openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /ConeHomogeneousEquations.lean
| 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 | |