Geometric Properties of Inverse Polynomial Images

Abstract

Given a polynomial ${\mathcal{T}}_{n}$ of degree n, consider the inverse image of $\mathbb{R}$ and [−1,1], denoted by ${\mathcal{T}}_{n}^{-1}(\mathbb{R})$ and ${\mathcal{T}}_{n}^{-1}([-1,1])$, respectively. It is well known that ${\mathcal{T}}_{n}^{-1}(\mathbb{R})$ consists of n analytic Jordan arcs moving from ∞ to ∞. In this paper, we give a necessary and sufficient condition such that (1)${\mathcal{T}}_{n}^{-1}([-1,1])$ consists of ν analytic Jordan arcs and (2)${\mathcal{T}}_{n}^{-1}([-1,1])$ is connected, respectively.