Computing Polynomial Conformal Models for Low-Degree Blaschke Products

Abstract

For any finite Blaschke product B, there is an injective analytic map ϕ : D → C and a polynomial p of the same degree as B such that B = p•ϕ on 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 ϕ. 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.

Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence

“…In 2019, T. J. Richards and M. Younsi [24] gave a first constructive result, describing an explicit construction for the polynomial conformal model for finite Blaschke products of degree at most 3. They also gave the following formula for the polynomial conformal model p and associated injective analytic map ϕ for a finite Blaschke product of arbitrarily high degree, whose zeros are evenly distributed on a circle centered at the origin.…”