What question did this study set out to answer?

To establish an improved asymptotic lower bound of (1/log 2) * n^2 log n for Hilbert numbers H(n) in real planar polynomial vector fields across all sufficiently large degrees.

August 16, 2026Open Access

The coefficient 1/log 2 in an n^2 log n lower bound for Hilbert numbers via two-axis recursive bifurcation

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To establish an improved asymptotic lower bound of (1/log 2) * n^2 log n for Hilbert numbers H(n) in real planar polynomial vector fields across all sufficiently large degrees.
Constructed dynamical systems using asymptotically dense boundary tangencies, quadratic Chebyshev pullbacks, reversible fold centers, and parity perturbations.
Introduced degree-specific seed choices to extend the asymptotic lower bound universally rather than strictly along dyadic subsequences.
Verified polynomial bookkeeping, recurrence relations, and algebraic identities using the Lean 4 proof assistant and an independent Python checker.
Demonstrated that the asymptotic limit infimum liminf H(n)/(n^2 log n) is at least 1/log 2 for real planar polynomial vector fields of degree at most n.
Proved that the n^2 log n lower bound applies universally to all sufficiently large integer degrees n.

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Implication

Theoretical analysis establishes an asymptotic lower bound for Hilbert numbers in planar polynomial vector fields, demonstrating cycle growth across all sufficiently large degrees.

Cite This Study

Haibo Lu (2026) studied this question.

synapsesocial.com/papers/6a819d36f2fb91fc834aed0f https://doi.org/10.5281/zenodo.21945583

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Implication

Theoretical analysis establishes an asymptotic lower bound for Hilbert numbers in planar polynomial vector fields, demonstrating cycle growth across all sufficiently large degrees.

The coefficient 1/log 2 in an n^2 log n lower bound for Hilbert numbers via two-axis recursive bifurcation

Key Points

Discussion

Implication

Cite This Study

The coefficient 1/log 2 in an n^2 log n lower bound for Hilbert numbers via two-axis recursive bifurcation

Key Points

Discussion

Implication

Cite This Study

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