Chow polynomials and chain polynomials of matroids are not log-concave

Abstract

The Hilbert--Poincaré series of the Chow rings of the toric varieties of matroids, often called the \emph{Chow polynomials}, and the $f$-polynomial of their boundary complexes, called the \emph{chain polynomials}, have been conjectured to only have real zeros. We construct several explicit infinite families of matroids from finite geometries, whose multichain counts fail to be log-concave. They share a common geometric construction called \emph{hyperplane coalescence}, together with several classical operations in projective geometry and combinatorial designs. This induces failures of log-concavity for Chow, augmented Chow, chain, and $h$-polynomials of several explicit infinite families of matroids, disproving the real-rootedness conjectures of Ferroni--Schröter, Huh--Stevens, and Athanasiadis--Kalampogia-Evangelinou from the lowest possible ranks. As an application of the same mechanism, we also give counterexamples to the log-concavity of $h$-polynomials of graph associahedra, answering in the negative a question of Sack.