{
  "abstract": "The well-known open problem of Bell and Skandera asks whether a real-rooted polynomial $f(t)$ with positive integer coefficients and constant term one is the $f$-polynomial of a simplicial complex. Mu and Welker proved that if the recursive decomposition $f(t)=g(t)+th(t)$ satisfies the corresponding coefficient inequality $h_i<g_i$, then this open problem has an affirmative answer. Mu and Welker also conjectured that the real-rootedness of $f(t)$ implies that of $g(t)$ and $h(t)$. We give counterexamples to the conjecture of Mu and Welker for every degree at least three, and prove that the assertion holds in degrees $1$ and $2$. Moreover, each polynomial we construct is the $f$-polynomial of a simplicial complex.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23694",
  "author_total": 3,
  "authors": [
    {
      "name": "Feihu Liu"
    },
    {
      "name": "Ying Wang"
    },
    {
      "name": "Zihao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0td8yhx3nrk69.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23694",
  "fulltext_url": null,
  "id": "0td8yhx3nrk69",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23694",
  "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": "W7214032222",
  "pdf_url": "https://arxiv.org/pdf/2609.23694",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0td8yhx3nrk69.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.23694",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23694"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counterexamples to the Mu--Welker recursive decomposition in every degree",
  "title_text": "Counterexamples to the Mu--Welker recursive decomposition in every degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/0td8yhx3nrk69",
  "venue": "arXiv",
  "versions": []
}