| 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 |
|
|
