The following is a known result due to Carl Gustav Jacob Jacobi (1841):
Let $F$ be any field, $C,D \in F[x,y]$.
(1) If $C$ and $D$ are algebraically dependent over $F$,
then $\operatorname{Jac}(C,D)=0$.
(2) Assume that $F$ is of characteristic zero.
If $\operatorname{Jac}(C,D)=0$, then $C$ and $D$ are algebraically dependent over $F$.
Proofs that I have found on-line are due to N. Saxena (page 10, with a slightly more general form) and L. Makar-Limanov (it seems that $\mathbb{C}$ can be replaced by $F$).
In particular, the above result says the following: Suppose that $k$ is a field of characteristic zero and $C,D \in k[x,y]$. Then: $\operatorname{Jac}(C,D)=0$ if and only if $C$ and $D$ are algebraically dependent over $k$.
Is this result still valid if we replace the field of characteristic zero $k$, by the UFD $\mathbb{Z}$? More elaborately, is the following claim true: Suppose that $C,D \in \mathbb{Z}[x,y]$. Then, $\operatorname{Jac}(C,D)=0$ if and only if $C$ and $D$ are algebraically dependent over $\mathbb{Z}$.
Edit: Perhaps there is no meaning for algebraic dependence over a UFD which is not a field? I just meant that there exist $e_{ij} \in \mathbb{Z}$ not all zero, such that $\sum \sum e_{ij} C^iD^j=0$. (This definition looks fine to me; am I missing something?).
Any comments are welcome!