{
  "abstract": "Let $X$ be a smooth complex affine variety of dimension $n$, and let $F=(f_1,\\ldots,f_r)$ be a tuple of nonzero regular functions on $X$ such that $f:=\\prod_{i=1}^r f_i$ is not invertible. We study the zero loci of the Bernstein-Sato ideals $B_F^{\\mathbf a}$ for nonnegative integral shifts $\\mathbf a$. For a fixed log resolution, every codimension-one irreducible component of $Z(B_F^{\\mathbf a})$ is a hyperplane of the form $L_E(\\mathbf s)+k_E+c=0$ with $c$ a positive integer. We give a new proof of this result using localized maximal and minimal extensions of relative D-modules. We also prove that $c\\leq L_E(\\mathbf a)+(n-1-δ_f)L_E(\\mathbf 1)-k_E$, where $δ_f=\\min\\{n-1,α_f\\}$ and $α_f$ is the minimal exponent of $f$. The problem of obtaining such an upper bound for arbitrary tuples (in particular, for $r>1$) was raised by Budur, van der Veer, and Van Werde, and the above inequality resolves it. To obtain the upper bound, we compare the diagonal slice of $Z(B_F^{\\mathbf 1})$ with the root set of $b_f$. A finite covering by translates, combined with diagonal specialization and the log-resolution description, shows that these sets have the same least and greatest points. Saito's root estimate at their common least point then yields the upper bound. We further establish a divisor-valued formulation of the local index comparison, recovering the detection of monodromy support via monodromy zeta functions and the multivariable A'Campo formula.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19869",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenzong Guo"
    },
    {
      "name": "Fanghan Xiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/csej99sxqcdd8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19869",
  "fulltext_url": null,
  "id": "csej99sxqcdd8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19869",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213612886",
  "pdf_url": "https://arxiv.org/pdf/2609.19869",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/csej99sxqcdd8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.19869",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19869"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Bounds for Codimension-One Components of Zero Loci of Bernstein-Sato Ideals",
  "title_text": "Bounds for Codimension-One Components of Zero Loci of Bernstein-Sato Ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/csej99sxqcdd8",
  "venue": "arXiv",
  "versions": []
}