Geodesic
Minimal degree rational open up mappings and related questions
Sergei Kalmykov1 ; Béla Nagy2 ; Olivier Sète3
1 Shanghai Jiao Tong University, School of Mathematical Sciences, and Russian Academy of Sciences, Keldysh Institute of Applied Mathematics
2 University of Szeged, Bolyai Institute, Department of Analysis
3 Universität Greifswald, Institute of Mathematics and Computer Science
Annales Fennici Mathematici, Tome 48 (2023) no. 2, pp. 429-451 Cet article a éte moissonné depuis la source Journal.fi

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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.
DOI : 10.54330/afm.131296
Keywords: Conformal mapping, critical values, critical points, Hilbert's Nullstellensatz, solving polynomial equations, Riemann surfaces, Hurwitz numbers

Sergei Kalmykov 1 ; Béla Nagy 2 ; Olivier Sète 3

1 Shanghai Jiao Tong University, School of Mathematical Sciences, and Russian Academy of Sciences, Keldysh Institute of Applied Mathematics
2 University of Szeged, Bolyai Institute, Department of Analysis
3 Universität Greifswald, Institute of Mathematics and Computer Science 
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Sergei Kalmykov; Béla Nagy; Olivier Sète. Minimal degree rational open up mappings  and related questions. Annales Fennici Mathematici, Tome 48 (2023) no. 2, pp. 429-451. doi: 10.54330/afm.131296

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