{
  "abstract": "We study Yang's higher numerical invariants for the quasi-homogeneous submodules \\[ M_{k,\\ell}=[z^k-w^\\ell]\\subset H^2(\\mathbb D^2), \\qquad k,\\ell\\in\\mathbb N,\\quad k\\neq\\ell. \\] The Hilbert--Schmidt property and the low-order invariants $Σ_0$ and $Σ_1$ for this family follow from earlier results on $M_{θ,φ}$-type submodules. Our aim is to determine the complete higher-order sequence. Using the weighted homogeneous decomposition associated with $z^k-w^\\ell$, we construct explicit orthonormal bases for the two defect spaces and determine the exact support of the shifted defect inner products. For $j\\geq1$, set \\[ A_j=\\left\\lceil\\frac{j}{\\ell}\\right\\rceil, \\qquad B_j=\\left\\lceil\\frac{j}{k}\\right\\rceil. \\] We obtain \\[ Σ_0(M_{k,\\ell})=\\frac{π^2}{6}, \\qquad Σ_j(M_{k,\\ell})=F(A_j,B_j), \\] where $F$ is an explicitly evaluated symmetric function on $\\mathbb N^2$. The sequence is nonincreasing, tends to zero, and satisfies $Σ_j>Σ_{j+1} \\Longleftrightarrow k\\mid j\\ \\text{or}\\ \\ell\\mid j$. Hence its strict descent set is $k\\mathbb N\\cup\\ell\\mathbb N$. This descent set determines the additive semigroup $\\langle k,\\ell\\rangle$, while the complete numerical invariant sequence determines the unordered pair $\\{k,\\ell\\}$. The order of the two exponents is not recoverable because of coordinate-swap symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25640",
  "author_total": 3,
  "authors": [
    {
      "name": "Yin Liu"
    },
    {
      "name": "Yufeng Lu"
    },
    {
      "name": "Yixin Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa3frjytk8961.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25640",
  "fulltext_url": null,
  "id": "fa3frjytk8961",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25640",
  "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.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.25640",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa3frjytk8961.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25640",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25640"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules $[z^k-w^\\ell]$",
  "title_text": "Numerical Invariants and Parameter Recovery for Quasi-Homogeneous Submodules [z^k−w^ℓ]",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/fa3frjytk8961",
  "venue": "arXiv",
  "versions": []
}