{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17135",
  "author_total": 4,
  "authors": [
    {
      "name": "Gergely Bérczi"
    },
    {
      "name": "Basile Coron"
    },
    {
      "name": "Christopher Eur"
    },
    {
      "name": "Shiyue Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2m6mtpamnp3a9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17135",
  "fulltext_url": null,
  "id": "2m6mtpamnp3a9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17135",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221835440",
  "pdf_url": "https://arxiv.org/pdf/2610.17135",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2m6mtpamnp3a9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17135",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17135"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Chow polynomials and chain polynomials of matroids are not log-concave",
  "title_text": "Chow polynomials and chain polynomials of matroids are not log-concave",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2m6mtpamnp3a9",
  "venue": "arXiv",
  "versions": []
}