Complex geometry of Blaschke products and associated circumscribed conics (Developments in Computer Algebra : Recent Research and Re-Formation of Basic Theory)

Abstract

We study geometrical properties of finite Blashcke products. For a Blaschke product B of degree d and the d preimages z_{k}(k=1, cdots, d) of $lambda$inpartial mathrm{D} by B, let L_{ $lambda$} be the set of d lines tangent to partial mathrm{D} at the d preimages z_{1}, cdots, z_{d} . Here, we denote by T_{B} the trace of the intersection points of each two elements in L_{ $lambda$} as $lambda$ ranges over the unit circle. We show that the trace T forms an algebraic curve of degree at most d-1.