Algebro-gemetric aspects of Heine-Stieltjes theory

TL;DR

This paper extends Heine-Stieltjes theory to higher-order univariate differential operators with polynomial coefficients, establishing existence and enumeration of polynomial solutions and their spectral polynomials under mild conditions.

Contribution

It develops a generalized Heine-Stieltjes framework for higher-order differential operators, including existence, uniqueness, and enumeration of polynomial solutions.

Findings

Exactly ((n+r) choose n) such polynomials V_n,i(z) exist under mild conditions

Generalizes classical results in Heine-Stieltjes theory to higher-order operators

Discusses degeneracies and their impact on solutions

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=\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.