One of the answers to [this question](https://math.stackexchange.com/questions/196024/can-someone-explain-the-abc-conjecture-to-me/196123#196123) 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](https://mathoverflow.net/questions/287610/if-f-g-in-dx-y-are-algebraically-dependent-over-d-then-f-g-in-dh-fo) 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](http://www.ams.org/journals/proc/2011-139-04/S0002-9939-2010-10523-3/S0002-9939-2010-10523-3.pdf) 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.



An analogy between the ring of polynomials in two variables and another (commutative?) ring
This seems like a strange question. Of course there are many analogies between many different commutative rings. However, if you're looking for a similar analogy, you have to replicate the basic features of Lang's analogy - in particular, the fact that the rings have the same dimension. This immediately kills all your suggested answers but $R = \mathbb Z[t]$. 


The analogy between $R = \mathbb Z[t]$ and $\mathbb C[x,y]$ is perfectly good generalization of Lang's analogy. Indeed it is almost a special case of Lang's analogy - whenever rings $R\_1$ and $R\_2$ have similar structure, we should expect that $R\_1[t]$ and $R\_2[t]$ have similar structure.


Finally I note that for many purposes the analogy between $\mathbb Z$ and $\mathbb C[t]$ is really a composition of two analogies, the first a closer analogy between $\mathbb Z$ and $\mathbb F\_q[t]$ and the second an analogy between $\mathbb F\_q[t]$ and $\mathbb C[t]$. For some purposes to understand the analogous construction on the other side it is crucial to pass through the middle step and take advantage of the special structures that are only available there (like the global Frobenius).


