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. As already noted by Petronio, Zieve's conjecture would imply the prime-degree conjecture of Edmonds, Kulkarni, and Stong. Song, Wen, and Zhang proved the latter conjecture in a preprint posted on arXiv in September.

arXiv commentSubstantially shortened and reorganized. The proof in Section 3 has been rewritten, and the exposition and references have been updated. The main theorem is unchanged.The number of pages has been reduced from 39 to 23