Certain rational functions on the Riemann sphere with more than three branch points

Ji-jian Song, Bin Xu

Published 2015-10-20Version 1

Consider a collection $\Lambda$ of partitions of a positive integer $d$ with 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 with branch data $\Lambda$ if and only if $\max\bigl(m_1,\cdots,m_l\bigr)$ is less than $d/{\rm GCD}(a_1,\cdots, a_p,b_1,\cdots, b_q)$, which generalizes a theorem of G. Boccara in Discrete Math.(1982). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.