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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 | import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPolynomial.Homogeneous
namespace OAI
namespace PiExponentJets.W22
open scoped BigOperators
open MvPolynomial
variable {k σ : Type*} [CommRing k]
noncomputable def dehomogenize : MvPolynomial (Option σ) k →+* MvPolynomial σ k :=
MvPolynomial.eval₂Hom MvPolynomial.C (fun o : Option σ => o.elim 1 MvPolynomial.X)
@[simp] theorem dehomogenize_X_none :
dehomogenize (MvPolynomial.X none : MvPolynomial (Option σ) k) = 1 := by
simp [dehomogenize]
@[simp] theorem dehomogenize_rename (p : MvPolynomial σ k) :
dehomogenize (MvPolynomial.rename some p) = p := by
change MvPolynomial.eval₂Hom MvPolynomial.C
(fun o : Option σ => o.elim 1 MvPolynomial.X) (MvPolynomial.rename some p) = p
rw [MvPolynomial.eval₂Hom_rename]
change MvPolynomial.eval₂ MvPolynomial.C MvPolynomial.X p = p
exact MvPolynomial.eval₂_eta p
noncomputable def homogenize (p : MvPolynomial σ k) : MvPolynomial (Option σ) k :=
∑ i ∈ Finset.range (p.totalDegree + 1),
MvPolynomial.rename some (MvPolynomial.homogeneousComponent i p) *
MvPolynomial.X none ^ (p.totalDegree - i)
@[simp] theorem dehomogenize_homogenize (p : MvPolynomial σ k) :
dehomogenize (homogenize p) = p := by
simp only [homogenize, map_sum, map_mul, map_pow, dehomogenize_rename,
dehomogenize_X_none, one_pow, mul_one]
exact MvPolynomial.sum_homogeneousComponent p
theorem homogenize_isHomogeneous (p : MvPolynomial σ k) :
(homogenize p).IsHomogeneous p.totalDegree := by
apply MvPolynomial.IsHomogeneous.sum
intro i hi
have hi' : i ≤ p.totalDegree := Nat.lt_succ_iff.mp (Finset.mem_range.mp hi)
have hcomponent :=
(MvPolynomial.homogeneousComponent_isHomogeneous i p).rename_isHomogeneous
(f := (some : σ → Option σ))
simpa only [Nat.add_sub_of_le hi'] using
hcomponent.mul (MvPolynomial.isHomogeneous_X_pow (none : Option σ) (p.totalDegree - i))
theorem homogenize_totalDegree_le (p : MvPolynomial σ k) :
(homogenize p).totalDegree ≤ p.totalDegree :=
(homogenize_isHomogeneous p).totalDegree_le
theorem homogenize_ne_zero {p : MvPolynomial σ k} (hp : p ≠ 0) : homogenize p ≠ 0 := by
intro h
apply hp
have := congrArg dehomogenize h
simpa using this
theorem map_span_homogenize {ι : Type*} (f : ι → MvPolynomial σ k) :
Ideal.map dehomogenize (Ideal.span (Set.range (fun i => homogenize (f i)))) =
Ideal.span (Set.range f) := by
rw [Ideal.map_span]
congr 1
ext p
simp only [Set.mem_image, Set.mem_range]
constructor
· rintro ⟨q, ⟨i, rfl⟩, hp⟩
exact ⟨i, (dehomogenize_homogenize (f i)).symm.trans hp⟩
· rintro ⟨i, rfl⟩
exact ⟨homogenize (f i), ⟨i, rfl⟩, dehomogenize_homogenize (f i)⟩
end PiExponentJets.W22
end OAI
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