On the Rigidity of the Roots of Power Series with Constrained Coefficients

Here we consider the set $\Sigma_S$ of roots of power series whose coefficients lie in a given set $S$ and how such sets of roots vary as the set $S$ varies. We give an estimate of the depth that complex roots can reach into the disc, offer some criterion for the set of roots to be connected or disconnected, and show that for two finite symmetric sets $S$ and $T$ of integers containing $1$, if $\Sigma_S = \Sigma_T$ then all of their elements between $1$ and $2\sqrt{\max(S)}+1$ must agree.