Computing polynomial conformal models for low-degree Blaschke products

TL;DR

This paper presents methods to explicitly compute polynomial conformal models for low-degree Blaschke products, including degree three and certain symmetric cases, filling a gap in practical computation techniques.

Contribution

It introduces explicit computational techniques for the polynomial and conformal map associated with low-degree Blaschke products, which were previously only known theoretically.

Findings

Computed polynomial models for degree at most three Blaschke products.

Extended methods to Blaschke products with equally spaced zeros on a circle.

Provided explicit formulas for conformal maps in these cases.

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.