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Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants

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Lewański,  Danilo
Max Planck Institute for Mathematics, Max Planck Society;

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Kramer, R., Lewański, D., & Shadrin, S. (2019). Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants. Documenta Mathematica, 24, 857-898. doi:10.25537/dm.2019v24.857-898.


Cite as: https://hdl.handle.net/21.11116/0000-0005-41BF-B
Abstract
We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers



and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of



quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are the



necessary conditions for the Chekhov-Eynard-Orantin topological recursion.