{
  "abstract": "Let $S=\\mathbb{K}[x_1,\\ldots,x_n]$ be a polynomial ring over a field $\\mathbb{K}$. For a polymatroidal ideal $I\\subseteq S$ generated in degree $d$, we prove $$ω(I^{(s)})=\\operatorname{reg} I^{(s)}=\\operatorname{reg} I^s=sd\\qquad(s\\ge1),$$ where symbolic powers use all associated primes and $ω$ denotes the largest degree of a minimal generator. Thus every symbolic power has a degree resolution. The argument uses a regularity bound for supermodular prime intersections, obtained from Takayama's formula and a homological estimate for noncover complexes. For every nonzero proper squarefree monomial ideal $L$, we also show $\\operatorname{reg} L^{(s)}\\le s\\min\\{m,λ(L)-\\operatorname{ht}L+1\\}$, where $m$ counts the minimal primes and $λ(L)$ is the support size, and obtain equality for skeleton and pure skeleton ideals. Finally, for every $d\\ge3$, we construct a simple graphic matroid whose basis ideal is generated in degree $d$ and has a second symbolic power that is not componentwise linear.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38655",
  "author_total": 2,
  "authors": [
    {
      "name": "Chwas Ahmed"
    },
    {
      "name": "Mohammed Rafiq Namiq"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtmj96xw2a8w6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38655",
  "fulltext_url": null,
  "id": "dtmj96xw2a8w6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38655",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38655",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtmj96xw2a8w6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38655",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38655"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections",
  "title_text": "Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/dtmj96xw2a8w6",
  "venue": "arXiv",
  "versions": []
}