Jonas Meyer's comment:
Quote from arxiv.org/abs/0902.3961, Bjorn Poonen, Feb. 2009: "Harvey Friedman asked whether there exists a polynomial $f(x,y)\in Q[x,y]$ such that the induced map $Q × Q\to Q$ is injective. Heuristics suggest that most sufficiently complicated polynomials should do the trick. Don Zagier has speculated that a polynomial as simple as $x^7+3y^7$ might already be an example. But it seems very difficult to prove that any polynomial works. Our theorem gives a positive answer conditional on a small part of a well-known conjecture." – Jonas Meyer
Added June 2019 Poonen's paper is published as:
Bjorn Poonen, Multivariable polynomial injections on rational numbers, Acta Arith. 145 (2010), no. 2, pp 123-127, doi:10.4064/aa145-2-2, arXiv:0902.3961.
