The Polya-Chebotarev problem and inverse polynomial images

TL;DR

This paper links the Pólya-Chebotarev problem to inverse polynomial images, showing that inverse images of polynomials solve minimal capacity problems and providing a method to construct such polynomials.

Contribution

It demonstrates that connected inverse images of polynomials solve Pólya-Chebotarev problems and introduces a nonlinear system approach to construct these polynomials.

Findings

Inverse images of polynomials solve Pólya-Chebotarev problems.

A nonlinear system for zeros of aT_n^2-1 is used for construction.

Connected inverse images can be explicitly constructed.

Abstract

Consider the problem, usually called the P\'olya-Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected inverse image $\T_n^{-1}([-1,1])$ of a polynomial $\T_n$ is always the solution of a certain P\'olya-Chebotarev problem. By solving a nonlinear system of equations for the zeros of $\T_n^2-1$, we are able to construct polynomials $\T_n$ with a connected inverse image.