The two-dimensional Centralizer Conjecture

A result by C. C.-A. Cheng, J. H. Mckay and S. S.-S. Wang says the following: Suppose the Jacobian of $A$ and $B$ is invertible in $\mathbb{C}[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w\in \mathbb{C}[x,y]$. Then $w \in $\mathbb{C}$[A]$\mathrm{.}WeshowthatinCM{W}^{\mathrm{\prime }}sresultitispossibletoreplace$$\mathbb{C}$$ by any field of characteristic zero, and we conjecture the following ‘two-dimensional Centralizer Conjecture over $D$’: Suppose the Jacobian of $A$ and $B$ is invertible in $D[x,y]$ and the Jacobian of $A$ and $w$ is zero for $A,B,w\in D[x,y]$, $D$ is an integral domain of characteristic zero. Then $w\in D[A]$. We show that if the famous two-dimensional Jacobian Conjecture is true, then the two-dimensional Centralizer Conjecture is true.