| import OAI.Analysis.HyperbolicCones.PolynomialScaling |
| import OAI.Analysis.HyperbolicCones.LineEvaluation |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Paper256 |
|
|
| theorem linePolynomial_eval (x : Ambient) (t : ℝ) : |
| (linePolynomial x).eval t = |
| MvPolynomial.eval (coordinates (t • basePoint - x)) polynomial := by |
| rw [coordinates_sub, coordinates_smul] |
| change (Polynomial.evalRingHom t) ((MvPolynomial.eval₂Hom Polynomial.C |
| (fun i => Polynomial.C (coordinates basePoint i) * Polynomial.X - |
| Polynomial.C (coordinates x i))) polynomial) = |
| (MvPolynomial.eval₂Hom (RingHom.id ℝ) |
| (t • coordinates basePoint - coordinates x)) polynomial |
| rw [MvPolynomial.map_eval₂Hom] |
| congr 2 |
| · ext r |
| simp |
| · ext i |
| simp only [Polynomial.coe_evalRingHom, Polynomial.eval_sub, Polynomial.eval_mul, |
| Polynomial.eval_C, Polynomial.eval_X, Pi.sub_apply, Pi.smul_apply, smul_eq_mul] |
| rw [mul_comm] |
|
|
| theorem linePolynomial_eval_neg (x : Ambient) (t : ℝ) : |
| (linePolynomial x).eval (-t) = |
| MvPolynomial.eval (coordinates (x + t • basePoint)) polynomial := by |
| rw [linePolynomial_eval] |
| have h : (-t) • basePoint - x = -(x + t • basePoint) := by |
| simp [neg_add_rev, sub_eq_add_neg] |
| rw [h, polynomial_eval_neg] |
|
|
| theorem real_aeval (p : Polynomial ℝ) (t : ℝ) : |
| p.aeval (t : ℂ) = Complex.ofReal (p.eval t) := |
| Polynomial.aeval_algebraMap_apply_eq_algebraMap_eval t p |
|
|
| end Paper256 |
|
|
| end |
|
|
| end OAI |
|
|
