A New Proof of Wei's Theorem

In the paper \emph{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 $k>\mathrm{\ell }+1$, where $$\mathrm{\ell }=\underset{i}{\min }\mathrm{\mid }{\pi }_i\mathrm{\mid }$$ 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 $\mathrm{\ell }\ge 3$ to auxiliary data with either fewer factors or smaller shortest length, while preserving the total defect and the marked conditions. The base case $\mathrm{\ell }\le 2$ is quoted from the literature.