The commuting derivations conjecture

In this talk we shall consider the Commuting Derivations Conjecture in dimension three: if $D_1$ and $D_2 \in LND$, which are linearly independent and satisfy $[D_1; D_2] = 0$, then $\mathrm{ker} D_1 \cap \mathrm{ker} D_2 = \mathbb{C}[f]$, where $f$ is a coordinate. Then it is shown that if the Commuting Derivations Conjecture in dimension $n$, the Cancellation Problem and Abhyankar–Sataye Conjecture in dimension $n-1$, all have an affirmative answer, then we can describe all coordinates of the form $p(X)Y + q(X; Z_1; \ldots ; Z_{n-1})$. Also, conjectures about possible generalisations of the concept of “coordinate” for elements of general rings are made. This talk will be based on the paper of Stefan Maubach [1].

References:
[1] Stefan Maubach, The commuting derivations conjecture, Journal of Pure and Applied Algebra 179 (2003) 159 – 168