A density result concerning inverse polynomial images

TL;DR

This paper investigates the geometric structure of inverse polynomial images, proving that the endpoints of certain inverse images form an increasingly fine net in the complex plane as polynomial degree grows.

Contribution

It establishes a quantitative density result for the endpoints of inverse polynomial images of [-1,1], advancing understanding of their geometric distribution.

Findings

Endpoints form an O(1/n)-net in the complex plane

Inverse images consist of two Jordan arcs

Quantitative density of endpoints as degree increases

Abstract

In this paper, we consider polynomials of degree $n$, for which the inverse image of $[-1,1]$ consists of two Jordan arcs. We prove that the four endpoints of these arcs form an ${\cal O}(1/n)$-net in the complex plane.