Minimal degree rational open up mappings and related questions

Kirjoittajat

  • 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

Avainsanat:

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

Abstrakti

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.
Numero
Vol 48 Nro 2 (2023)
Osasto
Articles

Julkaistu

2023-06-26

Viittaaminen

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

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