I would like to ask the following question. This is related to one lemma that I need in a recent research on the arithmetic behavior of transcendental functions with integer coefficients.


*Let $\alpha, \beta\in (-1,1)$, with $\alpha\neq \pm \beta$. I would like to prove that there exists a function $f\in \mathbb{Z}[[z]]$ analytic in the unit ball and such that $f(\beta)=0$ and $f(\alpha)\neq 0$.*


I am able to prove this when $\alpha\cdot \beta<0$ and when $|\beta|>|\alpha|$. 


Any help?



Analytic functions with integer coefficients
We may assume w.l.o.g $0<\beta<1$. Write $\beta^{-1}$ in the [$ \beta^{-2}$ expansion](https://en.wikipedia.org/wiki/Non-integer_representation) as:
$$\beta^{-1}=\sum\_{k=0}^\infty d\_k\beta^{2k}$$
with integer digits $0\le d\_k<\beta^{-2}$, and define
$$f(x):=-1+\sum\_{k=0}^\infty d\_kx^{2k+1}$$
Then $f\in\mathbb{Z}[[x]]$ is analytic in the unit disk, strictly increasing on the interval $(-1,1)$, and $f(\beta)=0$.


