The existential Hilbert sixteenth problem: a uniform bound on the number of limit cycles of planar polynomial vector fields
Hilbert's sixteenth problem asks, in its second part, for the maximum number and the relative position of the limit cycles of a real planar polynomial vector field , of degree . Its existential form, which Ilyashenko's centennial survey separates from finiteness for each individual field, asks whether that maximum is finite at all. Even the quadratic case was open. Is there, for every , a finite number such that every real planar polynomial vector field of degree at most has at most limit cycles?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Planar polynomial vector fields; limit cycles
- Posed by
- David Hilbert (1900), Problem 16, second part; the uniform-boundedness form as distinguished by Yu. Ilyashenko, Centennial history of Hilbert's 16th problem (Bull. AMS, 2002)
- Year posed
- 1900
- Years open
- 126y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 85 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for each there is a finite integer such that every real planar polynomial vector field of degree at most has at most limit cycles in the whole plane, with no restriction on coefficients, location, stability or hyperbolicity. The proof works with matching systems of local passages and shows that each fixed auxiliary system has finitely many isolated solutions, uniformly as coefficients vary. It does NOT give an explicit or effective , so it says nothing about the value of , about Smale's question whether , or about the relative positions of cycles asked for in Hilbert's full second part. The exact quintic Lienard bound in the same family is a separate entry.
What the AI did
The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 24, 2026; the uniform-bound manuscript is the principal one. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript were read against the problem as the manuscript's own historical section describes it. The theorem states the uniform bound in full generality: every degree, the whole plane, no genericity, cycles counted as geometric images regardless of stability or hyperbolicity. The proof (eleven sections of asymptotic separation, implicit-function calculus, subanalytic preparation and a projection-counting argument) was not refereed. There is no Lean formalization of this manuscript; the family's Lean file covers only the quintic Lienard companion. The manuscript says the bound is not effective. It also notes that Yeung (2025) reported a gap in one leading-term argument of Ilyashenko's 1991 monograph on individual finiteness, and states that its own proof does not use individual finiteness as a premise. A claim of this size should wait for an expert reader before it is shown at any tier.