# Theory Poly_Roots

theory Poly_Roots
imports Complex_Main
```(*  Author: John Harrison and Valentina Bruno
Ported from "hol_light/Multivariate/complexes.ml" by L C Paulson
*)

section ‹polynomial functions: extremal behaviour and root counts›

theory Poly_Roots
imports Complex_Main
begin

subsection‹Basics about polynomial functions: extremal behaviour and root counts.›

lemma sub_polyfun:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows   "(∑i≤n. a i * x^i) - (∑i≤n. a i * y^i) =
(x - y) * (∑j<n. ∑k= Suc j..n. a k * y^(k - Suc j) * x^j)"
proof -
have "(∑i≤n. a i * x^i) - (∑i≤n. a i * y^i) =
(∑i≤n. a i * (x^i - y^i))"
by (simp add: algebra_simps sum_subtractf [symmetric])
also have "... = (∑i≤n. a i * (x - y) * (∑j<i. y^(i - Suc j) * x^j))"
also have "... = (x - y) * (∑i≤n. (∑j<i. a i * y^(i - Suc j) * x^j))"
also have "... = (x - y) * (∑j<n. (∑i=Suc j..n. a i * y^(i - Suc j) * x^j))"
finally show ?thesis .
qed

lemma sub_polyfun_alt:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows   "(∑i≤n. a i * x^i) - (∑i≤n. a i * y^i) =
(x - y) * (∑j<n. ∑k<n-j. a (j+k+1) * y^k * x^j)"
proof -
{ fix j
have "(∑k = Suc j..n. a k * y^(k - Suc j) * x^j) =
(∑k <n - j. a (Suc (j + k)) * y^k * x^j)"
by (rule sum.reindex_bij_witness[where i="λi. i + Suc j" and j="λi. i - Suc j"]) auto }
then show ?thesis
qed

lemma polyfun_linear_factor:
fixes a :: "'a::{comm_ring,monoid_mult}"
shows  "∃b. ∀z. (∑i≤n. c i * z^i) =
(z-a) * (∑i<n. b i * z^i) + (∑i≤n. c i * a^i)"
proof -
{ fix z
have "(∑i≤n. c i * z^i) - (∑i≤n. c i * a^i) =
(z - a) * (∑j<n. (∑k = Suc j..n. c k * a^(k - Suc j)) * z^j)"
then have "(∑i≤n. c i * z^i) =
(z - a) * (∑j<n. (∑k = Suc j..n. c k * a^(k - Suc j)) * z^j)
+ (∑i≤n. c i * a^i)"
then show ?thesis
by (intro exI allI)
qed

lemma polyfun_linear_factor_root:
fixes a :: "'a::{comm_ring,monoid_mult}"
assumes "(∑i≤n. c i * a^i) = 0"
shows  "∃b. ∀z. (∑i≤n. c i * z^i) = (z-a) * (∑i<n. b i * z^i)"
using polyfun_linear_factor [of c n a] assms
by simp

lemma adhoc_norm_triangle: "a + norm(y) ≤ b ==> norm(x) ≤ a ==> norm(x + y) ≤ b"
by (metis norm_triangle_mono order.trans order_refl)

lemma polyfun_extremal_lemma:
fixes c :: "nat ⇒ 'a::real_normed_div_algebra"
assumes "e > 0"
shows "∃M. ∀z. M ≤ norm z ⟶ norm(∑i≤n. c i * z^i) ≤ e * norm(z) ^ Suc n"
proof (induction n)
case 0
show ?case
by (rule exI [where x="norm (c 0) / e"]) (auto simp: mult.commute pos_divide_le_eq assms)
next
case (Suc n)
then obtain M where M: "∀z. M ≤ norm z ⟶ norm (∑i≤n. c i * z^i) ≤ e * norm z ^ Suc n" ..
show ?case
proof (rule exI [where x="max 1 (max M ((e + norm(c(Suc n))) / e))"], clarify)
fix z::'a
assume "max 1 (max M ((e + norm (c (Suc n))) / e)) ≤ norm z"
then have norm1: "0 < norm z" "M ≤ norm z" "(e + norm (c (Suc n))) / e ≤ norm z"
by auto
then have norm2: "(e + norm (c (Suc n))) ≤ e * norm z"  "(norm z * norm z ^ n) > 0"
apply (metis assms less_divide_eq mult.commute not_le)
using norm1 apply (metis mult_pos_pos zero_less_power)
done
have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n)) =
(e + norm (c (Suc n))) * (norm z * norm z ^ n)"
by (simp add: norm_mult norm_power algebra_simps)
also have "... ≤ (e * norm z) * (norm z * norm z ^ n)"
using norm2 by (metis real_mult_le_cancel_iff1)
also have "... = e * (norm z * (norm z * norm z ^ n))"
finally have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n))
≤ e * (norm z * (norm z * norm z ^ n))" .
then show "norm (∑i≤Suc n. c i * z^i) ≤ e * norm z ^ Suc (Suc n)" using M norm1
by (drule_tac x=z in spec) (auto simp: intro!: adhoc_norm_triangle)
qed
qed

lemma norm_lemma_xy: assumes "¦b¦ + 1 ≤ norm(y) - a" "norm(x) ≤ a" shows "b ≤ norm(x + y)"
proof -
have "b ≤ norm y - norm x"
using assms by linarith
then show ?thesis
by (metis (no_types) add.commute norm_diff_ineq order_trans)
qed

lemma polyfun_extremal:
fixes c :: "nat ⇒ 'a::real_normed_div_algebra"
assumes "∃k. k ≠ 0 ∧ k ≤ n ∧ c k ≠ 0"
shows "eventually (λz. norm(∑i≤n. c i * z^i) ≥ B) at_infinity"
using assms
proof (induction n)
case 0 then show ?case
by simp
next
case (Suc n)
show ?case
proof (cases "c (Suc n) = 0")
case True
with Suc show ?thesis
by auto (metis diff_is_0_eq diffs0_imp_equal less_Suc_eq_le not_less_eq)
next
case False
with polyfun_extremal_lemma [of "norm(c (Suc n)) / 2" c n]
obtain M where M: "⋀z. M ≤ norm z ⟹
norm (∑i≤n. c i * z^i) ≤ norm (c (Suc n)) / 2 * norm z ^ Suc n"
by auto
show ?thesis
unfolding eventually_at_infinity
proof (rule exI [where x="max M (max 1 ((¦B¦ + 1) / (norm (c (Suc n)) / 2)))"], clarsimp)
fix z::'a
assume les: "M ≤ norm z"  "1 ≤ norm z"  "(¦B¦ * 2 + 2) / norm (c (Suc n)) ≤ norm z"
then have "¦B¦ * 2 + 2 ≤ norm z * norm (c (Suc n))"
by (metis False pos_divide_le_eq zero_less_norm_iff)
then have "¦B¦ * 2 + 2 ≤ norm z ^ (Suc n) * norm (c (Suc n))"
by (metis ‹1 ≤ norm z› order.trans mult_right_mono norm_ge_zero self_le_power zero_less_Suc)
then show "B ≤ norm ((∑i≤n. c i * z^i) + c (Suc n) * (z * z ^ n))" using M les
apply auto
apply (rule norm_lemma_xy [where a = "norm (c (Suc n)) * norm z ^ (Suc n) / 2"])
done
qed
qed
qed

lemma polyfun_rootbound:
fixes c :: "nat ⇒ 'a::{comm_ring,real_normed_div_algebra}"
assumes "∃k. k ≤ n ∧ c k ≠ 0"
shows "finite {z. (∑i≤n. c i * z^i) = 0} ∧ card {z. (∑i≤n. c i * z^i) = 0} ≤ n"
using assms
proof (induction n arbitrary: c)
case (Suc n) show ?case
proof (cases "{z. (∑i≤Suc n. c i * z^i) = 0} = {}")
case False
then obtain a where a: "(∑i≤Suc n. c i * a^i) = 0"
by auto
from polyfun_linear_factor_root [OF this]
obtain b where "⋀z. (∑i≤Suc n. c i * z^i) = (z - a) * (∑i< Suc n. b i * z^i)"
by auto
then have b: "⋀z. (∑i≤Suc n. c i * z^i) = (z - a) * (∑i≤n. b i * z^i)"
by (metis lessThan_Suc_atMost)
then have ins_ab: "{z. (∑i≤Suc n. c i * z^i) = 0} = insert a {z. (∑i≤n. b i * z^i) = 0}"
by auto
have c0: "c 0 = - (a * b 0)" using  b [of 0]
by simp
then have extr_prem: "~ (∃k≤n. b k ≠ 0) ⟹ ∃k. k ≠ 0 ∧ k ≤ Suc n ∧ c k ≠ 0"
by (metis Suc.prems le0 minus_zero mult_zero_right)
have "∃k≤n. b k ≠ 0"
apply (rule ccontr)
using polyfun_extremal [OF extr_prem, of 1]
apply (auto simp: eventually_at_infinity b simp del: sum_atMost_Suc)
apply (drule_tac x="of_real ba" in spec, simp)
done
then show ?thesis using Suc.IH [of b] ins_ab
by (auto simp: card_insert_if)
qed simp
qed simp

corollary
fixes c :: "nat ⇒ 'a::{comm_ring,real_normed_div_algebra}"
assumes "∃k. k ≤ n ∧ c k ≠ 0"
shows polyfun_rootbound_finite: "finite {z. (∑i≤n. c i * z^i) = 0}"
and polyfun_rootbound_card:   "card {z. (∑i≤n. c i * z^i) = 0} ≤ n"
using polyfun_rootbound [OF assms] by auto

lemma polyfun_finite_roots:
fixes c :: "nat ⇒ 'a::{comm_ring,real_normed_div_algebra}"
shows  "finite {z. (∑i≤n. c i * z^i) = 0} ⟷ (∃k. k ≤ n ∧ c k ≠ 0)"
proof (cases " ∃k≤n. c k ≠ 0")
case True then show ?thesis
by (blast intro: polyfun_rootbound_finite)
next
case False then show ?thesis
by (auto simp: infinite_UNIV_char_0)
qed

lemma polyfun_eq_0:
fixes c :: "nat ⇒ 'a::{comm_ring,real_normed_div_algebra}"
shows  "(∀z. (∑i≤n. c i * z^i) = 0) ⟷ (∀k. k ≤ n ⟶ c k = 0)"
proof (cases "(∀z. (∑i≤n. c i * z^i) = 0)")
case True
then have "~ finite {z. (∑i≤n. c i * z^i) = 0}"
with True show ?thesis
by (metis (poly_guards_query) polyfun_rootbound_finite)
next
case False
then show ?thesis
by auto
qed

lemma polyfun_eq_const:
fixes c :: "nat ⇒ 'a::{comm_ring,real_normed_div_algebra}"
shows  "(∀z. (∑i≤n. c i * z^i) = k) ⟷ c 0 = k ∧ (∀k. k ≠ 0 ∧ k ≤ n ⟶ c k = 0)"
proof -
{fix z
have "(∑i≤n. c i * z^i) = (∑i≤n. (if i = 0 then c 0 - k else c i) * z^i) + k"
by (induct n) auto
} then
have "(∀z. (∑i≤n. c i * z^i) = k) ⟷ (∀z. (∑i≤n. (if i = 0 then c 0 - k else c i) * z^i) = 0)"
by auto
also have "... ⟷  c 0 = k ∧ (∀k. k ≠ 0 ∧ k ≤ n ⟶ c k = 0)"
by (auto simp: polyfun_eq_0)
finally show ?thesis .
qed

end

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