{
  "abstract": "A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal $I$ of a polynomial ring $S$, the regularities of $S/I$ and $S/\\textrm{GIN}(I)$ are the same under the reverse lexicographic monomial ordering, where $\\textrm{GIN}(I)$ is the generic initial ideal. If $R$ is a polynomial ring whose variables are subdivided into disjoint blocks of variables $X_1,\\dots,X_c$, there is a natural multi-grading on $R$, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal $I$ of $R$. However, the full strength of the Bayer--Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals $I$ such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of $R/I$ in terms of almost regular sequences of the multi-graded generic initial ideal of $I$ restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for $d$-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25891",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel McGinnis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/01a5qxbb140h5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25891",
  "fulltext_url": null,
  "id": "01a5qxbb140h5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25891",
  "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"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7204436317",
  "pdf_url": "https://arxiv.org/pdf/2608.25891",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/01a5qxbb140h5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.25891",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25891"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for $d$-Leray complexes",
  "title_text": "Multi-graded generic initial ideals, regularity, and the optimal colorful fractional Helly theorem for d-Leray complexes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/01a5qxbb140h5",
  "venue": "arXiv",
  "versions": []
}