectures about possible generalisations of the concept of “coordinate” for elements of general rings are made.