{
  "abstract": "The second part of Hilbert's 16th problem is one of the most fundamental open problems in the qualitative theory of ordinary differential equations and dynamical systems. A central unresolved question is whether the Hilbert number $H(d)$ is finite for arbitrary polynomial degree $d$. (A very recent manuscript from OpenAI asserts the finiteness of $H(d)$; see the remark at the end of this paper.) In the absence of a general upper bound theory, most existing work constructs explicit perturbed polynomial systems to obtain improved lower bounds for $H(d)$, while rigorous upper bound estimates remain largely unavailable. This paper focuses on improving the quantitative lower bound for polynomial systems through a systematic Hierarchical Melnikov Realization framework. We construct perturbed dynamics based on an anisotropic Chebyshev Hamiltonian, where hierarchically selected perturbation coefficients generate simple zeros of the first-order Melnikov function simultaneously across multiple disjoint period annuli. Two distinct 3-adic filtrations supply the necessary logarithmic correction factors, and a Borel--Gauss analysis establishes the required local rank condition under sufficiently strong anisotropy. As the main result of this work, we prove that $H(d)=Ω\\bigl(d^2\\ln^2 d\\bigr)$, which improves the previously known lower bound by a factor of $\\ln d$. The full construction of limit cycles and the resulting dynamical bounds are formally certified using Lean 4 formal verification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12218",
  "author_total": 2,
  "authors": [
    {
      "name": "Chaoyang Qin"
    },
    {
      "name": "Xiaoming Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9taccgck8rb41.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12218",
  "fulltext_url": null,
  "id": "9taccgck8rb41",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12218",
  "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.CA"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7221083407",
  "pdf_url": "https://arxiv.org/pdf/2610.12218",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9taccgck8rb41.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12218",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12218"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds",
  "title_text": "Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/9taccgck8rb41",
  "venue": "arXiv",
  "versions": []
}