ALGEBRO-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) = Pk i=1 Qi(z) di dzi with polynomial coefficients set r = maxi=1,...,k(degQi(z)−i). If d(z) satisfies the conditions: i) r   0 and ii) degQk(z) = k + r we call it a non-degenerate higher Lam´e 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 Vn,i(z) whose corresponding eigenpolynomials Sn,i(z) are of degree n. We generalize a number of well-known results in this area and discuss occurring degeneracies.