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,



Analytic functions with integer coefficients with prescribed zeros
David Harbater wrote a paper devoted to this ring (denoted by $\mathbf{Z}\{t\}$) and similar ones, see "Convergent arithmetic power series", Amer. J. Math., 106 (1984), no. 4, pp. 801-846.


In particular, in Lemma 1.5, he proves that for each $r \in (0,1)$ and each $\lambda \in \mathbf{C}$ of absolute value at most $r$, there exists $f\in \mathbf{Z}\{t\}$ with a zero of order 1 at $\lambda$ and $\bar\lambda$ and no other zero on the closed disk of center 0 and radius $r$.


