# An analogy between the ring of polynomials in two variables and another (commutative?) ring

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.

• Since $\mathbb{Z}$ is the Grothendieck ring of $\omega$, perhaps the Grothendieck ring of $\omega^\omega$ would be suitable? – Alec Rhea Mar 11 '18 at 6:54
• Are you looking for a ring where one can prove an analogue of the Jacobian conjecture? Or why did you tag your question with that tag? – Zach Teitler Mar 11 '18 at 14:47
• Thanks. There are two reasons: (1) The one you mentioned; actually, in my second remark I mention Formanek's paper, which shows a connection between the two-dimensional JC and a polynomial ring with one variable (Formanek relies on results of Moh and Abhyankar that already show such a connection, but unfortunately I have not yet understood how exactly to obtain that connection for a specific Jacobain pair). (2) I have mentioned a case where things over $\mathbb{Z}$ are more complicated than over other integral domains, and that example is closely related to the JC. – user237522 Mar 11 '18 at 15:07

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).

• To make it more "numberish" one could take something like $\mathbb Z[\pi]$... – მამუკა ჯიბლაძე Mar 11 '18 at 13:08
• Thank you @Will Sawin (and thanks for the two commenters). Can you (or someone else) please give a specific example of a result and its analogy? (For $\mathbb{C}[x,y]$ and $\mathbb{Z}[t]$). – user237522 Mar 11 '18 at 13:16
• I do not know how to generalize Lang's results about $R_1=\mathbb{Z}$ and $R_2=\mathbb{C}[x]$ to $R_1[t]$ and $R_2[t]$, since he worked with a polynomial ring in one variable over an algebraically closed field of characteristic zero, but $R_2$ is not (an algebraically closed) field. Am I missing something trivial? – user237522 Mar 11 '18 at 14:43
• @user237522 I don't know any great example specifically for $\mathbb C[x,y]$. For general rings of functions on algebraic surfaces, there's any non-isotrivial surface mapping to a curve whose fibers have genus $>1$ having only finitely many sections and Falting's theorem. There's also going to be various local statements involving various kinds of cohomology groups. – Will Sawin Mar 12 '18 at 6:13
• @user237522 I don't mean to imply anything new about that particular theorem, the ABC conjecture. (This isn't actually Lang's result, it's due to Mason and Stothers.) Or what other "Lang's results" do you mean? I only meant to say something about the analogy Lang is discussing. – Will Sawin Mar 12 '18 at 6:16