INTERSECTIONS OF SCHUBERT VARIETIES AND CRITICAL POINTS OF THE GENERATING FUNCTION

Abstract

The theory of order $p$ linear differential equations with univalued solutions can be presented in terms of the Schubert calculus in the Grassmannian of $p$-dimensional subspaces of the vector space of complex polynomials in one variable. The regular singular points and the exponents of an equation determine an intersection of Schubert varieties.

Heine and Stieltjes in their studies of order $p = 2$ linear differential equations with polynomial coefficients having a polynomial solution of a prescribed degree discovered that such a solution was given by a critical point of a certain remarkable symmetric function. In terms of the Schubert calculus, the critical points determine the elements of the intersection of Schubert varieties.

The case $p > 2$ is studied. A function whose critical points determine the non-degenerate elements in the intersection of Schubert varieties is presented. The number of Fuchsian equations with univalued solutions and prescribed exponents at singular points is estimated from above by the intersection number of the corresponding Schubert classes. Conjecturally, for generic disposition of singular points, these numbers coincide, that is, the intersection of Schubert varieties is transversal and consists of non-degenerate elements only. For $p = 2$ the conjecture was proved earlier by the author.