Hyperbolicity preservers and majorization

TL;DR

This paper characterizes linear operators on polynomials that preserve majorization, showing that such operators are essentially those that preserve hyperbolicity, thus linking majorization order with hyperbolic polynomial transformations.

Contribution

It provides a complete characterization of linear hyperbolicity preservers that also preserve the majorization order on univariate hyperbolic polynomials.

Findings

Linear operators preserving majorization are characterized.

Preserving hyperbolicity is sufficient for majorization preservation.

The work links majorization order with hyperbolic polynomial transformations.

Abstract

The majorization order on $\RR^n$ induces a natural partial ordering on the space of univariate hyperbolic polynomials of degree $n$. We characterize all linear operators on polynomials that preserve majorization, and show that it is sufficient (modulo obvious degree constraints) to preserve hyperbolicity.