Minimal degree rational open up mappings and related questions

Authors

  • Sergei Kalmykov Shanghai Jiao Tong University, School of Mathematical Sciences, and Russian Academy of Sciences, Keldysh Institute of Applied Mathematics
  • Béla Nagy University of Szeged, Bolyai Institute, Department of Analysis
  • Olivier Sète Universität Greifswald, Institute of Mathematics and Computer Science

DOI:

https://doi.org/10.54330/afm.131296

Keywords:

Conformal mapping, critical values, critical points, Hilbert's Nullstellensatz, solving polynomial equations, Riemann surfaces, Hurwitz numbers

Abstract

We establish the existence and uniqueness of rational conformal maps of minimal degree \(n+1\) for opening up \(n\) arcs. In earlier results, the degree was exponential in \(n\). We also discuss two related problems. (a) We establish existence of rational functions of minimal degree with prescribed critical values, and show that the number of (suitably normalized) rational functions is given in terms of the Hurwitz numbers. (b) We consider the problem of finding rational functions of minimal degree with prescribed critical points, where we establish existence of solutions by considering certain polynomial equations, and where the number of normalized solutions is bounded from above by a Catalan number. We illustrate our results with two examples.

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Published

2023-06-26

Issue

Vol. 48 No. 2 (2023)

Section

Articles

License

Copyright (c) 2023 Annales Fennici Mathematici

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

How to Cite

Kalmykov, S., Nagy, B., & Sète, O. (2023). Minimal degree rational open up mappings and related questions. Annales Fennici Mathematici, 48(2), 429-451. https://doi.org/10.54330/afm.131296