Finite capture and the closure of roots of restricted polynomials

TL;DR

This paper investigates the roots of restricted polynomials and their connection to fractal connectedness loci, providing geometric constructions and finite-capture sets to characterize the roots outside the unit disk.

Contribution

It introduces a novel geometric framework for understanding roots of polynomials with restricted coefficients and establishes a precise relationship between these roots and finite-capture loci.

Findings

The non-real roots outside the unit disk form a fractal connectedness locus.

Finite-capture sets precisely describe the roots for large enough n.

Explicit trap geometry enables certified inverse search for roots.

Abstract

We study how a countable algebraic root set passes to a fractal connectedness locus. Let $D_n=\{-n+1,-n+2,\ldots,n-1\}$, and let $R_n$ be the set of roots of monic polynomials whose non-leading coefficients lie in $D_n$. We study $\overline{R_n}\setminus\overline{\mathbb{D}}$. Outside the closed unit disk this set equals a connectedness locus $M_n$ for a collinear affine iterated function system, or equivalently the zero set of reciprocal power series $1+\sum_{k\ge1} d_k c^{-k}$ with $d_k\in D_n$. For non-real parameters in the lens $X_n=\{\,c\in\mathbb{C}\setminus\overline{\mathbb{D}}:\ |c\pm1|<\sqrt{2n}\,\}$ we construct a canonical trap and enclosure for the associated difference attractor and use them to define finite-capture sets $\Theta_k(n)$ for the marked point $2c$. Our main result is the uniform inclusion $\overline{\Theta_k(n)}\cap(X_n\setminus\mathbb{R})\subset\Theta_{k+2}(n)$ for every $k\ge0$. Consequently, $(M_n\cap X_n)\setminus\mathbb{R}$ is exactly the closure of the finite-capture locus. The paper combines explicit trap geometry with certified inverse search. Moreover, $M_n\setminus\mathbb{R}\subset X_n$ for every $n\ge20$, and this is sharp for $2\le n\le19$. Thus, for $n\ge20$, the non-real part of $\overline{R_n}\setminus\overline{\mathbb{D}}$ is exactly the closure of the finite-capture locus.