[#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.
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
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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
- Positive metric entropy for the standard mapdynamical systems
- Generic analytic Arnold diffusion in a priori stable systemsdynamical systems
- Persistent exponential stretching of a material line in two-dimensional Euler flowdynamical systems
How to cite
TheoremDB contributors, “Hilbert's sixteenth problem, second part,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/hilberts-sixteenth-problem-second-partThis page as plain text: hilberts-sixteenth-problem-second-part.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.