Geometric Aspects of Heine - Stieltjes Theory

The goal of the paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = P k i=1 Q i (z) d i dz i with polynomial coefficients set r = max i=1,...,k (deg Q i (z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Q k (z) = k + r we call it a non-degenerate higher Lamé operator. Following the classical approach of E. Heine and T. Stieltjes, see [18], [41] we study the multiparameter spectral problem of finding all polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild non-degeneracy assumptions there exist exactly`n+r n ´ such polynomials V n,i (z) whose corresponding eigenpolynomials S n,i (z) are of degree n. We generalize a number of well-known results in this area and discuss occurring degeneracies.