Geometric properties of inverse polynomial images

TL;DR

This paper investigates the geometric structure of inverse polynomial images, providing conditions for their number of analytic Jordan arcs and connectivity, which enhances understanding of polynomial inverse images in complex analysis.

Contribution

It offers necessary and sufficient conditions for the number and connectivity of Jordan arcs in inverse polynomial images, advancing the geometric theory of polynomial mappings.

Findings

Conditions for the number of Jordan arcs in inverse images

Criteria for the connectedness of inverse polynomial images

Characterization of inverse images of [-1,1] under polynomials

Abstract

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