| import OAI.NumberTheory.PiExponent.Polynomials.HilbertPolynomialBase | |
| namespace OAI | |
| namespace PiExponentJets.W64 | |
| variable {k σ : Type*} [Field k] [Fintype σ] | |
| theorem quotientSection_finrank_zero_of_no_variables | |
| (hσ : Fintype.card σ = 0) (I : Ideal (MvPolynomial σ k)) (n : ℕ) : | |
| Module.finrank k (quotientSection I (n+1)) = 0 := by | |
| have h := Submodule.finrank_map_le (Ideal.Quotient.mkₐ k I).toLinearMap | |
| (MvPolynomial.homogeneousSubmodule σ k (n+1)) | |
| rw [homogeneousSection_finrank, card_exponents_degree, hσ, | |
| Nat.multichoose_zero_succ] at h | |
| exact Nat.eq_zero_of_le_zero h | |
| theorem sectionHilbertSeries_no_variables | |
| (hσ : Fintype.card σ = 0) (I : Ideal (MvPolynomial σ k)) : | |
| sectionHilbertSeries I = PowerSeries.C (Module.finrank k (quotientSection I 0) : ℤ) := by | |
| ext n | |
| cases n with | |
| | zero => simp | |
| | succ n => | |
| rw [coeff_sectionHilbertSeries, quotientSection_finrank_zero_of_no_variables hσ I n] | |
| simp only [PowerSeries.coeff_C, Nat.succ_ne_zero, ite_false, Nat.cast_zero] | |
| theorem exists_hilbert_numerator_no_variables | |
| (hσ : Fintype.card σ = 0) (I : Ideal (MvPolynomial σ k)) : | |
| ∃ P : Polynomial ℤ, sectionHilbertSeries I = | |
| (P : PowerSeries ℤ) * (PowerSeries.invOneSubPow ℤ (Fintype.card σ)).val := by | |
| refine ⟨Polynomial.C (Module.finrank k (quotientSection I 0) : ℤ), ?_⟩ | |
| simpa only [hσ, PowerSeries.invOneSubPow_zero, Units.val_one, mul_one, | |
| Polynomial.coe_C] using sectionHilbertSeries_no_variables hσ I | |
| end PiExponentJets.W64 | |
| end OAI | |