Reduction of Zieve's Conjecture on the Hurwitz problem to Three Branch Points

Abstract

M. E. Zieve conjectured that a candidate datum over the sphere is realizable whenever the gcd of the parts of every branch partition is one and it satisfies the lcm condition. We prove that this conjecture reduces to its three-branch-point case. More precisely, if every Zieve-admissible triple is realizable, then every Zieve-admissible datum with at least four branch points is realizable. In particular, Zieve's conjecture implies the prime-degree conjecture posed by A. L. Edmonds, R. S. Kulkarni and R. E. Stong. Generative-AI disclosure. OpenAI's GPT-5.6 Sol and GPT-6 Astra assisted throughout the research and preparation of this article, including the exploration of reduction strategies, proof development and checking, consistency checks on notation and terminology, and manuscript drafting and revision. The authors reviewed all AI-assisted material with careful scrutiny and take full responsibility for all mathematical claims and for the final text.

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BibTeX RIS

BibTeX

@misc{liu2026reduction,
  title = {{Reduction of Zieve's Conjecture on the Hurwitz problem to Three Branch Points}},
  author = {Xitaiyu Liu and Tianyang Sun and Bin Xu and Yu Ye and Yi Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23663},
  eprint = {2609.23663},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23663},
}

RIS

TY  - UNPB
TI  - Reduction of Zieve's Conjecture on the Hurwitz problem to Three Branch Points
AU  - Xitaiyu Liu
AU  - Tianyang Sun
AU  - Bin Xu
AU  - Yu Ye
AU  - Yi Zhou
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23663
UR  - https://arxiv.org/abs/2609.23663
AB  - M. E. Zieve conjectured that a candidate datum over the sphere is realizable whenever the gcd of the parts of every branch partition is one and it satisfies the lcm condition. We prove that this conjecture reduces to its three-branch-point case. More precisely, if every Zieve-admissible triple is realizable, then every Zieve-admissible datum with at least four branch points is realizable. In particular, Zieve's conjecture implies the prime-degree conjecture posed by A. L. Edmonds, R. S. Kulkarni and R. E. Stong. Generative-AI disclosure. OpenAI's GPT-5.6 Sol and GPT-6 Astra assisted throughout the research and preparation of this article, including the exploration of reduction strategies, proof development and checking, consistency checks on notation and terminology, and manuscript drafting and revision. The authors reviewed all AI-assisted material with careful scrutiny and take full responsibility for all mathematical claims and for the final text.
ER  -

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