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Fixed Points of Conjugated Blaschke Products with Applications to Gravitational Lensing

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Abstract

A conjecture in astronomy was recently resolved as an accidental corollary to a theorem regarding zeros of certain planar harmonic maps. As a step towards extending the Fundamental Theorem of Algebra, the theorem gave a bound of 5nt 5 for the number of zeros of a function of the form \(r(z) - \bar{z}\), where r(z) is rational of degree n. In this paper, we will investigate the case when r(z) is a Blaschke product. The resulting (sharp) bound is n + 3 and the proof is simple. We discuss an application to gravitational lenses consisting of collinear point masses.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Florida, 4204 E. Fowler Ave., PHY114, Tampa, FL, 33620, USA

    Ludwig Kuznia & Erik Lundberg

Authors
  1. Ludwig Kuznia
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  2. Erik Lundberg
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Correspondence to Ludwig Kuznia.

Additional information

The first author was supported by USF Thrust Area: Computational tools for discovery grant (Co-P.I. — D. Khavinson). The second author was supported by NSF grant #DMS-0701873 (P.I. — D. Khavinson).

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Kuznia, L., Lundberg, E. Fixed Points of Conjugated Blaschke Products with Applications to Gravitational Lensing. Comput. Methods Funct. Theory 9, 435–442 (2009). https://doi.org/10.1007/BF03321738

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  • DOI: https://doi.org/10.1007/BF03321738

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