What does this research mean for the field?

For real planar polynomial vector fields of degree at most n, the asymptotic lower bound for the Hilbert number H(n) satisfies liminf H(n)/(n^2 log n) ≥ 1/log 2. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Establish a sharper asymptotic lower bound for the Hilbert number H(n), representing the maximum number of limit cycles in real planar polynomial vector fields of degree at most n.

August 27, 2026Open Access

An n^2 log n lower bound with coefficient 1/log 2 for the Hilbert number of polynomial systems

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Establish a sharper asymptotic lower bound for the Hilbert number H(n), representing the maximum number of limit cycles in real planar polynomial vector fields of degree at most n.
Constructed planar vector fields utilizing asymptotically dense quadratic boundary tangencies, quadratic Chebyshev pullbacks, reversible centers, and simultaneous odd-polynomial Melnikov bifurcations.
Applied two successive boundary-vanishing perturbations tailored to each target degree to establish bounds for all large degrees rather than dyadic subsequences.
Verified recurrence relations, Chebyshev identities, and polynomial algebra via a Python finite checker and Lean 4 formalization layer.
Proved the lower bound limit infimum H(n)/(n^2 log n) >= 1/log 2 for planar polynomial systems of degree at most n across every sufficiently large degree.
Demonstrated the persistence of generated limit cycles using nonlinear center criteria and Melnikov expansion techniques.

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Implication

Mathematical analysis demonstrates an asymptotic lower bound for Hilbert numbers in real planar polynomial vector fields, indicating higher limit cycle capacity across large degrees.

Cite This Study

Haibo Lu (2026) studied this question.

synapsesocial.com/papers/6a8fe98110c91c1e92621237 https://doi.org/10.5281/zenodo.22099310

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Implication

Mathematical analysis demonstrates an asymptotic lower bound for Hilbert numbers in real planar polynomial vector fields, indicating higher limit cycle capacity across large degrees.

An n^2 log n lower bound with coefficient 1/log 2 for the Hilbert number of polynomial systems

Key Points

Discussion

Implication

Cite This Study

An n^2 log n lower bound with coefficient 1/log 2 for the Hilbert number of polynomial systems

Key Points

Discussion

Implication

Cite This Study

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