Supppose there are integers $a\_1,a\_2,\dots$ and a polynomial $p$ so that the integers $p(a\_1),p(a\_2)...$ satisfy some linear recurrence, i.e. $\sum p(a\_i)x^i$ is a rational function of $x$. Must integers $b\_i\in p^{-1}(p(a\_i))$ so that $\sum b\_ix^i$ is a rational function, necessarily exist?


(The answer is no if we ask for the function $\sum a\_i x^i$ to be rational, as can be seen when $p(t)=t^2$ and $a\_i$ being a random sequence of $\pm1$)



Does an inverse polynomial map on the taylor coefficients of a rational function preserve rationality?
If $p(x)=x^d$ then this question coincides with Pisot's d'th root conjecture. A proof is given in [this](http://www.emis.ams.org/journals/Annals/151_1/zannier.pdf) paper of Zannier, I'm not sure if it's the one that Qiaochu was referring to in the comments.


Edit: A more general question was answered in the subsequent article [Equations in the Hadamard ring of rational functions](http://arxiv.org/abs/math/0701772) by Andrea Ferretti and Umberto Zannier.


