Algebro-gemetric aspects of Heine-Stieltjes theory

Details Algebro-gemetric aspects of Heine-Stieltjes theory Algebro-gemetric aspects of Heine-Stieltjes theory

2008-12-22

arxiv.org/abs/0812.4193v1

Physics

Mathematical Physics

25 pages, 2 figures, contains a translation of a difficult and often quoted passage from Heine's book 'Handbuch der Kugelfunct

Scientific paper

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=\sum_i Q_i(z)d^i/dz^i with polynomial coefficients Q_i(z) set r=max_i (deg Q_i(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 V_n,i(z). We generalize a number of classically known results in this area and discuss occurring degeneracies.

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