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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
Software:
Risa/Asir
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References:
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