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

TL;DR

This paper investigates the roots of power series with coefficients constrained to specific sets, providing bounds, connectivity criteria, and conditions for root set equality, revealing structural rigidity in these roots.

Contribution

It introduces new estimates, criteria for connectivity, and conditions for root set equality, advancing understanding of roots of constrained power series.

Findings

Bounds on how deep roots can reach into the complex plane

Criteria for the connectedness of root sets

Conditions under which root sets of different coefficient sets coincide

Abstract

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.