| import OAI.NumberTheory.PiExponent.LocalAlgebra.ColonHilbertStep |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PositiveDimensionalHilbert |
|
|
| namespace OAI |
|
|
| namespace PiExponentJets.W64 |
|
|
| attribute [local instance] MvPolynomial.gradedAlgebra |
| variable {k σ : Type*} [Field k] [Finite σ] |
|
|
| noncomputable def shiftHilbertPolynomial (p : Polynomial ℚ) (d : ℕ) : Polynomial ℚ := |
| p.comp (Polynomial.X - Polynomial.C (d : ℚ)) |
|
|
| @[simp] theorem shiftHilbertPolynomial_eval (p : Polynomial ℚ) (d n : ℕ) |
| (hdn : d ≤ n) : |
| (shiftHilbertPolynomial p d).eval (n : ℚ) = p.eval ((n-d : ℕ) : ℚ) := by |
| simp [shiftHilbertPolynomial, Polynomial.eval_comp, Nat.cast_sub hdn] |
|
|
| @[simp] theorem shiftHilbertPolynomial_natDegree (p : Polynomial ℚ) (d : ℕ) : |
| (shiftHilbertPolynomial p d).natDegree = p.natDegree := by |
| rw [shiftHilbertPolynomial, Polynomial.natDegree_comp, |
| Polynomial.natDegree_X_sub_C, mul_one] |
|
|
| @[simp] theorem shiftHilbertPolynomial_leadingCoeff (p : Polynomial ℚ) (d : ℕ) : |
| (shiftHilbertPolynomial p d).leadingCoeff = p.leadingCoeff := by |
| rw [shiftHilbertPolynomial, Polynomial.leadingCoeff_comp] |
| · rw [Polynomial.leadingCoeff_X_sub_C, one_pow, mul_one] |
| · rw [Polynomial.natDegree_X_sub_C] |
| exact one_ne_zero |
|
|
| theorem shiftHilbertPolynomial_top_coeff (p : Polynomial ℚ) (d t : ℕ) |
| (hp : p.natDegree ≤ t) : |
| (shiftHilbertPolynomial p d).coeff t = p.coeff t := by |
| by_cases he : p.natDegree = t |
| · rw [← he] |
| calc |
| (shiftHilbertPolynomial p d).coeff p.natDegree = |
| (shiftHilbertPolynomial p d).coeff (shiftHilbertPolynomial p d).natDegree := by |
| rw [shiftHilbertPolynomial_natDegree] |
| _ = (shiftHilbertPolynomial p d).leadingCoeff := Polynomial.coeff_natDegree |
| _ = p.leadingCoeff := shiftHilbertPolynomial_leadingCoeff p d |
| _ = p.coeff p.natDegree := Polynomial.coeff_natDegree.symm |
| · have ht : p.natDegree < t := lt_of_le_of_ne hp he |
| rw [Polynomial.coeff_eq_zero_of_natDegree_lt |
| (by simpa using ht : (shiftHilbertPolynomial p d).natDegree < t), |
| Polynomial.coeff_eq_zero_of_natDegree_lt ht] |
|
|
| theorem hilbertMultiplicity_shift (p : Polynomial ℚ) (d r : ℕ) |
| (hp : p.natDegree ≤ r-1) : |
| hilbertMultiplicity (shiftHilbertPolynomial p d) r = hilbertMultiplicity p r := by |
| unfold hilbertMultiplicity |
| rw [shiftHilbertPolynomial_top_coeff p d (r-1) hp] |
|
|
| theorem hilbertMultiplicity_add (p q : Polynomial ℚ) (r : ℕ) : |
| hilbertMultiplicity (p+q) r = hilbertMultiplicity p r + hilbertMultiplicity q r := by |
| simp [hilbertMultiplicity, add_mul] |
|
|
| theorem colon_hilbert_polynomial_eventually |
| (I : Ideal (MvPolynomial σ k)) |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) |
| {d : ℕ} (f : MvPolynomial σ k) (hf : f.IsHomogeneous d) |
| (pJ pP : Polynomial ℚ) (NJ NP : ℕ) |
| (hJ : ∀ n : ℕ, NJ < n → |
| (Module.finrank k (quotientSection (Ideal.span {f} ⊔ I) n) : ℚ) = pJ.eval (n : ℚ)) |
| (hP : ∀ n : ℕ, NP < n → |
| (Module.finrank k (quotientSection (I.colon {f}) n) : ℚ) = pP.eval (n : ℚ)) : |
| ∀ n : ℕ, NJ + NP + d < n → |
| (Module.finrank k (quotientSection I n) : ℚ) = |
| (pJ + shiftHilbertPolynomial pP d).eval (n : ℚ) := by |
| intro n hn |
| have hdn : d ≤ n := by omega |
| have hnJ : NJ < n := by omega |
| have hnP : NP < n-d := by omega |
| have h := colon_hilbert_step I hI f hf (n-d) |
| rw [Nat.sub_add_cancel hdn] at h |
| have hc : (Module.finrank k (quotientSection I n) : ℚ) = |
| (Module.finrank k (quotientSection (Ideal.span {f} ⊔ I) n) : ℚ) + |
| (Module.finrank k (quotientSection (I.colon {f}) (n-d)) : ℚ) := by |
| exact_mod_cast h.symm |
| rw [hc, hJ n hnJ, hP (n-d) hnP, |
| Polynomial.eval_add, shiftHilbertPolynomial_eval pP d n hdn] |
|
|
| theorem colon_hilbert_polynomial_multiplicity |
| (pJ pP : Polynomial ℚ) (d r : ℕ) (hP : pP.natDegree ≤ r-1) : |
| hilbertMultiplicity (pJ + shiftHilbertPolynomial pP d) r = |
| hilbertMultiplicity pJ r + hilbertMultiplicity pP r := by |
| rw [hilbertMultiplicity_add, hilbertMultiplicity_shift pP d r hP] |
|
|
| end PiExponentJets.W64 |
|
|
| end OAI |
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|
