A polynomial bijection from Q × Q to Q

Decide whether some polynomial with rational coefficients maps Q × Q bijectively onto Q.

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1	# A polynomial bijection from Q × Q to Q
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3	Is there a polynomial in two variables with rational coefficients that maps pairs of rational numbers one-to-one onto the rational numbers? Polynomial injections probably exist—Zagier suggested that $x^7 + 3y^7$ might be one—but even that is known only under a deep conjecture, and the same conjecture rules out bijections. Unconditionally, both questions are open.
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5	## Problem
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7	Is there a polynomial $f \in \mathbb{Q}[x, y]$ such that the map
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9	$$\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}, \qquad (a, b) \mapsto f(a, b),$$
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11	is a bijection?
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13	A related question, attributed to Harvey Friedman: is there a polynomial $f \in \mathbb{Q}[x, y]$ that is injective on $\mathbb{Q} \times \mathbb{Q}$?
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15	## Background
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17	The bijection question was posted on MathOverflow in 2010 by a user signing as Z.H. Set-theoretic bijections $\mathbb{Q}^2 \to \mathbb{Q}$ exist since both sets are countable, and over the integers there are polynomial bijections $\mathbb{N}^2 \to \mathbb{N}$ (Cantor's pairing polynomial), so the question is how arithmetic rigidity of rational points constrains polynomial maps. The known conditional results come from the Bombieri–Lang conjecture, which predicts that rational points on surfaces of general type are not Zariski dense.
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19	## What is known
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21	- Poonen (2010): if the Bombieri–Lang conjecture holds for rational points on surfaces of general type, then for every number field $k$ there is a polynomial $f \in k[x, y]$ inducing an injection $k \times k \to k$. Poonen quotes Zagier's speculation that $x^7 + 3y^7$ might already be injective on $\mathbb{Q} \times \mathbb{Q}$.
22	- Bresciani (2021, published 2025): assuming the weak Bombieri–Lang conjecture, there is no polynomial bijection $\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$. The strategy follows a 2019 discussion led by Terence Tao on his blog.
23	- No unconditional result in either direction is known.
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25	## What would count as progress
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27	- An unconditional proof that no polynomial bijection exists, perhaps first for restricted families such as $f(x, y) = g(x) + h(y)$ or polynomials of low degree.
28	- An unconditional proof that some explicit polynomial, such as $x^7 + 3y^7$, is injective on $\mathbb{Q} \times \mathbb{Q}$.
29	- Reducing the conditional results to weaker or better understood hypotheses.
30	- A formal proof, or disproof, of the Lean statement `mathoverflow_21003` in [Mathoverflow/21003.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Mathoverflow/21003.lean).
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32	## References
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34	1. B. Poonen, [Multivariable polynomial injections on rational numbers](https://doi.org/10.4064/aa145-2-2), Acta Arith. 145 (2010), 123–127; [arXiv:0902.3961](https://arxiv.org/abs/0902.3961).
35	2. G. Bresciani, [A higher dimensional Hilbert irreducibility theorem](https://doi.org/10.1353/ajm.2025.a961346), Amer. J. Math. 147 (2025), 779–794; [arXiv:2101.01090](https://arxiv.org/abs/2101.01090).
36	3. T. Tao, [Ruling out polynomial bijections over the rationals via Bombieri–Lang?](https://terrytao.wordpress.com/2019/06/08/ruling-out-polynomial-bijections-over-the-rationals-via-bombieri-lang/), blog post (2019).
37	4. Z.H. and others, [Polynomial bijection from Q × Q to Q?](https://mathoverflow.net/questions/21003), MathOverflow question 21003 (2010).
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39	## Source
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41	Imported from [Formal Conjectures Mathoverflow/21003](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Mathoverflow/21003.lean), curated by Google DeepMind and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.