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