{
  "abstract": "Let $f=(f_1,\\dots,f_r)$ be a complete factorization of a central hyperplane arrangement $D$ in $X=\\mathbb{C}^n$. For a monoid ideal $K\\subseteq \\mathbb{N}^r$ we study the Bernstein-Sato ideal $B^K_f$ of $f$ along $K$, that is, the $\\mathbb{C}[s]$-annihilator of $\\mathcal{D}_X[s]f^s/\\sum_{m\\in K}\\mathcal{D}_X[s]f^{s+m}$. When $D$ is free we compute two families of these ideals with the help of AI. For the unit shift $K=\\langle e_i\\rangle$ we prove that $B^{-e_i}_f$ is generated by an explicit product of linear forms indexed by the dense edges of $D$ contained in $D_i$. This determines all the Bernstein-Sato ideals $B^{a,b}_f=\\operatorname{Ann}_{\\mathbb{C}[s]}\\mathcal{D}_X[s]f^{s-a}/\\mathcal{D}_X[s]f^{s-b}$, $a\\geq b$, of a free arrangement, generalizing formulas of Maisonobe (2016) and Bath (2020). The main new ingredient identifies the multiplicities of the relative characteristic cycle of $\\mathcal{D}_X[s]f^s/\\mathcal{D}_X[s]f^{s+e_i}$ along the conormal bundle of the origin with the coefficients of the Hilbert series of an Artinian complete intersection attached to a generic Ziegler restriction of $D$; the total multiplicity computed in Wu (2022) then forces all the resulting coefficientwise upper bounds to be equalities. For the coordinate monoid ideal $K=\\langle e_1,\\dots,e_r\\rangle$ we show that $B^K_f$ is generated by one Euler relation for each irreducible factor of the essential quotient of $D$. Finally, we show that the zero locus of a Bernstein-Sato ideal along a monoid ideal need not be a finite union of translated linear subvarieties, even for a reduced free arrangement in $\\mathbb{C}^2$: for $f=(x,y,x+y,x+2y)$ and $K=\\langle 3e_1,3e_2\\rangle$ we compute $B^K_f$ exactly and find an irreducible quadric component. This disproves a conjecture due to Budur.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26336",
  "author_total": 3,
  "authors": [
    {
      "name": "Wenzong Guo"
    },
    {
      "name": "Lei Wu"
    },
    {
      "name": "Fanghan Xiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d037wxkwtb91d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26336",
  "fulltext_url": null,
  "id": "d037wxkwtb91d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26336",
  "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.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7214099114",
  "pdf_url": "https://arxiv.org/pdf/2609.26336",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d037wxkwtb91d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26336",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26336"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Bernstein-Sato ideals for free hyperplane arrangements",
  "title_text": "Bernstein-Sato ideals for free hyperplane arrangements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/d037wxkwtb91d",
  "venue": "arXiv",
  "versions": []
}