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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | 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
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