The Homogeneous Polynomial Lyapunov Converse Conjecture
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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.
Context
A converse-Lyapunov question that underpins sum-of-squares stability certification, and the reason polynomial Lyapunov search is presumed complete for homogeneous systems.
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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.
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