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 $\Bbb Z$?



Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$?
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>


