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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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
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