Roots lie outside the circle of radius $1/(1 + B/|a_n|)$

Proved

MetodosNumericos.root_lower_bound

by Lucas · Sep 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

numerical-analysispolynomials

Let $P(z) = a_0z^n + \\dots + a_n$ have real coefficients with $a_n \\neq 0$ and $n \\ge 1$ , and put $B = \\max\\{|a_0|, \\dots, |a_{n-1}|\\}$ . Then every complex root satisfies $|z| \\ge 1/(1 + B/|a_n|)$ . This is Proposição 4.2.2, obtained in the source from the outer bound applied to the reversed polynomial.

Preamble

import Mathlib
import Definitions.Def_MetodosNumericos_polinomiosDefs

Formal statement

namespace MetodosNumericos

theorem root_lower_bound (a : ℕ → ℝ) (n : ℕ) (hn : 1 ≤ n) (han : a n ≠ 0)
    (B : ℝ) (hB : B = Finset.sup' (Finset.range n) (by simp [Finset.nonempty_range_iff]; omega)
      (fun i => |a i|))
    (z : ℂ) (hz : polyValC a n z = 0) :
    1 / (1 + B / |a n|) ≤ ‖z‖ := by sorry

end MetodosNumericos

Source

S. R. Freitas, Métodos Numéricos (UFMS, 2000), Cap. 4, Proposição 4.2.2, p. 75.

Read-back

What the Lean code literally says, in plain math · self-authored-by-drafting-agent (non-blind)

Disclosure: this read-back is not blind. It was written by the same agent that drafted the Lean statement, at the explicit instruction of the mission's human owner, and not by an independent auditor with fresh context.

The statement fixes $a : \\mathbb{N} \\to \\mathbb{R}$ , a natural number $n$ with $n \\ge 1$ , and a real number $B$ , under the hypotheses $a_n \\neq 0$ and

B = \\max_{0 \\le i \\le n-1} |a_i|,

the maximum being taken over the nonempty finite index set $\\{0, 1, \\dots, n-1\\}$ . For a complex $z$ with

\\sum_{i=0}^{n} a_i\\, z^{\\,n-i} = 0,

the conclusion is

\\frac{1}{1 + B/|a_n|} \\;\\le\\; \\lVert z \\rVert,

where $\\lVert z \\rVert$ is the modulus of $z$ . The inequality is non-strict. Since $|a_n| > 0$ and $B \\ge 0$ , the left-hand side is a positive real number, so the statement in particular forbids $z = 0$ from being a root.

Human review

Endorsed by Shuze Chen · Sep 24, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Sep 24, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Roots lie outside the circle of radius 1/(1+B/∣an∣)1/(1 + B/|a_n|)1/(1+B/∣an​∣)

Roots lie outside the circle of radius $1/(1 + B/|a_n|)$