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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 | import Mathlib
namespace OAI
section
noncomputable section
open Polynomial Module
open scoped Classical
namespace CriticalPercolation.LinearGrowth
lemma multiset_prod_dvd_pow_card {R : Type*} [CommMonoid R] (s : Multiset R) (a : R)
(h : ∀ b∈s, b∣a) : s.prod∣a^s.card := by
induction s using Multiset.induction_on with
| empty => simp
| cons b s ih =>
simp only [Multiset.prod_cons,Multiset.card_cons,pow_succ']
exact mul_dvd_mul (h b (Multiset.mem_cons_self _ _))
(ih (fun c hc => h c (Multiset.mem_cons_of_mem hc)))
lemma nilpotent_aeval_of_vanishing_roots {K R : Type*} [Field K] [Ring R] [Algebra K R]
(a : R) (hp : (minpoly K a).Monic) (hs : (minpoly K a).Splits)
(q : K[X]) (hq : ∀ x ∈ (minpoly K a).roots, q.IsRoot x) : IsNilpotent (aeval a q) := by
have hd : minpoly K a ∣ q^(minpoly K a).roots.card := by
nth_rw 1 [hs.eq_prod_roots_of_monic hp]
convert multiset_prod_dvd_pow_card ((minpoly K a).roots.map (fun x => X-C x)) q ?_ using 1
· simp only [Multiset.card_map]
· intro b hb
obtain ⟨x,hx,rfl⟩ := Multiset.mem_map.mp hb
exact dvd_iff_isRoot.mpr (hq x hx)
exact ⟨_,by rw [←map_pow]; exact minpoly.dvd_iff.mp hd⟩
lemma exists_scalar_unipotent_power {K V : Type*} [Field K] [IsAlgClosed K]
[AddCommGroup V] [Module K V] [FiniteDimensional K V]
(A : End K V) (a : K) (ha : a≠0)
(hr : ∀ r, (minpoly K A).IsRoot r → ∃ m : ℕ, 0 < m ∧ (r/a)^m=1) :
∃ m : ℕ, 0 < m ∧ IsNilpotent (A^m-a^m • (1:End K V)) := by
let s := (minpoly K A).roots.toFinset
have he (r : s) : ∃ m : ℕ, 0 < m ∧ (r.val/a)^m=1 :=
hr r.val (isRoot_of_mem_roots (Multiset.mem_toFinset.mp r.property))
choose m hm hm1 using he
let N := ∏ r : s, m r
have hN : 0 < N := Finset.prod_pos (fun r _ => hm r)
have hpow (r : s) : (r.val/a)^N=1 := by
have hd : m r∣N := Finset.dvd_prod_of_mem m (Finset.mem_univ r)
obtain ⟨k,hk⟩ := hd
rw [hk,pow_mul,hm1 r,one_pow]
refine ⟨N,hN,?_⟩
have hn := nilpotent_aeval_of_vanishing_roots A (minpoly.monic (Algebra.IsIntegral.isIntegral A))
(IsAlgClosed.splits (minpoly K A)) (X^N-C (a^N)) ?_
· simpa only [map_sub,map_pow,aeval_X,aeval_C,Algebra.algebraMap_eq_smul_one,_root_.smul_pow,one_pow] using hn
· intro r hr'
have hrN := hpow ⟨r,Multiset.mem_toFinset.mpr hr'⟩
have hep : r^N=a^N := by
apply (div_eq_one_iff_eq (pow_ne_zero _ ha)).mp
simpa only [div_pow] using hrN
simp only [IsRoot,eval_sub,eval_pow,eval_X,eval_C,hep,sub_self]
end CriticalPercolation.LinearGrowth
end
end
end OAI
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