Walsh's Conformal Map onto Lemniscatic Domains for Polynomial Pre-images II

Abstract

We consider Walsh's conformal map from the exterior of a set $E=\bigcup_{j=1}^\ell E_j$ consisting of $\ell$ compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when $E$ is a polynomial preimage of $[-1,1]$, i.e., when $E=P^{-1}([-1,1])$, where $P$ is an algebraic polynomial of degree $n$. Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when $E$ is the union of $\ell$ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if $E$ consists of $\ell=2$ or $\ell=3$ components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. All our theorems are illustrated with analytical or numerical examples.