File size: 3,230 Bytes
f205175 | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Matrix
namespace CriticalPercolation.SparseGrowth
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
lemma intMatrix_mahlerMeasure_eq_one (M : Matrix ι ι ℤ)
(hroots : ∀ z∈(M.map (Int.castRingHom ℂ)).charpoly.roots, ‖z‖≤1) :
(M.charpoly.map (Int.castRingHom ℂ)).mahlerMeasure=1 := by
rw [←Matrix.charpoly_map,Polynomial.mahlerMeasure_eq_leadingCoeff_mul_prod_roots,
(Matrix.charpoly_monic _).leadingCoeff,norm_one,one_mul]
apply Multiset.prod_eq_one
intro x hx
obtain ⟨z,hz,rfl⟩ := Multiset.mem_map.mp hx
exact max_eq_left (hroots z hz)
lemma intMatrix_root_is_rootOfUnity (M : Matrix ι ι ℤ)
(hroots : ∀ z∈(M.map (Int.castRingHom ℂ)).charpoly.roots, ‖z‖≤1)
(z : ℂ) (hz : z∈(M.map (Int.castRingHom ℂ)).charpoly.roots) (hz0 : z≠0) :
∃ m : ℕ, 0 < m ∧ z ^ m = 1 := by
apply Polynomial.pow_eq_one_of_mahlerMeasure_eq_one
(intMatrix_mahlerMeasure_eq_one M hroots) hz0
have he : Int.castRingHom ℂ = algebraMap ℤ ℂ := by ext; simp
simpa only [Polynomial.aroots_def,Matrix.charpoly_map,he] using hz
theorem intMatrix_exists_periodic_vector [Nonempty ι] (M : Matrix ι ι ℤ)
(hunit : IsUnit M)
(hroots : ∀ z∈(M.map (Int.castRingHom ℂ)).charpoly.roots, ‖z‖≤1) :
∃ m : ℕ, 0 < m ∧ ∃ v : ι → ℤ, v≠0 ∧ (M^m).mulVec v=v := by
let C := M.map (Int.castRingHom ℂ)
have hcunit : IsUnit C := hunit.map (Int.castRingHom ℂ).mapMatrix
have hdeg : C.charpoly.degree ≠ 0 := by
apply ne_of_gt
apply Polynomial.natDegree_pos_iff_degree_pos.mp
rw [Matrix.charpoly_natDegree_eq_dim]
exact Fintype.card_pos
obtain ⟨z,hz⟩ := IsAlgClosed.exists_root C.charpoly hdeg
have heig : Module.End.HasEigenvalue C.mulVecLin z := by
apply (Module.End.hasEigenvalue_iff_isRoot_charpoly _ _).mpr
rwa [Matrix.charpoly_mulVecLin]
have hz0 : z≠0 := by
intro hz0
obtain ⟨w,hw⟩ := heig.exists_hasEigenvector
apply hw.2
apply Matrix.mulVec_injective_iff_isUnit.mpr hcunit
have he := hw.apply_eq_smul
simpa [hz0] using he
have hzmem := (Polynomial.mem_roots (Matrix.charpoly_monic C).ne_zero).mpr hz
obtain ⟨m,hm,hmz⟩ := intMatrix_root_is_rootOfUnity M hroots z hzmem hz0
have heigpow := heig.pow m
rw [hmz] at heigpow
have hroot : (C^m).charpoly.IsRoot 1 := by
have he : (C^m).mulVecLin = C.mulVecLin^m := Matrix.toLin'_pow C m
rw [←Matrix.charpoly_mulVecLin,he]
exact (Module.End.hasEigenvalue_iff_isRoot_charpoly _ _).mp heigpow
have hrootZ : (M^m).charpoly.IsRoot 1 := by
apply Polynomial.IsRoot.of_map (f := Int.castRingHom ℂ) ?_ (Int.castRingHom ℂ).injective_int
simpa only [map_one,←Matrix.charpoly_map,Matrix.map_pow] using hroot
have heigZ : Module.End.HasEigenvalue (M^m).mulVecLin 1 := by
apply (Module.End.hasEigenvalue_iff_isRoot_charpoly _ _).mpr
rwa [Matrix.charpoly_mulVecLin]
obtain ⟨v,hv⟩ := heigZ.exists_hasEigenvector
refine ⟨m,hm,v,hv.2,?_⟩
simpa only [Matrix.mulVecLin_apply,one_smul] using hv.apply_eq_smul
end CriticalPercolation.SparseGrowth
end
end
end OAI
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