| import OAI.Analysis.HyperbolicCones.Pencil |
| import OAI.Analysis.HyperbolicCones.TangentObstruction |
| import OAI.Analysis.HyperbolicCones.Deformation |
| import OAI.Analysis.HyperbolicCones.LinePolynomial |
| import OAI.Analysis.HyperbolicCones.ConeZero |
| import OAI.Analysis.HyperbolicCones.Coordinates |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Matrix |
| open scoped Matrix.Norms.L2Operator |
|
|
| namespace Paper256 |
|
|
| theorem cone_not_spectrahedral (N : ℕ) (hN : 0 < N) (L : Ambient →ₗ[ℝ] Sym N) : |
| cone ≠ {x | (L x : Mat N ℝ).PosSemidef} := by |
| intro hL |
| obtain ⟨P⟩ := pencil_normalization N hN L hL |
| exact P.impossible |
|
|
| / |
| whose closed hyperbolicity cone is not spectrahedral. -/ |
| theorem main_result : |
| Module.finrank ℝ Ambient = 23 ∧ |
| polynomial.IsHomogeneous 20 ∧ |
| polynomial.totalDegree = 20 ∧ |
| MvPolynomial.eval (coordinates basePoint) polynomial = 1 ∧ |
| (∀ (x : Ambient) (z : ℂ), (linePolynomial x).aeval z = 0 → z.im = 0) ∧ |
| ∀ (N : ℕ) (_hN : 0 < N) (L : Ambient →ₗ[ℝ] Sym N), |
| cone ≠ {x | (L x : Mat N ℝ).PosSemidef} := by |
| exact ⟨ambient_finrank, polynomial_homogeneous, polynomial_totalDegree, |
| polynomial_basePoint, linePolynomial_complex_roots_real, cone_not_spectrahedral⟩ |
|
|
| end Paper256 |
|
|
| end |
|
|
| end OAI |
|
|
