{
  "abstract": "Recently in [6] the authors proved a lower bound for the $\\GSV$ index of a logarithmic vector field along an isolated hypersurface singularity and proposed a conjecture that the weighted homogeneity of this hypersurface singularity can be detected by the existence of either a transverse or a non-degenerate holomorphic logarithmic vector fields. In this paper we prove this conjecture affirmatively. We also prove that the $\\GSV$ index attains the lower bound if and only if the hypersurface singularity is weighted homogeneous.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.29886",
  "author_total": 2,
  "authors": [
    {
      "name": "Jihao Liu"
    },
    {
      "name": "Xiping Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5f8paaph8ehmf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.29886",
  "fulltext_url": null,
  "id": "5f8paaph8ehmf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.29886",
  "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": "W7166690602",
  "pdf_url": "https://arxiv.org/pdf/2606.29886",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5f8paaph8ehmf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2606.29886",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.29886"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Criteria for Weighted Homogeneity via Logarithmic Vector Fields",
  "title_text": "Criteria for Weighted Homogeneity via Logarithmic Vector Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5f8paaph8ehmf",
  "venue": "arXiv",
  "versions": []
}