I would like to ask the following question. Maybe some of you can help me at least with a hint.

*Let $\alpha, \beta\in B(0,1)$ (unit ball), with $\alpha\neq \beta, \overline{\beta}$. 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 $f(\alpha)\neq 0$.*

Yesterday, Pietro Majer was able to help me in the case $\alpha,\beta\in \mathbb{R}$. In fact, I was able to adapt Pietro's idea for solving the case $\beta\in (-1,1)$ and $\alpha\in B(0,1)$. However, I can not deal with a complex $\beta$ since his idea depends on writing $1/\beta$ in basis $1/\beta^2$.

This question is related to one of my works on transcendental functions with integer coefficients with prescribed arithmetic behavior in algebraic points.

Thanks in advance,

muchsmaller for $\beta$ than for $\alpha$. Write $f$ for the polynomial giving the first terms, of degree $n$ say. Now $-f(\beta)$ is your "target" - it's what you want the remainder of the series to sum to. Is there an integer multiple of $\beta^{n+1}$ that gets you closer to the target? If so, add/subtract it; if not move on to the next power of $\beta$ etc. You should be able to prove that there is a constant $M$ such that the distance from the target when you are adding multiples of $\beta^n$ is at most $M|\beta|^n$. $\endgroup$ – Anthony Quas Nov 18 '16 at 22:37