A New Proof of Wei's Theorem
Abstract
In the paper \emph{Structure and realizability for rational maps}, Wei proved the following. Let k nontrivial partitions of an integer d≥ 3 be given, satisfying the Riemann--Hurwitz formula for the case of maps between two spheres. If k>ℓ+1, where ℓ=mini|πi| 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 ℓ≥3 to auxiliary data with either fewer factors or smaller shortest length, while preserving the total defect and the marked conditions. The base case ℓ≤2 is quoted from the literature.
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BibTeX
@misc{wei2026proof,
title = {{A New Proof of Wei's Theorem}},
author = {Zhiqaing Wei and Mengyan Xing and Xingya Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25838},
eprint = {2610.25838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25838},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A New Proof of Wei's Theorem
AU - Zhiqaing Wei
AU - Mengyan Xing
AU - Xingya Zhao
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25838
UR - https://arxiv.org/abs/2610.25838
AB - In the paper \emph{Structure and realizability for rational maps}, Wei proved the following. Let k nontrivial partitions of an integer d≥ 3 be given, satisfying the Riemann--Hurwitz formula for the case of maps between two spheres. If k>ℓ+1, where ℓ=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 ℓ≥3 to auxiliary data with either fewer factors or smaller shortest length, while preserving the total defect and the marked conditions. The base case ℓ≤2 is quoted from the literature.
ER -