A realizability criterion for rational maps with three branch points

Abstract

Let \[ π_{0}=[\underbrace{2,\ldots,2}_{2s-1}],\quad π_{\infty}=[\underbrace{3,\ldots,3}_{s},\underbrace{1,\ldots,1}_{s-2}],\quad π_{1}=[γ_{1},γ_{2},γ_{3}] \] be three nontrivial partitions of the integer \(d=4s-2\), where \(s\ge 3\). We establish a necessary and sufficient condition for the candidate datum \(\{π_{0},π_{\infty},π_{1}\}\) to be realizable by a rational map.