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.