section {* polynomial functions: extremal behaviour and root counts *}
(* Author: John Harrison and Valentina Bruno
Ported from "hol_light/Multivariate/complexes.ml" by L C Paulson
*)
theory PolyRoots
imports Complex_Main
begin
subsection{*Geometric progressions*}
lemma setsum_gp_basic:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "(1 - x) * (\<Sum>i\<le>n. x^i) = 1 - x^Suc n"
by (simp only: one_diff_power_eq [of "Suc n" x] lessThan_Suc_atMost)
lemma setsum_gp0:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
using setsum_gp_basic[of x n]
by (simp add: real_of_nat_def mult.commute divide_simps)
lemma setsum_power_add:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"
by (simp add: setsum_right_distrib power_add)
lemma setsum_power_shift:
fixes x :: "'a::{comm_ring,monoid_mult}"
assumes "m \<le> n"
shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"
proof -
have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"
by (simp add: setsum_right_distrib power_add [symmetric])
also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"
using `m \<le> n` by (intro setsum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
finally show ?thesis .
qed
lemma setsum_gp_multiplied:
fixes x :: "'a::{comm_ring,monoid_mult}"
assumes "m \<le> n"
shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"
proof -
have "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"
by (metis mult.assoc mult.commute assms setsum_power_shift)
also have "... =x^m * (1 - x^Suc(n-m))"
by (metis mult.assoc setsum_gp_basic)
also have "... = x^m - x^Suc n"
using assms
by (simp add: algebra_simps) (metis le_add_diff_inverse power_add)
finally show ?thesis .
qed
lemma setsum_gp:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(\<Sum>i=m..n. x^i) =
(if n < m then 0
else if x = 1 then of_nat((n + 1) - m)
else (x^m - x^Suc n) / (1 - x))"
using setsum_gp_multiplied [of m n x]
apply (auto simp: real_of_nat_def)
by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)
lemma setsum_gp_offset:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(\<Sum>i=m..m+n. x^i) =
(if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
using setsum_gp [of x m "m+n"]
by (auto simp: power_add algebra_simps)
lemma setsum_gp_strict:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
by (induct n) (auto simp: algebra_simps divide_simps)
subsection{*Basics about polynomial functions: extremal behaviour and root counts.*}
lemma sub_polyfun:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
(x - y) * (\<Sum>j<n. \<Sum>k= Suc j..n. a k * y^(k - Suc j) * x^j)"
proof -
have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
(\<Sum>i\<le>n. a i * (x^i - y^i))"
by (simp add: algebra_simps setsum_subtractf [symmetric])
also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
by (simp add: power_diff_sumr2 ac_simps)
also have "... = (x - y) * (\<Sum>i\<le>n. (\<Sum>j<i. a i * y^(i - Suc j) * x^j))"
by (simp add: setsum_right_distrib ac_simps)
also have "... = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - Suc j) * x^j))"
by (simp add: nested_setsum_swap')
finally show ?thesis .
qed
lemma sub_polyfun_alt:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
(x - y) * (\<Sum>j<n. \<Sum>k<n-j. a (j+k+1) * y^k * x^j)"
proof -
{ fix j
have "(\<Sum>k = Suc j..n. a k * y^(k - Suc j) * x^j) =
(\<Sum>k <n - j. a (Suc (j + k)) * y^k * x^j)"
by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + Suc j" and j="\<lambda>i. i - Suc j"]) auto }
then show ?thesis
by (simp add: sub_polyfun)
qed
lemma polyfun_linear_factor:
fixes a :: "'a::{comm_ring,monoid_mult}"
shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) =
(z-a) * (\<Sum>i<n. b i * z^i) + (\<Sum>i\<le>n. c i * a^i)"
proof -
{ fix z
have "(\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) =
(z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)"
by (simp add: sub_polyfun setsum_left_distrib)
then have "(\<Sum>i\<le>n. c i * z^i) =
(z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)
+ (\<Sum>i\<le>n. c i * a^i)"
by (simp add: algebra_simps) }
then show ?thesis
by (intro exI allI)
qed
lemma polyfun_linear_factor_root:
fixes a :: "'a::{comm_ring,monoid_mult}"
assumes "(\<Sum>i\<le>n. c i * a^i) = 0"
shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) = (z-a) * (\<Sum>i<n. b i * z^i)"
using polyfun_linear_factor [of c n a] assms
by simp
lemma adhoc_norm_triangle: "a + norm(y) \<le> b ==> norm(x) \<le> a ==> norm(x + y) \<le> b"
by (metis norm_triangle_mono order.trans order_refl)
lemma polyfun_extremal_lemma:
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
assumes "e > 0"
shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm(\<Sum>i\<le>n. c i * z^i) \<le> 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: "\<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> 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)) \<le> norm z"
then have norm1: "0 < norm z" "M \<le> norm z" "(e + norm (c (Suc n))) / e \<le> norm z"
by auto
then have norm2: "(e + norm (c (Suc n))) \<le> 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 "... \<le> (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))"
by (simp add: algebra_simps)
finally have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n))
\<le> e * (norm z * (norm z * norm z ^ n))" .
then show "norm (\<Sum>i\<le>Suc n. c i * z^i) \<le> 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: "\<lbrakk>abs b + 1 \<le> norm(y) - a; norm(x) \<le> a\<rbrakk> \<Longrightarrow> b \<le> norm(x + y)"
by (metis abs_add_one_not_less_self add.commute diff_le_eq dual_order.trans le_less_linear
norm_diff_ineq)
lemma polyfun_extremal:
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
assumes "\<exists>k. k \<noteq> 0 \<and> k \<le> n \<and> c k \<noteq> 0"
shows "eventually (\<lambda>z. norm(\<Sum>i\<le>n. c i * z^i) \<ge> 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: "\<And>z. M \<le> norm z \<Longrightarrow>
norm (\<Sum>i\<le>n. c i * z^i) \<le> 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 ((abs B + 1) / (norm (c (Suc n)) / 2)))"], clarsimp)
fix z::'a
assume les: "M \<le> norm z" "1 \<le> norm z" "(\<bar>B\<bar> * 2 + 2) / norm (c (Suc n)) \<le> norm z"
then have "\<bar>B\<bar> * 2 + 2 \<le> norm z * norm (c (Suc n))"
by (metis False pos_divide_le_eq zero_less_norm_iff)
then have "\<bar>B\<bar> * 2 + 2 \<le> norm z ^ (Suc n) * norm (c (Suc n))"
by (metis `1 \<le> norm z` order.trans mult_right_mono norm_ge_zero self_le_power zero_less_Suc)
then show "B \<le> norm ((\<Sum>i\<le>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"])
apply (simp_all add: norm_mult norm_power)
done
qed
qed
qed
lemma polyfun_rootbound:
fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
assumes "\<exists>k. k \<le> n \<and> c k \<noteq> 0"
shows "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<and> card {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<le> n"
using assms
proof (induction n arbitrary: c)
case (Suc n) show ?case
proof (cases "{z. (\<Sum>i\<le>Suc n. c i * z^i) = 0} = {}")
case False
then obtain a where a: "(\<Sum>i\<le>Suc n. c i * a^i) = 0"
by auto
from polyfun_linear_factor_root [OF this]
obtain b where "\<And>z. (\<Sum>i\<le>Suc n. c i * z^i) = (z - a) * (\<Sum>i< Suc n. b i * z^i)"
by auto
then have b: "\<And>z. (\<Sum>i\<le>Suc n. c i * z^i) = (z - a) * (\<Sum>i\<le>n. b i * z^i)"
by (metis lessThan_Suc_atMost)
then have ins_ab: "{z. (\<Sum>i\<le>Suc n. c i * z^i) = 0} = insert a {z. (\<Sum>i\<le>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: "~ (\<exists>k\<le>n. b k \<noteq> 0) \<Longrightarrow> \<exists>k. k \<noteq> 0 \<and> k \<le> Suc n \<and> c k \<noteq> 0"
by (metis Suc.prems le0 minus_zero mult_zero_right)
have "\<exists>k\<le>n. b k \<noteq> 0"
apply (rule ccontr)
using polyfun_extremal [OF extr_prem, of 1]
apply (auto simp: eventually_at_infinity b simp del: setsum_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 \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
assumes "\<exists>k. k \<le> n \<and> c k \<noteq> 0"
shows polyfun_rootbound_finite: "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0}"
and polyfun_rootbound_card: "card {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<le> n"
using polyfun_rootbound [OF assms] by auto
lemma polyfun_finite_roots:
fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
shows "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<longleftrightarrow> (\<exists>k. k \<le> n \<and> c k \<noteq> 0)"
proof (cases " \<exists>k\<le>n. c k \<noteq> 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 \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
shows "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = 0) \<longleftrightarrow> (\<forall>k. k \<le> n \<longrightarrow> c k = 0)"
proof (cases "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = 0)")
case True
then have "~ finite {z. (\<Sum>i\<le>n. c i * z^i) = 0}"
by (simp add: infinite_UNIV_char_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 \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
shows "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = k) \<longleftrightarrow> c 0 = k \<and> (\<forall>k. k \<noteq> 0 \<and> k \<le> n \<longrightarrow> c k = 0)"
proof -
{fix z
have "(\<Sum>i\<le>n. c i * z^i) = (\<Sum>i\<le>n. (if i = 0 then c 0 - k else c i) * z^i) + k"
by (induct n) auto
} then
have "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = k) \<longleftrightarrow> (\<forall>z. (\<Sum>i\<le>n. (if i = 0 then c 0 - k else c i) * z^i) = 0)"
by auto
also have "... \<longleftrightarrow> c 0 = k \<and> (\<forall>k. k \<noteq> 0 \<and> k \<le> n \<longrightarrow> c k = 0)"
by (auto simp: polyfun_eq_0)
finally show ?thesis .
qed
end