Section 1.3 of https://swc-math.github.io/aws/2008/08BeukersNotesDraft.pdf claims there exists a non-constant power series $f(x)$ with positive radius of convergence $\rho,$ such that for any algebraic number $a$ with $|a| < \rho,$ the value $f(a)$ is rational. The author says a proof can be found in [], presumably meaning they meant to add a citation but forgot.
Where can I find a reference to this power series?
Given any sequence $\{x_n\}_{n\ge 1}$ of distinct reals, I will define a non-constant entire function $f$ that takes $S$ into the rationals. This will be of the form $$f(z) = \sum_{n=1}^\infty c_n \prod_{k < n} (z - x_k) \tag 1$$ Note that for such a function, $f(x_j) = \sum_{n=1}^{j} c_n \prod_{k < n} (x_j - x_k)$, and given real numbers $c_1, \ldots, c_{j-1}$, there is a dense set of reals $c_j$ for which this will be rational. We just need to ensure that the series (1) converges uniformly on compact sets: this can be done by taking $|c_n| \le b_n$, where $b_n > 0$ is small enough that, say, $b_n \prod_{k < n} |z - x_k| < 2^{-n}$ for $|z| \le n$.
