A generalization of the Heine--Stieltjes theorem

TL;DR

This paper generalizes the Heine-Stieltjes theorem to a broader class of differential operators that preserve real zeros, solving a longstanding conjecture and revealing new interlacing properties of solutions.

Contribution

It extends the classical theorem to all non-degenerate operators preserving real zeros, addressing a conjecture and introducing new methods for analyzing zero interlacing.

Findings

Extended the Heine-Stieltjes theorem to all relevant differential operators.

Described intricate interlacing relations between zeros of solutions.

Provided new results even for classical cases.

Abstract

We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question on how to generalize their results to higher degrees. Many of the results are new even for the classical case.