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Uniform bounds for planar polynomial limit cycles
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Abstract
For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Chaoyang Qin and Xiaoming Sun, Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds (arXiv:2610.12218v1, 8 October 2026). This AI-assisted preprint cites OpenAI's uniform-bound manuscript while claiming the complementary lower bound $H(d)=\Omega(d^2\log^2 d)$ and explicitly states that its results were obtained independently before the release.
Formalization
- OpenAI math release, Hilbert's sixteenth problem: uniform bounds for limit cycles — Lean scope notes (lean/docs/143.md, release of 6 October 2026). The family's Lean scope note covers only the exact two-cycle bound and its attainment for quintic Liénard systems; no formalization of this paper's uniform bound B(d) (Theorem 1.1) is listed. [family-level]
- franzhusch, Hilberts 16th Problem (google-deepmind/formal-conjectures issue #1805, opened 20 January 2026). An open request, labelled "needs-prerequisites", to formalize the statement of both parts of Hilbert's sixteenth problem; it predates the release and does not refer to this paper.
Possibly relevant literature (found by the agent, not checked)
- Patrick Speissegger, Limit cycles of planar vector fields: Hilbert's 16th problem and o-minimality (arXiv:1804.03585, April 2018). A survey of the o-minimality approach to the second part of Hilbert's sixteenth problem, the line of work (Kaiser–Rolin–Speissegger, Galal–Kaiser–Speissegger) that the paper's introduction compares its own separation and closure arguments to.
- Jaume Llibre and Pablo Pedregal, Hilbert's 16th problem. I. When differential systems meet variational methods (arXiv:1904.01292, v1 April 2019; v4 of October 2020 is marked withdrawn). An earlier preprint that also claimed a uniform bound depending only on the degree, using a variational/Morse-theoretic approach; relevant as context for the paper's claim to resolve uniform boundedness (Theorem 1.1).
- Armengol Gasull and Paulo Santana, A note on Hilbert 16th Problem (arXiv:2407.13465v2, 1 October 2024). Its result that a finite Hilbert number is realizable by structurally stable fields with only hyperbolic limit cycles is relevant to the manuscript's final reduction from isolated cycles to hyperbolic cycles through a signed rotation.
Digests and expositions
- Singh-AP, ai-math-papers-explained: Uniform bounds for planar polynomial limit cycles (GitHub, accessed 2026-10-21). A beginner-level explainer of this paper and the quintic Liénard companion, with proof outline, caveats and an errata list; the README states that the slides, infographics, report and audio were generated with NotebookLM and that the main text was written with AI assistance. [AI-generated per source] [family-level]
News and blogs
- Azmat, Hilbert's 16th Problem: OpenAI's Limit-Cycle Proof Explained (Binary Verse AI, 10 October 2026). A blog explainer of this paper's claimed uniform bound and of the quintic Liénard companion; it notes that the bound is not effective and that independent verification is pending, and quotes no outside experts. [family-level]
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (self-published audit, findings dated 10 October 2026). Finding G1 distinguishes the family's general uniform-bound claim from the quintic Liénard theorem covered by its Lean statements at the audited commit
fd4aeeb2. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 143
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026,
author = {{OpenAI}},
title = {{Uniform bounds for planar polynomial limit cycles}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf}{OAI:uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/143.md. The Lean statements have not been checked by people against the paper's claims.
Family 143: Hilbert's sixteenth problem: uniform bounds for limit cycles
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.
Other papers in this family:
- Two limit cycles for quintic Liénard systems · September 24, 2026