{
  "abstract": "We prove that the Ehrhart $h^*$-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and $(4,a,a+29)$, for all sufficiently large integers $a$, have $h^*$-polynomials that are not real-rooted. These counterexamples disprove Ferroni's real-rootedness conjecture. Their duals are cycle matroids of theta graphs and have the same $h^*$-polynomials. Nevertheless, every matroid of rank two or corank two has a positive $h^*$-coefficient sequence that is ultra log-concave of order equal to the polynomial's degree. In particular, the unimodality conjecture holds in both cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37439",
  "author_total": 1,
  "authors": [
    {
      "name": "Houshan Fu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fmv69s30aaw2j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37439",
  "fulltext_url": null,
  "id": "fmv69s30aaw2j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37439",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214925599",
  "pdf_url": "https://arxiv.org/pdf/2609.37439",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fmv69s30aaw2j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37439",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37439"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart $h^*$-polynomials",
  "title_text": "Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart h^*-polynomials",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/fmv69s30aaw2j",
  "venue": "arXiv",
  "versions": []
}