There is a new manuscript on the arXiv by Giulio Bresciani (https://arxiv.org/pdf/2101.01090.pdf) 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/

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/

There is a new manuscript on the arXiv by Giulio Bresciani (https://arxiv.org/pdf/2101.01090.pdf) 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 write 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/

There is a new manuscript on the arXiv by Giulio Bresciani (https://arxiv.org/pdf/2101.01090.pdf) 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/