| import OAI.Analysis.HyperbolicCones.PolynomialBasic |
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| namespace OAI |
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| noncomputable section |
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| open Polynomial |
| open scoped BigOperators |
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| universe u |
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| namespace Paper256 |
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| theorem polynomial_prod_top {ι : Type u} (s : Finset ι) |
| (f : ι → Polynomial ℝ) (d : ι → ℕ) |
| (h : ∀ i ∈ s, (f i).natDegree ≤ d i) : |
| (∏ i ∈ s, f i).natDegree ≤ ∑ i ∈ s, d i ∧ |
| (∏ i ∈ s, f i).coeff (∑ i ∈ s, d i) = ∏ i ∈ s, (f i).coeff (d i) := by |
| classical |
| revert h |
| induction s using Finset.induction_on with |
| | empty => simp |
| | @insert a s ha ih => |
| intro h |
| have hfa := h a (Finset.mem_insert_self a s) |
| have hs := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) |
| simp only [Finset.prod_insert ha, Finset.sum_insert ha] |
| refine ⟨natDegree_mul_le_of_le hfa hs.1, ?_⟩ |
| rw [coeff_mul_add_eq_of_natDegree_le hfa hs.1, hs.2] |
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| theorem affine_substitution_top {σ : Type u} [Fintype σ] |
| (p : MvPolynomial σ ℝ) (d : ℕ) (hp : p.IsHomogeneous d) (e x : σ → ℝ) : |
| let q := MvPolynomial.eval₂ Polynomial.C |
| (fun i => Polynomial.C (e i) * Polynomial.X - Polynomial.C (x i)) p |
| q.natDegree ≤ d ∧ q.coeff d = MvPolynomial.eval e p := by |
| classical |
| let l : σ → Polynomial ℝ := fun i => C (e i) * X - C (x i) |
| have hl (i : σ) : (l i).natDegree ≤ 1 := by |
| apply (natDegree_sub_le _ _).trans |
| exact max_le (le_trans (natDegree_C_mul_le _ _) (by simp)) (by simp) |
| have hc (i : σ) : (l i).coeff 1 = e i := by simp [l] |
| have hm (a : σ →₀ ℕ) : |
| (∏ i, l i ^ a i).natDegree ≤ a.degree ∧ |
| (∏ i, l i ^ a i).coeff a.degree = ∏ i, e i ^ a i := by |
| have h := polynomial_prod_top Finset.univ (fun i => l i ^ a i) a |
| (fun i _ => by simpa using natDegree_pow_le_of_le (a i) (hl i)) |
| rw [Finsupp.degree_eq_sum] |
| refine ⟨h.1, h.2.trans ?_⟩ |
| apply Finset.prod_congr rfl |
| intro i _ |
| simpa [hc] using (coeff_pow_of_natDegree_le (m := a i) (hl i)) |
| dsimp |
| change (MvPolynomial.eval₂ C l p).natDegree ≤ d ∧ |
| (MvPolynomial.eval₂ C l p).coeff d = MvPolynomial.eval e p |
| rw [MvPolynomial.eval₂_eq', MvPolynomial.eval_eq'] |
| constructor |
| · apply natDegree_sum_le_of_forall_le |
| intro a ha |
| have hd : a.degree = d := by |
| simpa only [Finsupp.degree_eq_weight_one, Pi.one_def] using |
| hp (MvPolynomial.mem_support_iff.mp ha) |
| exact (natDegree_C_mul_le _ _).trans ((hm a).1.trans_eq hd) |
| · rw [finsetSum_coeff] |
| apply Finset.sum_congr rfl |
| intro a ha |
| have hd : a.degree = d := by |
| simpa only [Finsupp.degree_eq_weight_one, Pi.one_def] using |
| hp (MvPolynomial.mem_support_iff.mp ha) |
| rw [coeff_C_mul, ← hd, (hm a).2] |
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| theorem linePolynomial_top (x : Ambient) : |
| (linePolynomial x).natDegree ≤ 20 ∧ (linePolynomial x).coeff 20 = 1 := by |
| have h := affine_substitution_top polynomial 20 polynomial_homogeneous |
| (coordinates basePoint) (coordinates x) |
| simpa [linePolynomial, polynomial_basePoint] using h |
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| theorem linePolynomial_natDegree (x : Ambient) : (linePolynomial x).natDegree = 20 := by |
| apply le_antisymm (linePolynomial_top x).1 |
| exact le_natDegree_of_ne_zero (by rw [(linePolynomial_top x).2]; norm_num) |
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| theorem linePolynomial_monic (x : Ambient) : (linePolynomial x).Monic := by |
| change (linePolynomial x).leadingCoeff = 1 |
| rw [← coeff_natDegree, linePolynomial_natDegree, (linePolynomial_top x).2] |
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| end Paper256 |
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| end |
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| end OAI |
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