Compatibility of Real-Rooted Polynomials with Mixed Signs
Combinatorics
2024-08-06 v2
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.
Cite
@article{arxiv.2407.16226,
title = {Compatibility of Real-Rooted Polynomials with Mixed Signs},
author = {Jonathan Leake and Nick Ryder},
journal= {arXiv preprint arXiv:2407.16226},
year = {2024}
}