Minimal degree rational open up mappings and related questions

TL;DR

This paper proves the existence and uniqueness of minimal degree rational conformal maps for opening up multiple arcs, and explores related problems involving prescribed critical values and points, with results linked to Hurwitz numbers and Catalan numbers.

Contribution

It introduces new minimal degree rational maps for opening arcs and addresses related critical value and point problems, providing existence proofs and combinatorial counts.

Findings

Existence and uniqueness of minimal degree rational conformal maps for opening n arcs.

Number of rational functions with prescribed critical values related to Hurwitz numbers.

Number of solutions with prescribed critical points bounded by Catalan 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.