| import Mathlib.Algebra.Polynomial.Roots |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.HilbertGrowthComparison |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeDegreeExistence |
| import OAI.NumberTheory.PiExponent.Polynomials.ColonHilbertPolynomial |
| import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousHilbertPolynomial |
| import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousPrimeFactor |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W22 |
|
|
| open PiExponentJets.W64 |
| attribute [local instance] MvPolynomial.gradedAlgebra |
|
|
| variable {k σ : Type*} [Field k] [Fintype σ] |
|
|
| theorem polynomial_eq_of_nat_tail (p q : Polynomial ℚ) (N : ℕ) |
| (h : ∀ n : ℕ, N < n → p.eval (n : ℚ) = q.eval (n : ℚ)) : p = q := by |
| have hinj : Function.Injective (fun n : ℕ => ((N + 1 + n : ℕ) : ℚ)) := by |
| intro a b hab |
| change ((N + 1 + a : ℕ) : ℚ) = ((N + 1 + b : ℕ) : ℚ) at hab |
| have hab' : N + 1 + a = N + 1 + b := by exact_mod_cast hab |
| omega |
| apply Polynomial.eq_of_infinite_eval_eq |
| apply (Set.infinite_range_of_injective hinj).mono |
| rintro x ⟨n, rfl⟩ |
| exact h _ (by omega) |
|
|
| noncomputable def actualHP (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : Polynomial ℚ := |
| (exists_homogeneous_hilbert_polynomial σ I hI).choose |
|
|
| theorem actualHP_eventually (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : |
| ∃ N : ℕ, ∀ n : ℕ, N < n → |
| (Module.finrank k (quotientSection I n) : ℚ) = (actualHP I hI).eval (n : ℚ) := |
| (exists_homogeneous_hilbert_polynomial σ I hI).choose_spec |
|
|
| theorem actualHP_unique (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| (p : Polynomial ℚ) (N : ℕ) |
| (hp : ∀ n : ℕ, N < n → |
| (Module.finrank k (quotientSection I n) : ℚ) = p.eval (n : ℚ)) : |
| actualHP I hI = p := by |
| obtain ⟨M, hM⟩ := actualHP_eventually I hI |
| apply polynomial_eq_of_nat_tail _ _ (M + N) |
| intro n hn |
| exact (hM n (by omega)).symm.trans (hp n (by omega)) |
|
|
| theorem actualHP_congr {I J : Ideal (MvPolynomial σ k)} |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| (hJ : J.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) (h : I = J) : |
| actualHP I hI = actualHP J hJ := by |
| subst J |
| rfl |
|
|
| theorem actualHP_top |
| (h : (⊤ : Ideal (MvPolynomial σ k)).IsHomogeneous |
| (MvPolynomial.homogeneousSubmodule σ k)) : actualHP ⊤ h = 0 := by |
| apply actualHP_unique _ h 0 0 |
| intro n _ |
| have hz : Module.finrank k (quotientSection (⊤ : Ideal (MvPolynomial σ k)) n) = 0 := |
| Module.finrank_eq_zero_of_subsingleton k _ |
| simp only [hz, Nat.cast_zero, Polynomial.eval_zero] |
|
|
| theorem actualHP_leadingCoeff_nonneg (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : |
| 0 ≤ (actualHP I hI).leadingCoeff := by |
| obtain ⟨N, hN⟩ := actualHP_eventually I hI |
| apply W27.polynomial_leadingCoeff_nonneg_of_eventually_nonneg _ (N + 1) |
| intro n hn |
| rw [← hN n (by omega)] |
| positivity |
|
|
| theorem actualHP_natDegree_antitone {I J : Ideal (MvPolynomial σ k)} |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| (hJ : J.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) (hIJ : I ≤ J) : |
| (actualHP J hJ).natDegree ≤ (actualHP I hI).natDegree := by |
| by_cases hz : actualHP J hJ = 0 |
| · simp only [hz, Polynomial.natDegree_zero, Nat.zero_le] |
| have hpos : 0 < (actualHP J hJ).leadingCoeff := |
| lt_of_le_of_ne (actualHP_leadingCoeff_nonneg J hJ) |
| (Ne.symm (Polynomial.leadingCoeff_ne_zero.mpr hz)) |
| obtain ⟨NI, hNI⟩ := actualHP_eventually I hI |
| obtain ⟨NJ, hNJ⟩ := actualHP_eventually J hJ |
| apply W24.natDegree_le_of_eventual_polynomial_le _ _ hpos (NI + NJ + 1) |
| intro n hn |
| rw [← hNI n (by omega), ← hNJ n (by omega)] |
| have hdim := (quotientSectionFactor I J hIJ n).finrank_range_le |
| rw [LinearMap.range_eq_top.mpr (quotientSectionFactor_surjective I J hIJ n), |
| finrank_top] at hdim |
| exact_mod_cast hdim |
|
|
| theorem actualHP_colon_step (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| {d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) |
| (hJ : (Ideal.span {f} ⊔ I).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| (hP : (I.colon {f}).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : |
| actualHP I hI = actualHP (Ideal.span {f} ⊔ I) hJ + |
| shiftHilbertPolynomial (actualHP (I.colon {f}) hP) d := by |
| obtain ⟨NJ, hNJ⟩ := actualHP_eventually (Ideal.span {f} ⊔ I) hJ |
| obtain ⟨NP, hNP⟩ := actualHP_eventually (I.colon {f}) hP |
| apply actualHP_unique I hI _ (NJ + NP + d) |
| exact colon_hilbert_polynomial_eventually I hI f hf _ _ NJ NP hNJ hNP |
|
|
| end PiExponentJets.W22 |
|
|
| end OAI |
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|
