Geodesic
Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants
Documenta mathematica, Tome 24 (2019), pp. 857-898
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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 necessary conditions for the Chekhov-Eynard-Orantin topological recursion.
DOI : 10.4171/dm/695
Classification : 05E05, 14H57, 14N10
Mots-clés : enumerative geometry, dessins d'enfants, Hurwitz numbers, spectral curves 
@article{10_4171_dm_695,
     author = {Danilo Lewa\'nski and Reinier Kramer and Sergey Shadrin},
     title = {Quasi-Polynomiality of {Monotone} {Orbifold} {Hurwitz} {Numbers} and {Grothendieck's} {Dessins} {d'Enfants}},
     journal = {Documenta mathematica},
     pages = {857--898},
     year = {2019},
     volume = {24},
     doi = {10.4171/dm/695},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/695/}
}
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Danilo Lewański; Reinier Kramer; Sergey Shadrin. Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants. Documenta mathematica, Tome 24 (2019), pp. 857-898. doi: 10.4171/dm/695

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