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Mathematics > Complex Variables
Title: A note on Rational Functions with three branching points on the Riemann sphere
(Submitted on 13 Jan 2024 (v1), revised 15 May 2024 (this version, v7), latest version 28 May 2024 (v9))
Abstract: Studying the existence of rational functions with given branching datum is a classical problem in the field of complex analysis and algebraic geometry. This problem dates back to Hurwitz and remains open to this day. In this paper, we utilize complex analysis to establish a property of rational functions with 3 branching points on the Riemann sphere. Given two compact Riemann surfaces $M$ and $N$, a pair $(d,\mathcal{D})$ of an integer $d\geq2$ and a collection $\mathcal{D}$ of nontrivial partitions of $d$ is called a candidate branching datum if it satisfies the Riemann-Hurwitz formula. And a candidate branching datum is exceptional if there does not exist a rational function realization it. As applications, we present some new types of exceptional branching datum. These results cover some previous results mentioned in \cite{EKS84,PP06,Zhu19}. We also deduce the realizability of a certain type of candidate branching datum on the Riemann sphere.
Submission history
From: Zhi Qiang Wei [view email][v1] Sat, 13 Jan 2024 02:38:44 GMT (10kb)
[v2] Wed, 24 Jan 2024 03:59:20 GMT (10kb)
[v3] Mon, 29 Jan 2024 10:35:00 GMT (10kb)
[v4] Thu, 1 Feb 2024 03:46:10 GMT (10kb)
[v5] Wed, 27 Mar 2024 05:00:59 GMT (10kb)
[v6] Tue, 2 Apr 2024 06:29:13 GMT (10kb)
[v7] Wed, 15 May 2024 13:56:38 GMT (11kb)
[v8] Wed, 22 May 2024 03:23:05 GMT (11kb)
[v9] Tue, 28 May 2024 13:37:26 GMT (11kb)
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