{
  "abstract": "Explicit sequences of forms with the Hilbert series predicted by Fröberg's conjecture are difficult to construct beyond the almost complete intersection case. Over a field of characteristic zero, we prove that six forms of degree $d\\ge4$ in four variables cannot have the predicted series if each is divisible by a power of a linear form with exponent at least $(15d+91)/16$. A power of the product of six quadric cones gives a nonzero element of the inverse system in a degree where the predicted series vanishes. This obstruction implies that every fixed-order Taylor truncation of the homogeneous Rogers--Szegő polynomial at $q=1$ fails to complete five powers of linear forms to the predicted series in all sufficiently large degrees. As a fixed-degree counterpart, we use exact certificates to prove the conjecture for any number of general quintics or septics; together with known results, this establishes the equal-degree conjecture in four variables for $1\\le d\\le9$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24797",
  "author_total": 1,
  "authors": [
    {
      "name": "Qihang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0r9e7vbkmgw81.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24797",
  "fulltext_url": null,
  "id": "0r9e7vbkmgw81",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24797",
  "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.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7204162060",
  "pdf_url": "https://arxiv.org/pdf/2608.24797",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0r9e7vbkmgw81.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2608.24797",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24797"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hilbert Series in Four Variables and Linear Factors of High Multiplicity",
  "title_text": "Hilbert Series in Four Variables and Linear Factors of High Multiplicity",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:14Z",
  "url": "https://research.draconitic.com/paper/0r9e7vbkmgw81",
  "venue": "arXiv",
  "versions": []
}