On rational functions with more than three branch points

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 $$\max (m_1,\cdots ,m_l)<\frac{d}{\mathrm{GCD}(a_1,\cdots ,a_p,b_1,\cdots ,b_q)}.$$ As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.

19 pages We change the title. We improve exposition the manuscript greatly. Any comments are welcome!

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