{
  "abstract": "Let $f$ be an isolated hypersurface singularity and let $J(f)$ be its Jacobian ideal. We study the higher Milnor numbers $μ_k(f)=\\dim_{\\mathbb C}\\mathcal O_n/\\mathfrak m^kJ(f).$ Our main result concerns an ordinary semi-homogeneous singularity $f=f_r+f_{r+1}+\\cdots$, where the projective hypersurface defined by $f_r$ is smooth. We prove that $μ_k(f)=μ_k(f_r) \\qquad\\text{for every }k\\geq0.$ This gives the equal-weight case of a conjecture of Shi, Yau, and Zuo. The main algebraic input is the common-order case of the Valabrega--Valla criterion. From it, we obtain the graded Jacobian module and the exact intersection formula. As a consequence, we obtain a closed formula for $μ_k$ and determine the sharp stabilization index of $\\mathfrak m^kJ(f)$. We also determine the optimal uniform range in which the Tjurina formula holds for all ordinary semi-homogeneous germs with fixed $n$ and $r$. We next consider the log-concavity property. For ordinary semi-homogeneous singularities, the higher Milnor sequence has an initial log-convex part and a log-concave tail. For an arbitrary isolated hypersurface singularity in $n\\geq2$ variables, both the higher Milnor and higher Tjurina sequences are eventually strictly log-concave. In the plane curve case, we prove that the log-concavity defects change sign at most once, and we characterize the sequences which are log-concave at every positive index. Finally, we give examples showing that the higher Milnor sequence does not determine the contact-equivalence class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32523",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhiwen Liu"
    },
    {
      "name": "Stephen S. -T. Yau"
    },
    {
      "name": "Mingyi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f9ps9hfb8pm82.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32523",
  "fulltext_url": null,
  "id": "f9ps9hfb8pm82",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32523",
  "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": "W7223650389",
  "pdf_url": "https://arxiv.org/pdf/2610.32523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f9ps9hfb8pm82.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Jacobian layers and higher Milnor sequences: rigidity, stabilization, and log-concavity",
  "title_text": "Jacobian layers and higher Milnor sequences: rigidity, stabilization, and log-concavity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/f9ps9hfb8pm82",
  "venue": "arXiv",
  "versions": []
}