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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 40 41 42 43 44 45 46 47 48 | import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeHilbertDegree
namespace OAI
namespace PiExponentJets.W24
open Polynomial
theorem natDegree_le_of_eventual_polynomial_le (p q : Polynomial ℚ)
(hp : 0 < p.leadingCoeff) (N : ℕ)
(hle : ∀ n : ℕ, N ≤ n → p.eval (n : ℚ) ≤ q.eval (n : ℚ)) :
p.natDegree ≤ q.natDegree := by
by_contra hn
have hdeg : q.degree < p.degree := degree_lt_degree (Nat.lt_of_not_ge hn)
have hnonneg := W27.polynomial_leadingCoeff_nonneg_of_eventually_nonneg
(q - p) N (fun n hn => by simpa only [eval_sub] using sub_nonneg.mpr (hle n hn))
rw [leadingCoeff_sub_of_degree_lt' hdeg] at hnonneg
exact (not_le_of_gt hp) (neg_nonneg.mp hnonneg)
theorem natDegree_affine_comp_le (p : Polynomial ℚ) (a b : ℚ) :
(p.comp (C a * X + C b)).natDegree ≤ p.natDegree := by
have ha : (C a * X + C b : Polynomial ℚ).natDegree ≤ 1 := by
rw [natDegree_add_C]
simpa only [natDegree_X] using natDegree_C_mul_le a (X : Polynomial ℚ)
exact (natDegree_comp_le).trans ((Nat.mul_le_mul_left _ ha).trans_eq (Nat.mul_one _))
theorem natDegree_eq_of_polynomial_growth_bounds
(q lower upper : Polynomial ℚ) (d : ℕ)
(hql : 0 < q.leadingCoeff) (hll : 0 < lower.leadingCoeff)
(hld : lower.natDegree = d) (hud : upper.natDegree ≤ d)
(a b c e : ℚ) (N : ℕ)
(hlo : ∀ n : ℕ, N ≤ n → lower.eval (n : ℚ) ≤ q.eval (a * n + b))
(hup : ∀ n : ℕ, N ≤ n → q.eval (n : ℚ) ≤ upper.eval (c * n + e)) :
q.natDegree = d := by
have hlow : lower.natDegree ≤ (q.comp (C a * X + C b)).natDegree :=
natDegree_le_of_eventual_polynomial_le lower _ hll N (by
simpa only [eval_comp, eval_add, eval_mul, eval_C, eval_X] using hlo)
have hupp : q.natDegree ≤ (upper.comp (C c * X + C e)).natDegree :=
natDegree_le_of_eventual_polynomial_le q _ hql N (by
simpa only [eval_comp, eval_add, eval_mul, eval_C, eval_X] using hup)
exact le_antisymm
(hupp.trans ((natDegree_affine_comp_le upper c e).trans hud))
(hld ▸ hlow.trans (natDegree_affine_comp_le q a b))
end PiExponentJets.W24
end OAI
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