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In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices.

The function constructed though is by necessity complex and certainly not elementary. I think it would be interesting to know if an example exists with simpler functions. Hence the following question:

Does there exist a finite set of polynomials whose iterates generate $\mathbb{Q}$?

I'm not certain of the answer to this even for linear polynomials. For example $x/2+1$, $x/2-1$ and $2x$ gets you close by generating all non-repeating binary fractions, a subset of $\mathbb{Q}$. However it seems impossible to extend this to obtain all of $\mathbb{Q}$.

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    $\begingroup$ There are only finitely many primes dividing the denominators of the coefficients. So you will stay in the ring generated by the inverses of these primes. $\endgroup$
    – user473423
    Commented Jan 4, 2022 at 10:06
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    $\begingroup$ Yes, thank you, I think you are right - too easy - I should have realised that! Add this as an answer which I will accept if you like. $\endgroup$
    – Ivan Meir
    Commented Jan 4, 2022 at 10:25

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