Rational maps admit a canonical decomposition of their pullback metrics into footballs, which implies that branch data consisting of k nontrivial partitions are realizable whenever k exceeds the shortest partition length by more than one.
Hurwitz existence problem and fiber products.arXiv:2408.10874v1 [math.CV] (2024)
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Structure and realizability for rational maps
Rational maps admit a canonical decomposition of their pullback metrics into footballs, which implies that branch data consisting of k nontrivial partitions are realizable whenever k exceeds the shortest partition length by more than one.