Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$.

Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$ as there are analogies between $\Bbb F_q[x]$ and $\Bbb Z$?

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    $\begingroup$ The obvious answer is that $\mathbb F_q[X_1,\ldots,X_n]$ is analogous to $\mathbb Z[X_1,\ldots,X_{n-1}]$. Of course, this is really just saying that affine $N$ space over $\mathbb F_q[x]$ and over $\mathbb Z$ are analogous, so really it's the original analogy. But you may want to rephrase your question to eliminate obvious "base change" examples of this sort. $\endgroup$ – Joe Silverman Jun 10 '16 at 11:43

In the sense that $\mathbb{F}_q[x_1,x_2]\simeq \mathbb{F}_q[x_1]\otimes_{\mathbb{F}_q}\mathbb{F}_q[x_2]$, the analogous $\mathbb{Z}$-like object is $\mathbb{Z}\otimes_{\mathbb{F}_1}\mathbb{Z}$, which in the papers of James Borger is identified with the big Witt vectors $W(\mathbb{Z})$ of $\mathbb{Z}$.

For reference, see this paper: http://maths-people.anu.edu.au/~borger/preprints/01/lrfoe13.pdf


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