Rational functions with prescribed critical points

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 the postcomposition with a linear-fractional transformation. We give an explicit formula for the number of classes having a given degree d and given multiplicities m_1, ..., m_n of given n critical points, for generic positions of the critical points. This number is the multiplicity of the irreducible sl(2) representation with highest weight 2d-2-m_1- ... -m_n in the tensor product of the irreducible sl(2) representations with highest weights m_1, ..., 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.