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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.

30J10 Blaschke products
30C20 Conformal mappings of special domains
Full Text: DOI
[1] Becker, T., Weispfenning, V.: Gröbner Bases. Springer, New York (1993) · Zbl 0772.13010
[2] Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 4th edn. Springer, New York (2015) · Zbl 1335.13001
[3] Daepp, U; Gorkin, P; Mortini, R, Ellipses and finite Blaschke products, Am. Math. Mon., 109, 785-794, (2002) · Zbl 1022.30039
[4] Flatto, L.: Poncelet’s Theorem. Amer. Math. Soc, Providence (2008) · Zbl 1157.51001
[5] Fujimura, M, Inscribed ellipses and Blaschke products, Comput. Methods Funct. Theory, 13, 557-573, (2013) · Zbl 1291.30040
[6] Gau, HL; Wu, PY, Numerical range and Poncelet property, Taiwan. J. Math., 7, 173-193, (2003) · Zbl 1051.15019
[7] Gorkin, P; Skubak, E, Polynomials, ellipses, and matrices: two questions, one answer, Am. Math. Mon., 118, 522-533, (2011) · Zbl 1227.51013
[8] Gorkin, P; Wagner, N, Ellipses and compositions of finite Blaschke products, J. Math. Anal. Appl., 445, 1345-1366, (2017) · Zbl 1354.30053
[9] Mashreghi, J.: Derivatives of Inner Functions. Springer, New York (2013) · Zbl 1276.30005
[10] Mirman, B, Numerical ranges and Poncelet curves, Linear Algebra Appl., 281, 59-85, (1998) · Zbl 0936.15024
[11] Risa/Asir (Kobe Distribution), an open source general computer algebra system. http://www.math.kobe-u.ac.jp/Asir/asir.html
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