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arXiv · 1312.2538

Enumeration of Grothendieck's dessins and KP hierarchy

Peter Zograf
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Abstract

Branched covers of the complex projective line ramified over $0,1$ and $\infty$ (Grothendieck's {\em dessins d'enfant}) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over $\infty$ 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 PDE that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the KP (Kadomtsev-Petviashvili) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of {\em dessins} of given genus and degree, thus providing a fast algorithm for computing these numbers.

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Peter Zograf. 2014-03-22. Enumeration of Grothendieck's dessins and KP hierarchy. https://arxiv.org/abs/1312.2538

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