2024/07/23 by Leake, Jonathan, Ryder, Nick
#05A20 #26C10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.16226
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.