A New Proof of Wei's Theorem

In the paper Structure and realizability for rational maps, Wei proved the following. Let $k$ nontrivial partitions of an integer $d\ge 3$ be given, satisfying the Riemann–Hurwitz formula for the case of maps between two spheres. If $k>\ell+1$, where $\ell=\min_i|π_i|$ is the shortest partition length, then there exists a rational map whose branch datum is given by these partitions. In this paper we give a new proof of Wei's theorem on realizing a tuple of partitions as the ramification data of a rational function. The proof is inductive on the shortest partition length. Two auxiliary results are established: a two-cut partition theorem and a gluing theorem for marked spherical coverings. These reduce any counterexample with shortest length $\ell\ge3$ to auxiliary data with either fewer factors or smaller shortest length, while preserving the total defect and the marked conditions. The base case $\ell\le2$ is quoted from the literature.