On rational functions with more than three branch points

TL;DR

This paper characterizes when certain rational functions with multiple branch points exist on the Riemann sphere, providing a criterion based on partitions and gcd conditions, and introduces new realizable branch data via Belyi functions.

Contribution

It establishes a necessary and sufficient condition for the existence of rational functions with specified branch data involving more than three branch points.

Findings

Provides a clear criterion involving gcd for the existence of such rational functions.

Identifies a new class of branch data realizable by Belyi functions.

Enhances understanding of branched coverings on the Riemann sphere.

Abstract

Let $\Lambda$ be a collection of partitions of a positive integer $d$ of the form $$(a_1,\cdots, a_p),\,(b_1,\cdots, b_q),\,(m_1+1,1,\cdots,1),\cdots, (m_l+1,1,\cdots,1),$$ where $(m_1,\cdots, m_l)$ is a partition of $p+q-2>0$. We prove that there exists a rational function on the Riemann sphere $\overline{\mathbb{C}}$ with branch data $\Lambda$ if and only if $$\max\bigl(m_1,\cdots,m_l\bigr) < \frac{d}{{\rm GCD}(a_1,\cdots, a_p,b_1,\cdots, b_q)}.$$ As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.