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b797a5b | 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.NumberTheory.PiExponent.LocalAlgebra.ConePrimeLocalization
import OAI.NumberTheory.PiExponent.Polynomials.Homogenization
namespace OAI
noncomputable section
open scoped BigOperators
namespace PiExponentSiegel.W17.ConeLocalLength
section Scaling
variable {k B σ : Type*} [CommRing k] [CommRing B]
theorem eval₂_scale_of_isHomogeneous (F : MvPolynomial σ k) (d : ℕ)
(hF : F.IsHomogeneous d) (c : k →+* B) (x : σ → B) (t : B) :
MvPolynomial.eval₂ c (fun i => x i * t) F =
MvPolynomial.eval₂ c x F * t ^ d := by
classical
simp only [MvPolynomial.eval₂_eq, Finset.sum_mul]
apply Finset.sum_congr rfl
intro m hm
simp only [mul_pow, Finset.prod_mul_distrib]
rw [Finset.prod_pow_eq_pow_sum, ← hF.degree_eq_sum_deg_support hm, mul_assoc]
end Scaling
variable (k σ : Type*) [CommRing k]
private theorem coneToLaurent_scaled_eval (F : ConeRing k σ) :
coneToLaurent k σ F =
MvPolynomial.eval₂ (LaurentPolynomial.C.comp MvPolynomial.C)
(fun i : Option σ =>
LaurentPolynomial.C (i.elim 1 MvPolynomial.X) *
(LaurentPolynomial.T 1 : LaurentChart k σ)) F := by
have hv : (fun i : Option σ => i.elim (LaurentPolynomial.T 1 : LaurentChart k σ)
(fun j => LaurentPolynomial.C (MvPolynomial.X j) * LaurentPolynomial.T 1)) =
(fun i : Option σ => LaurentPolynomial.C (i.elim 1 MvPolynomial.X) *
(LaurentPolynomial.T 1 : LaurentChart k σ)) := by
funext i
cases i <;> simp only [Option.elim_none, Option.elim_some, map_one, one_mul]
change MvPolynomial.eval₂ (LaurentPolynomial.C.comp MvPolynomial.C)
(fun i : Option σ => i.elim (LaurentPolynomial.T 1)
(fun j => LaurentPolynomial.C (MvPolynomial.X j) * LaurentPolynomial.T 1)) F = _
rw [hv]
private theorem laurent_eval_dehomogenize (F : ConeRing k σ) :
MvPolynomial.eval₂ (LaurentPolynomial.C.comp MvPolynomial.C)
(fun i : Option σ => LaurentPolynomial.C (i.elim 1 MvPolynomial.X)) F =
(LaurentPolynomial.C (PiExponentJets.W22.dehomogenize F) : LaurentChart k σ) :=
(MvPolynomial.map_eval₂Hom MvPolynomial.C
(fun i : Option σ => i.elim 1 MvPolynomial.X) LaurentPolynomial.C F).symm
theorem coneToLaurent_homogeneous (F : ConeRing k σ) (d : ℕ)
(hF : F.IsHomogeneous d) :
coneToLaurent k σ F =
LaurentPolynomial.C (PiExponentJets.W22.dehomogenize F) *
(LaurentPolynomial.T 1 : LaurentChart k σ) ^ d := by
rw [coneToLaurent_scaled_eval]
rw [eval₂_scale_of_isHomogeneous F d hF, laurent_eval_dehomogenize]
theorem coneToLaurent_homogenize (f : AffineRing k σ) :
coneToLaurent k σ (PiExponentJets.W22.homogenize f) =
LaurentPolynomial.C f * (LaurentPolynomial.T 1 : LaurentChart k σ) ^ f.totalDegree := by
simpa using coneToLaurent_homogeneous k σ (PiExponentJets.W22.homogenize f)
f.totalDegree (PiExponentJets.W22.homogenize_isHomogeneous f)
theorem map_homogenized_span_eq_laurent_span {ι : Type*} (f : ι → AffineRing k σ) :
(Ideal.span (Set.range (fun i => PiExponentJets.W22.homogenize (f i)))).map
(coneToLaurent k σ) =
(Ideal.span (Set.range f)).map (LaurentPolynomial.C : AffineRing k σ →+* LaurentChart k σ) := by
apply le_antisymm
· apply Ideal.map_le_iff_le_comap.mpr
apply Ideal.span_le.mpr
rintro _ ⟨i, rfl⟩
change coneToLaurent k σ (PiExponentJets.W22.homogenize (f i)) ∈
(Ideal.span (Set.range f)).map
(LaurentPolynomial.C : AffineRing k σ →+* LaurentChart k σ)
rw [coneToLaurent_homogenize]
exact Ideal.mul_mem_right _ _ (Ideal.mem_map_of_mem _
(Ideal.subset_span (Set.mem_range_self i)))
· apply Ideal.map_le_iff_le_comap.mpr
apply Ideal.span_le.mpr
rintro _ ⟨i, rfl⟩
change (LaurentPolynomial.C (f i) : LaurentChart k σ) ∈
(Ideal.span (Set.range (fun j => PiExponentJets.W22.homogenize (f j)))).map
(coneToLaurent k σ)
apply (Ideal.mul_unit_mem_iff_mem _
((LaurentPolynomial.isUnit_T (R := AffineRing k σ) 1).pow (f i).totalDegree)).mp
rw [← coneToLaurent_homogenize]
exact Ideal.mem_map_of_mem _ (Ideal.subset_span (Set.mem_range_self i))
variable (P : Ideal (AffineRing k σ)) [P.IsPrime]
theorem homogeneous_mem_conePrime_iff (F : ConeRing k σ) (d : ℕ)
(hF : F.IsHomogeneous d) :
F ∈ conePrime k σ P ↔ PiExponentJets.W22.dehomogenize F ∈ P := by
change coneAwayToLaurent k σ (algebraMap (ConeRing k σ) (ConeAway k σ) F) ∈
laurentPrime (AffineRing k σ) P ↔ _
rw [coneAwayToLaurent_algebraMap, coneToLaurent_homogeneous k σ F d hF,
Ideal.mul_unit_mem_iff_mem _ ((LaurentPolynomial.isUnit_T 1).pow d)]
change PiExponentJets.W22.dehomogenize F ∈
(laurentPrime (AffineRing k σ) P).comap (algebraMap (AffineRing k σ) (LaurentChart k σ)) ↔ _
rw [laurentPrime_comap]
theorem homogenize_mem_conePrime_iff (f : AffineRing k σ) :
PiExponentJets.W22.homogenize f ∈ conePrime k σ P ↔ f ∈ P := by
simpa using homogeneous_mem_conePrime_iff k σ P
(PiExponentJets.W22.homogenize f) f.totalDegree
(PiExponentJets.W22.homogenize_isHomogeneous f)
end PiExponentSiegel.W17.ConeLocalLength
end
end OAI
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