Enumeration of Grothendieck's Dessins and KP Hierarchy

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Enumeration of Grothendieck's Dessins and KP Hierarchy. / Zograf, Peter.

In: International Mathematics Research Notices, Vol. 2015, No. 24, 2015, p. 13533-13544.

Research output: Contribution to journal › Article › peer-review

Author

Zograf, Peter. / Enumeration of Grothendieck's Dessins and KP Hierarchy. In: International Mathematics Research Notices. 2015 ; Vol. 2015, No. 24. pp. 13533-13544.

BibTeX

@article{78913c52f2614b879da1c9e60b072689,

title = "Enumeration of Grothendieck's Dessins and KP Hierarchy",

abstract = "Branched covers of the complex projective line ramified over 0, 1, and infinity (Grothendieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over infinity and given numbers of preimages of 0 and 1 are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev-Petviashvili (KP) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of dessins of given genus and degree, thus providing a fast algorithm for computing these numbers.",

author = "Peter Zograf",

year = "2015",

doi = "10.1093/imrn/rnv077",

language = "English",

volume = "2015",

pages = "13533--13544",

journal = "International Mathematics Research Notices",

issn = "1073-7928",

publisher = "Oxford University Press",

number = "24",

}

RIS

TY - JOUR

T1 - Enumeration of Grothendieck's Dessins and KP Hierarchy

AU - Zograf, Peter

PY - 2015

Y1 - 2015

N2 - Branched covers of the complex projective line ramified over 0, 1, and infinity (Grothendieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over infinity and given numbers of preimages of 0 and 1 are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev-Petviashvili (KP) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of dessins of given genus and degree, thus providing a fast algorithm for computing these numbers.

AB - Branched covers of the complex projective line ramified over 0, 1, and infinity (Grothendieck's dessins d'enfant) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over infinity and given numbers of preimages of 0 and 1 are considered. The generating function for the numbers of such covers is shown to satisfy a partial differential equation (PDE) that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the Kadomtsev-Petviashvili (KP) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of dessins of given genus and degree, thus providing a fast algorithm for computing these numbers.

U2 - 10.1093/imrn/rnv077

DO - 10.1093/imrn/rnv077

M3 - Article

VL - 2015

SP - 13533

EP - 13544

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 24

ER -

ID: 4007468