TITLE: Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?
QUESTION [516 upvotes]: Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?

REPLY [36 votes]: There is a new manuscript on the arXiv by Giulio Bresciani, A higher dimensional Hilbert irreducibility theorem, arXiv:2101.01090, which shows that assuming the weak Bombieri--Lang conjecture, there cannot be a polynomial bijection from $\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$.
The author writes that:

Our strategy is essentially the one followed in a "polymath project"
led by T. Tao, see [Tao19], hence this result should be credited to the polymath project as a whole.

[Tao19] https://terrytao.wordpress.com/2019/06/08/ruling-out-polynomial-bijections-over-the-rationals-via-bombieri-lang/