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Boris Bychkov

Publications and source records attributed to Boris Bychkov.

21 records · Page 2Linked to original sources

arXiv2019

Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion

In this paper we prove, in a purely combinatorial way, a structural quasi-polynomiality property for the Bousquet-Mélou-Schaeffer numbers. Conjecturally, this property should follow from the Chekhov-Eynard-Orantin topological recursion for these numbers (or, to be more precise, the Bouchard-Eynard version of the topological recursion for higher order critical points), which we derive in this paper from the recent result of Alexandrov-Chapuy-Eynard-Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental's theory for cohomological field theories.

math.CO↗

arXiv2016

Degrees of cohomological classes of multisinguliarities in Hurwitz spaces of rational functions

The main goal of the present paper are new formulae for degrees of strata in Hurwitz spaces of rational functions having two degenerate critical values with preimages of prescribed multiplicities. We consider the case where the multiplicities of the preimages of one critical value are arbitrary, while the second critical value has degeneracy of codimension~$1$. Our formulae are based on the universal cohomological expressions for codimension~$1$ strata in terms of certain basic cohomology classes in general Hurwitz spaces of rational functions obtained by M.~Kazarian and S.~Lando. We prove new relations valid in cohomology of Hurwitz spaces that were conjectured by M.~Kazarian on the base of computer experiments. As a corollary, we obtain new, previously unknown, explicit formulae for certain families of double Hurwitz numbers in genus~$0$. One may hope that the methods developed in the present paper are applicable to proving more general relations in cohomology rings of Hurwitz spaces and deducing more general formulae for double Hurwitz numbers.

math.AG↗

arXiv2014

On the number of coverings of the sphere ramified over given points

We present the generating function for the numbers of isomorphism classes of coverings of the two-dimensional sphere by the genus $g$ compact oriented surface not ramified outside of a given set of $m+1$ points in the target, fixed ramification type over one point, and arbitrary ramification types over the remaining $m$ points. We present the genus expansion of this generating function and prove, that the generating function of coverings of genus $0$ satisfies some system of differential equations. We show that this generating function is a specialization of the function from paper \cite{GJ} and, therefore, satisfies the KP-hierarchy.

math.CO↗