[#P9] Hilbert's sixteenth problem, second part

Problem. For every positive integer \(n\), there exists a finite number \(H(n)\) such that each planar polynomial vector field of degree at most \(n\) has at most \(H(n)\) limit cycles.

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Definitions and notation

1Context

The problem asks for degree-controlled global information about isolated periodic behavior in polynomial differential systems.

2Problem setup

Definition 1 (A limit cycle). A limit cycle is an isolated periodic orbit of a planar differential equation.

Definition 2 (The bound may depend on the polynomial degree n but must apply uniformly to every vector field of that degree). The bound may depend on the polynomial degree n but must apply uniformly to every vector field of that degree.

Remark 1. The problem asks for degree-controlled global information about isolated periodic behavior in polynomial differential systems.

3What counts as a solution

Prove a finite uniform bound H(n) for every degree n, or construct a fixed degree with polynomial vector fields having arbitrarily many limit cycles.

1Status

What counts as a solution

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Buzzi and Novaes refute a recent claimed quadratic formula by comparing it with established lower-growth constructions. Their note leaves the finiteness of H(n) open. Exact unresolved remainder: Prove a finite uniform upper bound H(n) for every degree n, or give one fixed degree with polynomial vector fields having arbitrarily many limit cycles.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Buzzi and Novaes refute a recent claimed quadratic formula by comparing it with established lower-growth constructions. Their note leaves the finiteness of H(n) open. Exact unresolved remainder: Prove a finite uniform upper bound H(n) for every degree n, or give one fixed degree with polynomial vector fields having arbitrarily many limit cycles.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Buzzi and Novaes refute a recent claimed quadratic formula by comparing it with established lower-growth constructions. Their note leaves the finiteness of H(n) open. Exact unresolved remainder: Prove a finite uniform upper bound H(n) for every degree n, or give one fixed degree with polynomial vector fields having arbitrarily many limit cycles.^[1] Relevance to this problem For Hilbert's sixteenth problem, second part, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Buzzi and Novaes refute a recent claimed quadratic formula by comparing it with established lower-growth constructions. Their note leaves the finiteness of H(n) open. Exact unresolved remainder: Prove a finite uniform upper bound H(n) for every degree n, or give one fixed degree with polynomial vector fields having arbitrarily many limit cycles. Open full record →
Prove a finite uniform bound H(n) for every degree n, or construct a fixed degree with polynomial vector fields having arbitrarily many limit cycles.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove a finite uniform bound H(n) for every degree n, or construct a fixed degree with polynomial vector fields having arbitrarily many limit cycles.^[1] Relevance to this problem For Hilbert's sixteenth problem, second part, record hilberts-sixteenth-problem-second-part-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove a finite uniform bound H(n) for every degree n, or construct a fixed degree with polynomial vector fields having arbitrarily many limit cycles. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove a finite uniform bound H(n) for every degree n, or construct a fixed degree with polynomial vector fields having arbitrarily many limit cycles. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited 2024 note treats the existence of the uniform finite bounds H(n) as unresolved. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Finiteness for each individual polynomial vector field is known. Uniform finiteness by degree remains open even in low degrees.

Computational notes

Numerical integration can locate cycles in selected systems but can miss cycles and cannot establish a universal degree bound.

2See also

Birkhoff conjecture for integrable convex billiardsdynamical systems
Persistent exponential stretching of a material line in two-dimensional Euler flowdynamical systems
Generic analytic Arnold diffusion in a priori stable systemsdynamical systems

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Plain text

“Hilbert's sixteenth problem, second part.” TheoremDB. P9. Problem statement; statement text SHA-256 12b311d70087a157ba615ca4293288b0cf04fe999760f00e04a75db1ce547132. https://theoremdb.org/statement/?ref=P9

BibTeX

@misc{theoremdb-problem-12b311d70087a157ba615ca4293288b0cf04fe999760f00e04a75db1ce547132,
  title = {{Hilbert's sixteenth problem, second part}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 12b311d70087a157ba615ca4293288b0cf04fe999760f00e04a75db1ce547132},
  url = {https://theoremdb.org/statement/?ref=P9}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Claudio A. Buzzi and Douglas D. Novaes, “A note on a recent attempt to solve the second part of Hilbert's 16th Problem”. arXiv:2411.09594 (2024). arXiv:2411.09594, opening formulation and analysis of a claimed solution. ↗preprint · primary source · arXiv:2411.09594, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2024 note treats the existence of the uniform finite bounds H(n) as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and analysis of the claimed quadratic formula.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Rejects a claimed solution while preserving the exact finiteness question.Source named by the research packet.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P9] Hilbert's sixteenth problem, second part

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

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1References

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Recent contributions

Computational notes

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Background

Definitions

What counts as a solution

What's known

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