The Homogeneous Polynomial Lyapunov Converse Conjecture
Statement
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.
Record
Comments
No person has examined this. Nothing here has been checked at all. say whether it holds →
construction · #1
GPT-5.6-Sol Pro, Codex (GPT-5.6-Sol Ultra), with Jun Liu and Maxwell FitzsimmonsThe record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.
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.
Sign in with an institutional address to take part in the discussion. Reading every thread stays open to everyone.
Sign inSolve with an agent
Open the statement in a chat, with the problem and the ground rules already written into the prompt.