The Eneström–Kakeya theorem

A polynomial ∑_{k=0}^n a_k z^k with real coefficients a_0 ≥ a_1 ≥ ⋯ ≥ a_n > 0 has no zero in the open unit disk. On the unit circle a zero z satisfies z^{n+1} = 1, and z^{k+1} = 1 at every strict descent a_{k+1} < a_k; with strictly decreasing coefficients there is no zero in the closed disk. The proof multiplies by 1 - z, telescopes, and applies the triangle inequality.

Main results

• Dilog.enestrom_kakeya: no zero in ‖z‖ < 1.
• Dilog.enestrom_kakeya_boundary: the zeros on ‖z‖ = 1 are roots of unity of the stated orders.
• Dilog.enestrom_kakeya_strict: strictly decreasing coefficients give no zero in ‖z‖ ≤ 1.
• Dilog.monotoneState_never_orthogonal: ∑ p_k e^{-ikt} ≠ 0 for every real t when p_0 > p_1 > ⋯ > p_n > 0.
• Dilog.enestrom_kakeya_boundary_series, Dilog.strictAnti_series_ne_zero: the same on the circle for summable non-increasing series (no truncation): zeros sit at roots of unity pinned by the descents, and strictly decreasing positive populations never orthogonalize. The proof takes real parts of the telescoped series instead of the equality case of the triangle inequality.

References

• A. Hurwitz, Über einen Satz des Herrn Kakeya, Tôhoku Math. J. 4 (1913) 89–93, §2: the zeros on the unit circle under non-increasing coefficients (roots of unity; coefficients in groups of equal consecutive values), with the converse. enestrom_kakeya_boundary is his argument.
• Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.

theorem Dilog.enestrom_kakeya {n : ℕ} (a : ℕ → ℝ) (hpos : ∀ k ≤ n, 0 < a k) (hmono : ∀ k < n, a (k + 1) ≤ a k) {z : ℂ} (hz : ‖z‖ < 1) : ∑ k ∈ Finset.range (n + 1), ↑(a k) * z ^ k ≠ 0

The Eneström–Kakeya theorem (reciprocal form): a polynomial with positive, non-increasing real coefficients has no zero strictly inside the unit disc.

Boundary refinement and the monotone clock

The base theorem localises zeros to ‖z‖ ≥ 1. On the circle ‖z‖ = 1 the same telescoped identity, read through real parts instead of norms, forces a rigid structure: a boundary zero can only sit at a root of unity. Strict decrease at every step then rules the circle out entirely.

theorem Dilog.enestrom_kakeya_boundary {n : ℕ} (a : ℕ → ℝ) (hpos : ∀ k ≤ n, 0 < a k) (hmono : ∀ k < n, a (k + 1) ≤ a k) {z : ℂ} (hz : ‖z‖ = 1) (hS : ∑ k ∈ Finset.range (n + 1), ↑(a k) * z ^ k = 0) : z ^ (n + 1) = 1 ∧ ∀ k < n, a (k + 1) < a k → z ^ (k + 1) = 1

Boundary Eneström–Kakeya. If the polynomial vanishes at a point of the unit circle, that point is a root of unity: zⁿ⁺¹ = 1, and zᵏ⁺¹ = 1 at every step where the coefficients strictly decrease.

theorem Dilog.enestrom_kakeya_strict {n : ℕ} (a : ℕ → ℝ) (hpos : ∀ k ≤ n, 0 < a k) (hmono : ∀ k < n, a (k + 1) < a k) {z : ℂ} (hz : ‖z‖ ≤ 1) : ∑ k ∈ Finset.range (n + 1), ↑(a k) * z ^ k ≠ 0

Strict Eneström–Kakeya. If the coefficients are strictly decreasing at every step, the polynomial has no zero on the closed unit disc ‖z‖ ≤ 1.

theorem Dilog.monotoneState_never_orthogonal {n : ℕ} (p : ℕ → ℝ) (hpos : ∀ k ≤ n, 0 < p k) (hmono : ∀ k < n, p (k + 1) < p k) (t : ℝ) : ∑ k ∈ Finset.range (n + 1), ↑(p k) * Complex.exp (-↑k * ↑t * Complex.I) ≠ 0

The monotone clock that never ticks. A quantum state with strictly decreasing populations p₀ > p₁ > ⋯ > pₙ > 0 on the equally spaced energy levels 0, 1, …, n never reaches an orthogonal state: its autocorrelation ∑ pₖ e^{-i k t} is nonzero for every time t. The finite, coefficient-side companion of zetaState_never_orthogonal.

theorem Dilog.eq_one_of_re_eq_one {w : ℂ} (hn : ‖w‖ = 1) (hr : w.re = 1) : w = 1

Re w = 1 and ‖w‖ = 1 force w = 1.

theorem Dilog.enestrom_kakeya_boundary_series (p : ℕ → ℝ) (hmono : Antitone p) (hsum : Summable p) {z : ℂ} (hz : ‖z‖ = 1) (hS : HasSum (fun (k : ℕ) => ↑(p k) * z ^ k) 0) (k : ℕ) : p (k + 1) < p k → z ^ (k + 1) = 1

Eneström–Kakeya on the circle, for series. Let p be summable and non-increasing. If ∑ pₖ zᵏ = 0 at a point ‖z‖ = 1, then z^{k+1} = 1 at every descent p_{k+1} < p_k: the zeros on the circle are roots of unity, pinned by where the populations drop (Remark 1 of the paper, for series).

theorem Dilog.strictAnti_series_ne_zero (p : ℕ → ℝ) (hpos : ∀ (k : ℕ), 0 < p k) (hanti : StrictAnti p) (hsum : Summable p) {z : ℂ} (hz : ‖z‖ = 1) : ¬HasSum (fun (k : ℕ) => ↑(p k) * z ^ k) 0

Strictly decreasing populations never orthogonalize (series form): for p > 0 summable and strictly decreasing, ∑ pₖ zᵏ ≠ 0 on the whole unit circle. With p_n = 6/(π² n²) this is Theorem 4 again, read from the coefficients as Remark 1 says.