Algebro-gemetric aspects of Heine-Stieltjes theory
Abstract
The goal of this paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given given operator T=Σi Qi(z)di/dzi with polynomial coefficients Qi(z) set r=maxi (deg Qi(z)-i). Following the classical approach of Heine and Stieltjes we study the multiparameter spectral problem of finding all polynomial V(z) of degree at most r such that the equation: T(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) choose n) such polynomials Vn,i(z). We generalize a number of classically known results in this area and discuss occurring degeneracies.
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