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Blaschke products and circumscribed conics. (English) Zbl 1394.30045
By a canonical Blaschke product of degree $$d$$, the author means a function of the form $B(z)=z\prod_{k=1}^{d-1}\frac{z-a_k}{1-\bar{a_k}z}\,, \qquad |a_k|<1, \quad k=1,2,\ldots,d-1.$ Given $$\lambda\in\mathbb{T}$$, where $$\mathbb T$$ is the unit circle, let $$L_\lambda$$ be the set of the $$d$$ lines which are tangent to $$\mathbb{T}$$ at the $$d$$ preimages of $$\lambda$$ by $$B$$. Denote by $$T_B$$ the trace of the intersection points of each pair of two lines in $$L_\lambda$$ as $$\lambda$$ ranges over the unit circle. Here is one of the main results of the paper.
Theorem. Let $$B$$ be a canonical Blaschke product of degree $$d$$. Then, the trace $$T_B$$ forms an algebraic curve of degree at most $$d-1$$.
When the degree is low, some additional information is available. For instance, if $$d=3$$, then the trace $$T_B$$ is a nondegenerate conic, that is, either an ellipse, a circle, a parabola, or a hyperbola. If $$d=4$$, then under certain assumptions on the zeros $$a_k$$, $$k=1,2,3$$, the trace $$T_B$$ forms a cubic algebraic curve.

##### MSC:
 30J10 Blaschke products 30C20 Conformal mappings of special domains
##### Keywords:
Blaschke product; algebraic curve
Risa/Asir
Full Text:
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