An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators

TL;DR

This paper investigates the properties of certain invariant subsets of the complex plane under linear differential operators with polynomial coefficients, focusing on their existence, minimality, and root-preserving characteristics.

Contribution

It introduces a framework for analyzing invariant subsets under differential operators, extending Polya-Schur theory to non-generate and degenerate cases.

Findings

Identifies conditions for invariant subsets under specific operators

Establishes existence of minimal invariant subsets

Provides insights into root-preserving properties of differential operators

Abstract

Given a linear ordinary differential operator T with polynomial coefficients, we study the class of closed subsets of the complex plane such that T sends any polynomial (resp. any polynomial of degree exceeding a given positive integer) with all roots in a given subset to a polynomial with all roots in the same subset or to 0. Below we discuss some general properties of such invariant subsets as well as the problem of existence of the minimal under inclusion invariant subset.