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, A1 ∩ A2 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, Z1, . . . , Zn−1). Also, conjectures about possible generalisations of the concept of “coordinate” for elements of general rings are made.