Fix a radius $r \leq 1$. I'm interested in any necessary conditions, or any sufficient conditions, for a subset $S$ of $B(0,r)$, the origin-centered open disk of radius $r$, for $S$ to be the set of zeros of a power series $f(z)$ which has integral coefficients and which converges on $B(0,r)$.
Even answers that might work some small variation on my question will also interest me. For example, $S$ might be a multiset with the formal multiplicities putting a condition on the zeros of $f$. I'd also be interested if the coefficients of $f$ can vary over the ring of integers of some number field bigger than the rationals.
