| import OAI.Geometry.HeilbronnTriangle.DigitEncoding | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem355.DesignatedCoefficient | |
| open Polynomial | |
| def position (T : ℕ) (i : Fin 3) (a : Fin T) : ℕ := | |
| ![a.val, T * a.val, T ^ 2 - (T + 1) * a.val] i | |
| theorem position_sum_matching (T : ℕ) (v : Fin 3 → Fin T) : | |
| position T 0 (v 0) + position T 1 (v 1) + position T 2 (v 2) = T ^ 2 ↔ | |
| v 0 = v 1 ∧ v 1 = v 2 := by | |
| have hle : (T + 1) * (v 2).val ≤ T ^ 2 := by | |
| have hv : (v 2).val + 1 ≤ T := (v 2).isLt | |
| have hh := Nat.mul_le_mul_left (T + 1) hv | |
| nlinarith | |
| have hsub := Nat.sub_add_cancel hle | |
| change (v 0).val + T * (v 1).val + (T ^ 2 - (T + 1) * (v 2).val) = T ^ 2 ↔ | |
| v 0 = v 1 ∧ v 1 = v 2 | |
| constructor | |
| · intro h | |
| have hm := DigitEncoding.matched_indices (v 0).isLt (v 2).isLt | |
| (show (v 0).val + T * (v 1).val = (T + 1) * (v 2).val by omega) | |
| exact ⟨Fin.ext (hm.1.trans hm.2.symm), Fin.ext hm.2⟩ | |
| · rintro ⟨h01, h12⟩ | |
| have hv0 : (v 0).val = (v 2).val := congrArg Fin.val (h01.trans h12) | |
| have hv1 : (v 1).val = (v 2).val := congrArg Fin.val h12 | |
| rw [hv0, hv1] | |
| nlinarith | |
| theorem sum_diagonal {R : Type*} [AddCommMonoid R] (T : ℕ) | |
| (F : (Fin 3 → Fin T) → R) : | |
| (∑ v : Fin 3 → Fin T, if v 0 = v 1 ∧ v 1 = v 2 then F v else 0) = | |
| ∑ a : Fin T, F (fun _ => a) := by | |
| classical | |
| rw [← Finset.sum_filter] | |
| symm | |
| apply Finset.sum_bij (fun a _ => fun _ : Fin 3 => a) | |
| · intro a ha | |
| simp | |
| · intro a ha b hb hab | |
| exact congrFun hab 0 | |
| · intro v hv | |
| rcases Finset.mem_filter.mp hv with ⟨_, h01, h12⟩ | |
| refine ⟨v 0, Finset.mem_univ _, ?_⟩ | |
| funext i | |
| fin_cases i <;> simp_all | |
| · intro a ha | |
| rfl | |
| theorem det_monomial_rows {R : Type*} [CommRing R] | |
| (v : Fin 3 → ℕ) (A : Matrix (Fin 3) (Fin 3) R) : | |
| Matrix.det (Matrix.of (fun i j => monomial (v i) (A i j))) = | |
| monomial (v 0 + v 1 + v 2) A.det := by | |
| simp only [Matrix.det_fin_three, Matrix.of_apply, Polynomial.monomial_mul_monomial, map_add, map_sub] | |
| def polynomialMatrix {R : Type*} [CommRing R] (T : ℕ) | |
| (f : Fin 3 → Fin T → Fin 3 → R) : Matrix (Fin 3) (Fin 3) R[X] := | |
| fun i j => ∑ a : Fin T, monomial (position T i a) (f i a j) | |
| theorem designated_coefficient {R : Type*} [CommRing R] (T : ℕ) | |
| (f : Fin 3 → Fin T → Fin 3 → R) : | |
| (Matrix.det (polynomialMatrix T f)).coeff (T ^ 2) = | |
| ∑ a : Fin T, Matrix.det (Matrix.of (fun i j => f i a j)) := by | |
| classical | |
| have hm := (Matrix.detRowAlternating (n := Fin 3) (R := R[X])).toMultilinearMap.map_sum_finset | |
| (fun i (a : Fin T) j => monomial (position T i a) (f i a j)) | |
| (fun _ => Finset.univ) | |
| change Matrix.det (Matrix.of (fun i => ∑ a : Fin T, fun j => | |
| monomial (position T i a) (f i a j))) = _ at hm | |
| have hrows : (fun i => ∑ a : Fin T, fun j => | |
| monomial (position T i a) (f i a j)) = polynomialMatrix T f := by | |
| funext i j | |
| simp only [Finset.sum_apply, polynomialMatrix] | |
| rw [hrows, Fintype.piFinset_univ] at hm | |
| change (polynomialMatrix T f).det = _ at hm | |
| rw [hm, Polynomial.finsetSum_coeff] | |
| change (∑ v : Fin 3 → Fin T, | |
| (Matrix.det (Matrix.of (fun i j => monomial (position T i (v i)) (f i (v i) j)))).coeff | |
| (T ^ 2)) = _ | |
| calc | |
| _ = ∑ v : Fin 3 → Fin T, | |
| if v 0 = v 1 ∧ v 1 = v 2 then Matrix.det (Matrix.of (fun i j => f i (v i) j)) else 0 := by | |
| apply Finset.sum_congr rfl | |
| intro v hv | |
| have hdet := det_monomial_rows (fun i => position T i (v i)) | |
| (Matrix.of (fun i j => f i (v i) j)) | |
| simp only [Matrix.of_apply] at hdet | |
| rw [hdet, Polynomial.coeff_monomial] | |
| by_cases h : v 0 = v 1 ∧ v 1 = v 2 | |
| · have hp := (position_sum_matching T v).mpr h | |
| rw [ite_eq_left hp, ite_eq_left h] | |
| · have hp : ¬ (position T 0 (v 0) + position T 1 (v 1) + | |
| position T 2 (v 2) = T ^ 2) := fun hp => h ((position_sum_matching T v).mp hp) | |
| rw [ite_eq_right hp, ite_eq_right h] | |
| _ = _ := sum_diagonal T (fun v => Matrix.det (Matrix.of (fun i j => f i (v i) j))) | |
| theorem map_designated_coefficient {R : Type*} [CommRing R] (φ : ℤ →+* R) | |
| (T : ℕ) (P : Matrix (Fin 3) (Fin 3) ℤ[X]) | |
| (f : Fin 3 → Fin T → Fin 3 → R) | |
| (hmap : ∀ i j, (P i j).map φ = polynomialMatrix T f i j) : | |
| φ (P.det.coeff (T ^ 2)) = ∑ a : Fin T, Matrix.det (Matrix.of (fun i j => f i a j)) := by | |
| have hm : P.det.map φ = Matrix.det (Matrix.of (fun i j => (P i j).map φ)) := | |
| (Polynomial.mapRingHom φ).map_det P | |
| have hmatrix : (fun i j => (P i j).map φ) = polynomialMatrix T f := by | |
| funext i j | |
| exact hmap i j | |
| rw [hmatrix] at hm | |
| rw [← Polynomial.coeff_map, hm] | |
| exact designated_coefficient T f | |
| theorem designated_coefficient_ne_zero {R : Type*} [CommRing R] (φ : ℤ →+* R) | |
| (T : ℕ) (P : Matrix (Fin 3) (Fin 3) ℤ[X]) | |
| (f : Fin 3 → Fin T → Fin 3 → R) | |
| (hmap : ∀ i j, (P i j).map φ = polynomialMatrix T f i j) | |
| (hne : (∑ a : Fin T, Matrix.det (Matrix.of (fun i j => f i a j))) ≠ 0) : | |
| P.det.coeff (T ^ 2) ≠ 0 := by | |
| intro hz | |
| apply hne | |
| rw [← map_designated_coefficient φ T P f hmap, hz, map_zero] | |
| end Problem355.DesignatedCoefficient | |
| end | |
| end OAI | |