Compatibility of Real-Rooted Polynomials with Mixed Signs

TL;DR

This paper characterizes families of real-rooted polynomials with mixed signs, extending previous work on same-sign polynomials and settling an open question using elementary linear algebra and interlacing polynomial theory.

Contribution

It provides a complete characterization of compatible real-rooted polynomial families with mixed signs, generalizing prior results on same-sign polynomials.

Findings

Generalizes the same-sign characterization to mixed signs

Settles an open question from Chudnovsky and Seymour's 2007 paper

Uses elementary methods with linear algebra and interlacing polynomials

Abstract

We characterize compatible families of real-rooted polynomials, allowing both positive and negative leading coefficients. Our characterization naturally generalizes the same-sign characterization used by Chudnovsky and Seymour in their famous 2007 paper proving the real-rootedness of independence polynomials of claw-free graphs, thus fully settling a question left open in their paper. Our methods are generally speaking elementary, utilizing mainly linear algebra and the established theory of interlacing polynomials, with a bit of invariant theory.