Finite semidefinite lifts of hyperbolicity cones
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Does every closed hyperbolicity cone have an exact finite affine semidefinite lift, allowing arbitrary real coefficients and finitely many auxiliary variables? A counterexample requires excluding all such lifts, rather than only matrix pencils in the original variables.
Why it matters and what counts as progress
Why it matters: It tests the projected Lax conjecture and the possibility of one universal finite semidefinite representation principle.
What counts as progress: Prove universal exact semidefinite representability with auxiliary variables, or exclude every finite lift for one hyperbolicity cone.
How the problem developed
- 2015 · Partial result [1]
Tim Netzer and Raman Sanyal Showed that every smooth hyperbolicity cone is a spectrahedral shadow, covering the smooth-boundary subclass.
- 5 Oct 2026 · Reported claim [2]
OpenAI The manuscript claims a closed hyperbolicity cone with no exact finite affine semidefinite lift, even with arbitrary finite auxiliary variables and real coefficients.
AI contribution
Hyperbolicity Cones Without Semidefinite Lifts
An OpenAI manuscript claims a real homogeneous hyperbolic polynomial whose closed cone has no exact finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI reports a hyperbolicity cone with no finite semidefinite lift