Long time ago there was a question on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt of answering it has been given, highly downvoted by the way. But this answer isn't obviously unsuccessful, because the following problem (for case $n=2$) remains open.
Problem. Let $f$ be a polynomial with rational (or even integer!) coefficients in $n$ variables $x_1,\dots,x_n$. Suppose there exist two distinct points $\boldsymbol a=(a_1,\dots,a_n)$ and $\boldsymbol b=(b_1,\dots,b_n)$ from $\mathbb R^n$ such that $f(\boldsymbol a)=f(\boldsymbol b)$. Does this imply the existence of two points $\boldsymbol a'$ and $\boldsymbol b'$ from $\mathbb Q^n$ satisfying $f(\boldsymbol a')=f(\boldsymbol b')$?
Even case $n=1$ seems to be non-obvious.
EDIT. Just because we have a very nice counter example (immediately highly rated by the MO community) by Hailong Dao in case $n=1$ and because for $n>1$ there are always points $\boldsymbol a,\boldsymbol b\in\mathbb R^n$ with the above property, the problem can be "simplified" as follows.
Is it true for a polynomial $f\in\mathbb Q[\boldsymbol x]$ in $n>1$ variables that there exist two points $\boldsymbol a,\boldsymbol b\in\mathbb Q^n$ such that $f(\boldsymbol a)=f(\boldsymbol b)$?
The existence of injective polynomials $\mathbb Q^2\to\mathbb Q$ is discussed in B. Poonen's preprint (and in comments to this question). What can be said for $n>2$?
FURTHER EDIT. The expected answer to the problem is in negative. In other words, there exist injective polynomials $\mathbb Q^n\to\mathbb Q$ for any $n$.
Thanks to the comments of Harry Altman and Will Jagy, case $n>1$ is now fully reduced to $n=2$. Namely, any injective polynomial $F(x_1,x_2)$ gives rise to the injective polynomial $F(F(x_1,x_2),x_3)$, and so on; in the other direction, any $F(x_1,\dots,x_n)$ in more than 2 variables can be specialized to $F(x_1,x_2,0,\dots,0)$.
In spite of Bjorn Poonen's verdict that case $n=2$ can be resolved by an appeal to the Bombieri--Lang conjecture for $k$-rational points on surfaces of general type (or even to the 4-variable version of the $abc$ conjecture), I remain with a hope that this can be done by simpler means. My vague attempt (for which I search in the literature) is to start with a homogeneous form $F(x,y)=ax^n+by^n$, or any other homogeneous form of odd degree $n$, which has the property that only finitely many integers are represented by $F(x,y)$ with $x,y\in\mathbb Z$ relatively prime. In order to avoid this finite set of "unpleasant" pairs $x,y$, one can replace them by other homogeneous forms $x=AX^m+BY^m$ and $y=CX^m+DY^m$ (again, for $m$ odd and sufficiently large, say), so that $x$ and $y$ escape the unpleasant values. Then the newer homogeneous form $G(X,Y)=F(AX^m+BY^m,CX^m+DY^m)$ will give the desired polynomial injection. So, can one suggest a homogeneous form $F(x,y)$ with the above property?
