{
  "abstract": "Following \\cite {KSW} we continue the study the cellular complexes formed by polynomials of a given degree having a given sequence of multiplicities of real roots. A computer-assisted calculation disproves the earlier conjecture that homology of one point compactification of the closure of such cell is concentrated in at most one degree. Namely, for $ω=(3,1,1,3)$ in degree $d=18$, the reduced homology is $\\ZZ^2$ in degree $7$. The obstruction is already visible in a signed cell count, whose value is $-2$. We relate that count to the rational signed weight enumerator $F_ω(t)=\\sum_{η\\preceqω}(-1)^{\\elln(η)}t^{|η|}$. For the counterexample, $F_ω(t)=-t^{14}/(1+t^2)^2$, which gives an exact linear formula for the Euler characteristic and forces the total rational Betti number to be unbounded. We prove a number of results and formulate a complete conjecture describing the above homology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19733",
  "author_total": 1,
  "authors": [
    {
      "name": "Boris Shapiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pxr8qkwn1trm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19733",
  "fulltext_url": null,
  "id": "5pxr8qkwn1trm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19733",
  "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.AT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7203949840",
  "pdf_url": "https://arxiv.org/pdf/2608.19733",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pxr8qkwn1trm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.19733",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19733"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Real polynomials with given multiplicities of real roots: Complete conjectural description of homology",
  "title_text": "Real polynomials with given multiplicities of real roots: Complete conjectural description of homology",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/5pxr8qkwn1trm",
  "venue": "arXiv",
  "versions": []
}