Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds
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(d2ln2 d), 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.
Related
- A Combinatorial Formula and a Hilbert Scheme Realization of a Rank-One Virasoro Irregular Conformal Block
- Hilbert Series and Logarithmic Degrees of A-Hypergeometric Series
- Zeros of Melnikov functions for periodically perturbed Hamiltonian systems with random coefficients
- Melnikov's Valency Variety Problem
- Algebraic realization of stable Poincaré-Reeb graphs
BibTeX
@misc{qin2026hierarchical,
title = {{Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds}},
author = {Chaoyang Qin and Xiaoming Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12218},
eprint = {2610.12218},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12218},
}
RIS
TY - UNPB
TI - Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds
AU - Chaoyang Qin
AU - Xiaoming Sun
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12218
UR - https://arxiv.org/abs/2610.12218
AB - 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(d^2ln^2 d), 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.
ER -