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.
Related
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- Characteristic Chow Polynomials of Layered-Permutation Intervals
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BibTeX
@misc{berczi2026chow,
title = {{Chow polynomials and chain polynomials of matroids are not log-concave}},
author = {Gergely Bérczi and Basile Coron and Christopher Eur and Shiyue Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17135},
eprint = {2610.17135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17135},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Chow polynomials and chain polynomials of matroids are not log-concave
AU - Gergely Bérczi
AU - Basile Coron
AU - Christopher Eur
AU - Shiyue Li
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17135
UR - https://arxiv.org/abs/2610.17135
AB - 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.
ER -