One of the answers to this question says:
"In Serge Lang's *Algebra*, he says: "One of the most fruitful analogies in mathematics is that between the integers $\mathbb{Z}$ and the ring of polynomials $F[t]$ over a field $F$". He then *proves* the *abc* conjecture *for polynomials*, and for good measure he proves Fermat's Last Theorem for polynomials. In other words, Lang is saying that if something is true for the ring of polynomials, one ought to check if it is true for that rather important ring called the integers. But it turns out that the ring of integers can be rather more troublesome, which may be surprising".

What happens if we replace $\mathbb{C}[t]$ by $\mathbb{C}[x,y]$? Namely, is there an interesting/a similar analogy between some (commutative?) ring $R$ and $\mathbb{C}[x,y]$?

Actually, here one can find an example that things over $\mathbb{Z}$ may become more complicated than over other integral domains ($R=k[u^3,v^3,u^2,v^2,uv]$).

**Remarks:**
(1) Of course one can suggest $R=A_1(\mathbb{C})$, the first Weyl algebra, and the
stable equivalence between the Dixmier and Jacobian Conjectures, but I think I actually prefer that $R$ will be a commutative ring which is 'simpler' than $\mathbb{C}[x,y]$ (Perhaps $R=\mathbb{Z}$ or $R=\mathbb{C}[t]$).
(2) Probably Formanek's paper is close to an answer I am looking for
(with $R=\mathbb{C}[t]$ or $\mathbb{Z}[t]$?).

Thank you very much for any comments/help!

**Edit:** I think I should have divided my question into two separate questions:
(1) Generalizing Lang's result concerning $R_1=\mathbb{Z}$ and $R_2=\mathbb{C}[x]$ to $R_1[t]$ and $R_2[t]$ (the below answer is nice, except that it is not elaborating what is exactly the generalized result of Lang's result).
(2) Finding connections between the (two-dimensional) Jacobian Conjecture and other theorems in number theory, similarly to what Formanek has done;
perhaps I will ask this in a new question.