Jonas Meyer's comment:

Quote from [arxiv.org/abs/0902.3961][1], 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

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**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](https://doi.org/10.4064/aa145-2-2), arXiv:[0902.3961](https://arxiv.org/abs/0902.3961).

[1]:http://arxiv.org/abs/0902.3961