| import Mathlib |
| import OAI.NumberTheory.PiExponent.Polynomials.PolynomialFrameDegree |
|
|
| namespace OAI |
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|
| noncomputable section |
| open scoped BigOperators |
| namespace PiExponent.PolynomialPoleBound |
|
|
| variable {K C σ : Type*} [Field K] [CommRing C] [Algebra C K] |
|
|
| def realValuation (v : AddValuation K (WithTop ℤ)) : AddValuation K (WithTop ℝ) := |
| v.map (Int.castAddHom ℝ).withTopMap rfl |
| ((show Monotone (fun z : ℤ => (z : ℝ)) from Int.cast_mono).withTop_map) |
|
|
| @[simp] theorem realValuation_apply (v : AddValuation K (WithTop ℤ)) (x : K) : |
| realValuation v x = WithTop.map (fun z : ℤ => (z : ℝ)) (v x) := rfl |
|
|
| theorem valuation_prod (v : AddValuation K (WithTop ℝ)) |
| (s : Finset σ) (f : σ → K) : v (∏ i ∈ s, f i) = ∑ i ∈ s, v (f i) := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => simp |
| | insert i s hi ih => simp [hi, ih] |
|
|
| theorem monomial_order_lower (v : AddValuation K (WithTop ℝ)) |
| (x : σ → K) (w : σ → ℝ) (D : ℝ) |
| (hx : ∀ i, ((-w i * D : ℝ) : WithTop ℝ) ≤ v (x i)) |
| (hc : ∀ c : C, (0 : WithTop ℝ) ≤ v (algebraMap C K c)) |
| (d : σ →₀ ℕ) (c : C) : |
| ((-Finsupp.weight w d * D : ℝ) : WithTop ℝ) ≤ |
| v (MvPolynomial.aeval x (MvPolynomial.monomial d c)) := by |
| classical |
| rw [MvPolynomial.aeval_monomial, v.map_mul, Finsupp.prod, valuation_prod] |
| have heq : ((-Finsupp.weight w d * D : ℝ) : WithTop ℝ) = |
| ∑ i ∈ d.support, d i • ((-w i * D : ℝ) : WithTop ℝ) := by |
| simp only [← WithTop.coe_nsmul, ← WithTop.coe_sum] |
| apply congrArg (fun r : ℝ => (r : WithTop ℝ)) |
| simp only [Finsupp.weight_apply, Finsupp.sum, nsmul_eq_mul, |
| Finset.sum_mul, neg_mul] |
| rw [← Finset.sum_neg_distrib] |
| apply Finset.sum_congr rfl |
| intro i hi |
| ring |
| rw [heq] |
| calc |
| (∑ i ∈ d.support, d i • ((-w i * D : ℝ) : WithTop ℝ)) ≤ |
| ∑ i ∈ d.support, v (x i ^ d i) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| rw [v.map_pow] |
| exact nsmul_le_nsmul_right (hx i) (d i) |
| _ ≤ v (algebraMap C K c) + ∑ i ∈ d.support, v (x i ^ d i) := |
| le_add_of_nonneg_left (hc c) |
|
|
| theorem polynomial_order_lower (v : AddValuation K (WithTop ℝ)) |
| (x : σ → K) (w : σ → ℝ) (D N : ℝ) (hD : 0 ≤ D) |
| (hx : ∀ i, ((-w i * D : ℝ) : WithTop ℝ) ≤ v (x i)) |
| (hc : ∀ c : C, (0 : WithTop ℝ) ≤ v (algebraMap C K c)) |
| (p : MvPolynomial σ C) |
| (hp : ∀ d ∈ p.support, Finsupp.weight w d ≤ N) : |
| ((-N * D : ℝ) : WithTop ℝ) ≤ v (MvPolynomial.aeval x p) := by |
| classical |
| rw [p.as_sum, map_sum] |
| apply v.map_le_sum |
| intro d hd |
| have hnum : -N * D ≤ -Finsupp.weight w d * D := |
| mul_le_mul_of_nonneg_right (neg_le_neg (hp d hd)) hD |
| exact (WithTop.coe_le_coe.mpr hnum).trans (monomial_order_lower v x w D hx hc d _) |
|
|
| theorem framePolynomial_order_lower {m : ℕ} [Algebra ℂ K] |
| (v : AddValuation K (WithTop ℝ)) (x : Fin (m+1) → K) |
| (w : Fin (m+1) → ℝ) (D N : ℝ) (hD : 0 ≤ D) |
| (hx : ∀ i, ((-w i * D : ℝ) : WithTop ℝ) ≤ v (x i)) |
| (hc : ∀ c : ℂ, (0 : WithTop ℝ) ≤ v (algebraMap ℂ K c)) |
| (p : MvPolynomial (Fin (m+1)) ℂ) (hp : PiExponentApprox.HasWeightedDegreeLE w N p) : |
| ((-N * D : ℝ) : WithTop ℝ) ≤ v (MvPolynomial.aeval x p) := by |
| apply polynomial_order_lower v x w D N hD hx hc p |
| intro d hd |
| simpa [Finsupp.weight_eq_sum, nsmul_eq_mul, PiExponentApprox.monomialWeight] using hp d hd |
|
|
| end PiExponent.PolynomialPoleBound |
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|
| end |
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| end OAI |
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|
