Rational Functions with Prescribed Critical Points

By I. Scherbak. 2002

52 citations recorded by the source corpus.

Abstract.A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into the other by postcomposition with a linear-fractional transformation. We give an explicit formula for the number of classes having a given degree d and given multiplicities m1,..., mn of given n critical points, for generic positions of the critical points. This number is the multiplicity of the irreducible sl2 representation with highest weight $2d - 2 - m_1 - \cdots - m_n $ in the tensor product of the irreducible sl2 representations with highest weights $m_1 , \ldots ,m_n .$The classes are labeled by the orbits of critical points of a remarkable symmetric function which first appeared in the XIX century in studies of Fuchsian differential equations, and then in the XX century in the theory of KZ equations.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/04687a15f81b6e5f2eafa8fdabf4e3e15429da8a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. V. Tarasov
2. Alexandre Eremenko
3. Andrei Gabrielov
4. Dmitri Panov
5. Gabriele Mondello
6. E. Mukhin
7. A. Varchenko

In Geometry and Topology

1. V. Tarasov
2. Alexandre Eremenko
3. Andrei Gabrielov
4. Dmitri Panov
5. Gabriele Mondello
6. Bin Xu
7. A. Varchenko
8. E. Mukhin

In Statistical and Nonlinear Physics

1. Dmitri Panov
2. Gabriele Mondello
3. V. Tarasov
4. A. Varchenko

Citations in the corpus

• Metrics with conic singularities and spherical polygons (2014, background): “For a generic choice of the critical points aj , the number of classes of rational functions is equal to the Kostka number, see [32], [8].”
• Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces (2013, background): “The beautiful theorem in [Scherbak 2002] says that there is a least upper bound given by the Schubert calculus for the number of equivalent classes of all the rational functions with ramification divisor D, which can be achieved by a generic choice of the support of D.”
• Spherical Metrics with Conical Singularities on a 2-Sphere: Angle Constraints (2015, background): “cs is also provided. Notice that the general case of χ(S,ϑ) >0 and ϑ ∈/(0,1]n is not covered by the above works. Another manifestation of the non-uniqueness of the solution is provided by Scherbak [15], who counted the exact number of such metrics for almost all configurations of x1,...,xn ∈ S, wh”