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f4b3ad1 | 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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import OAI.NumberTheory.Ostmann.Characters.IntegerPolynomialBounds
import OAI.NumberTheory.Ostmann.Arithmetic.SymbolicLineComparison
namespace OAI
/-! # Signed arithmetic comparison for actual integer polynomial tests -/
namespace Ostmann
open scoped BigOperators
noncomputable def integerTestValue {A : Type*} {n : ℕ} (value : A → ℤ)
(F : MvPolynomial (Fin n) ℤ) (x : Fin n → A) : ℤ :=
MvPolynomial.eval₂Hom (RingHom.id ℤ) (fun j => value (x j)) F
/-- The signed replacement estimate with all zero and divisor-count bridges
proved for the actual integer polynomial evaluations. -/
theorem integer_polynomial_symbolic_comparison_le {A I : Type*} [Fintype A] [Fintype I]
{n : ℕ} (value : A → ℤ) (hinj : Function.Injective value)
(F : I → MvPolynomial (Fin n) ℤ)
(μ : Fin n → A → ℝ) (hμ : ∀ i a, 0 ≤ μ i a)
(hmass : ∀ i, ∑ a, μ i a = 1)
(α : ℝ) (hα : 0 ≤ α) (hmax : ∀ i a, μ i a ≤ α)
(P : Finset ℕ) (hprime : ∀ p ∈ P, p.Prime)
(ν : ℕ → ℝ) (hν : ∀ p ∈ P, 0 ≤ ν p) (hνmass : ∑ p ∈ P, ν p = 1)
(β V H : ℝ) (hβ : 0 ≤ β) (hV : 0 < V) (hH : 1 ≤ H)
(hνmax : ∀ p ∈ P, ν p ≤ β) (hsize : ∀ p ∈ P, V ≤ Real.log (p : ℝ))
(hvalue : ∀ i x, |(integerTestValue value (F i) x : ℝ)| ≤ H)
(Ψ : (Fin n → A) → ℕ → (I → Bool) → ℂ)
(B : ℝ) (hB : 0 ≤ B) (hΨ : ∀ x p v, ‖Ψ x p v‖ ≤ B) :
‖∑ x, ∑ p ∈ P, ((productPrior μ x * ν p : ℝ) : ℂ) *
(Ψ x p (fun i => arithmeticTestFlag (p ∣ (integerTestValue value (F i) x).natAbs)) -
Ψ x p (fun i => arithmeticTestFlag (F i = 0)))‖ ≤
2 * B * ∑ i, (((F i).totalDegree : ℝ) * α + (Real.log H / V) * β) := by
classical
let Q : I → MvPolynomial (Fin n) ℚ := fun i => MvPolynomial.map (Int.castRingHom ℚ) (F i)
let N : I → (Fin n → A) → ℕ := fun i x => (integerTestValue value (F i) x).natAbs
have hzero (i : I) (x : Fin n → A) : N i x = 0 ↔
MvPolynomial.eval (fun j => (value (x j) : ℚ)) (Q i) = 0 := by
dsimp only [N, Q, integerTestValue]
rw [integerPolynomial_rational_eval]
simp only [Int.natAbs_eq_zero, Int.cast_eq_zero]
have hcard (i : I) (x : Fin n → A) (hx : N i x ≠ 0) :
((P.filter fun p => p ∣ N i x).card : ℝ) ≤ Real.log H / V := by
have hz : integerTestValue value (F i) x ≠ 0 := by
intro hz
apply hx
simp only [N, hz, Int.natAbs_zero]
exact int_prime_divisor_card_le P _ hz hprime V H hV hsize (hvalue i x)
have hk : 0 ≤ Real.log H / V := div_nonneg (Real.log_nonneg hH) hV.le
have h := polynomial_symbolic_comparison_le (fun a => (value a : ℚ))
(Int.cast_injective.comp hinj) Q μ hμ hmass α hα hmax P ν hν hνmass N hzero
β (Real.log H / V) hβ hk hνmax hcard Ψ B hB hΨ
have hflag (i : I) : arithmeticTestFlag (Q i = 0) = arithmeticTestFlag (F i = 0) :=
congrArg arithmeticTestFlag (propext (integerPolynomial_rational_zero_iff (F i)))
have hdegree (i : I) : (Q i).totalDegree = (F i).totalDegree :=
integerPolynomial_rational_degree (F i)
simpa only [hflag, hdegree, N] using h
end Ostmann
end OAI
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