The Homogeneous Polynomial Lyapunov Converse Conjecture
Does every globally asymptotically stable homogeneous polynomial vector field admit a homogeneous polynomial Lyapunov function? No. A planar homogeneous cubic vector field with integer coefficients is globally asymptotically stable yet admits no positive definite homogeneous polynomial with nonpositive Lie derivative, and no real-analytic Lyapunov function even locally, though it does have exponential and rational strict Lyapunov functions.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Control theory
- Posed by
- Amir Ali Ahmadi, Pablo Parrilo
- Year posed
- 2011
- Years open
- 15y
- Solved
- 2026-07-17
- Model
- GPT-5.6-Sol Pro, Codex (GPT-5.6-Sol Ultra)
- Vendor
- OpenAI
- Collaborators
- Jun Liu, Maxwell Fitzsimmons
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The acknowledgement is direct: the authors used GPT-5.6-Sol Pro to discover the counterexample and to produce initial versions of the mathematical arguments in Sections II to IV. Codex assisted with drafting and with the Lean 4 formalization. The authors independently verified all AI-generated content.
Verification
A Lean 4 development covering the four items of the main theorem accompanies the paper at the linked repository, following the proof architecture of the text. We have not compiled it. arXiv preprint, not yet peer-reviewed.