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