Blaschke Products and Circumscribed Conics

Abstract

We study geometrical properties of finite Blaschke products. For a Blaschke product B of degree d, let \(L_{\lambda }\) be the set of the lines tangent to the unit circle at the d preimages \( B^{-1}(\lambda ) \). We show that the trace of the intersection points of each pair of two elements in \( L_{\lambda } \) as \( \lambda \) ranges over the unit circle forms an algebraic curve of degree at most \( d-1 \). In case of low degree, we have more precise results. For instance, for \( d=3 \), the trace forms a conic section. For \( d=4 \), we provide a necessary and sufficient condition for Blaschke products whose trace include a conic section.