# Analytic functions with integer coefficients with prescribed zeros

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.

• Pick the first few integers so that substituting $\alpha$ you get something far from 0; and substituting $\beta$ you get something close to 0. Now pick the remaining coefficients, only using a term when you halve the remaining distance to 0. You can obtain a uniform bound on the size of the coefficients (depending on the argument and modulus of $\beta$). Provided the initial terms gave you something very large for $\alpha$ and very small for $\beta$, you will ensure that the $\beta$ sum converges to 0, while you will not have added enough to ensure that the $\alpha$ sum converges to 0. – Anthony Quas Nov 18 '16 at 20:46