Computing polynomial conformal models for low-degree Blaschke products

Abstract

For any finite Blaschke product $B$, there is an injective analytic map $\varphi:\mathbb{D}\to\mathbb{C}$ and a polynomial $p$ of the same degree as $B$ such that $B=p\circ\varphi$ on $\mathbb{D}$. Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial $p$ or the associated conformal map $\varphi$. In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary degree whose zeros are equally spaced on a circle centered at the origin.