| import OAI.Algebra.Determinantal.BorderPolars |
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| namespace OAI |
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| namespace PermanentBorder.AmbientPolar |
| open MvPolynomial Matrix PolarAnalysis KernelBilinear |
| open scoped Classical |
| variable {σ κ : Type} [Fintype σ] [Fintype κ] |
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|
| noncomputable def substitution (b : σ → ℂ) |
| (L : (Option κ → ℂ) →L[ℂ] (σ → ℂ)) : Poly σ →ₐ[ℂ] Poly (Option κ) := |
| affineSubstitution b (fun j i => L (Pi.single i 1) j) |
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|
| omit [Fintype σ] in |
| lemma eval_substitution (b : σ → ℂ) |
| (L : (Option κ → ℂ) →L[ℂ] (σ → ℂ)) (P : Poly σ) (z : Option κ → ℂ) : |
| eval z (substitution b L P) = eval (b + L z) P := by |
| classical |
| change (aeval z : Poly (Option κ) →ₐ[ℂ] ℂ) (aeval _ P) = _ |
| rw [comp_aeval_apply] |
| apply congrArg (fun g : σ → ℂ => aeval g P) |
| funext j |
| simp only [map_add, map_sum, map_mul, aeval_C, aeval_X, Algebra.algebraMap_self_apply] |
| have hz : z = ∑ i, z i • Pi.single i 1 := by |
| ext i |
| simp [Pi.single_apply] |
| have hL : L z j = ∑ i, L (Pi.single i 1) j * z i := by |
| conv_lhs => rw [hz, map_sum] |
| simp [mul_comm] |
| simpa only [Pi.add_apply] using congrArg (fun v : ℂ => b j + v) hL.symm |
|
|
| noncomputable def directional (h : κ → σ → ℂ) (i : κ) : Poly σ →ₗ[ℂ] Poly σ := |
| ∑ j, h i j • (pderiv j).toLinearMap |
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| noncomputable def operators (b : σ → ℂ) |
| (L : (Option κ → ℂ) →L[ℂ] (σ → ℂ)) (h : κ → σ → ℂ) : |
| Option κ → Poly σ →ₗ[ℂ] Poly (Option κ) := |
| Option.elim' (substitution b L).toLinearMap |
| (fun i => (substitution b L).toLinearMap.comp (directional h i)) |
|
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| omit [Fintype κ] in |
| lemma derivative_directional (h : κ → σ → ℂ) (P : Poly σ) (y : σ → ℂ) (i : κ) : |
| fderiv ℂ (fun y => eval y P) y (h i) = eval y (directional h i P) := by |
| rw [(CI.hasFDerivAt_eval P y).fderiv, CI.polynomialDerivative_apply] |
| simp [directional, mul_comm] |
|
|
| lemma polarPair_operators (b : σ → ℂ) |
| (L : (Option κ → ℂ) →L[ℂ] (σ → ℂ)) (h : κ → σ → ℂ) (P : Poly σ) : |
| polarPair (fun y => eval y P) (fun z => b + L (optionProduct.symm z)) h = |
| fun z => optionProduct (polynomialSystem (operators b L h) P (optionProduct.symm z)) := by |
| funext z |
| apply Prod.ext |
| · exact (eval_substitution b L P _).symm |
| · funext i |
| change fderiv ℂ (fun y => eval y P) (b + L (optionProduct.symm z)) (h i) = |
| eval (optionProduct.symm z) (substitution b L (directional h i P)) |
| rw [eval_substitution, derivative_directional] |
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|
| end PermanentBorder.AmbientPolar |
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| end OAI |
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