Zheng's realizability criterion for branch data

Let d=ddd=d'd”, where dd' is the minimal prime factor of dd and d>2d”>2, so that dd is neither prime nor equal to 44. Let D\mathcal{D} be branch data with kk partitions, and suppose it satisfies the Hurwitz condition. Zheng's realizability criterion. If k>d+1k>d”+1, then D\mathcal{D} is realizable. Moreover, this bound is sharp: the displayed collection of partitions in the statement, with k=d+1k=d”+1 and ν(D)=2d2\nu(\mathcal{D})=2d-2, is non-realizable. This criterion gives a sufficient condition for realizability together with an explicit sharpness example; the source does not provide a separate resolution status for it.

Sources & referencesView supporting material

Primary source

Zhiqiang Wei, “Structure and realizability for rational maps”, arXiv:2511.06784 (2026).

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