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Miña-Díaz–Rubin Equipotential Refutation

Abstract

A degree-one monic Faber polynomial is Chebyshev on a compact set but fails on an exterior equipotential.

Definition 1.1 (Monic sup-norm minimization).

Formalization. D5/S3/Analytic/Chebyshev/FaberEquipotentialRefutation.IsChebyshev (✓ std3).

Citation. Erwin Miña-Díaz; Olof Rubin (2025). Chebyshev polynomials on equipotential curves. URL: https://arxiv.org/abs/2505.03967v1.

Commentary.

A monic polynomial p of degree n is Chebyshev on E if every uniform real bound for any monic degree-n competitor q also bounds p on E. On a compact nonempty set this is sup-norm minimization. The definition uses pointwise bounds rather than a supremum operator.

Definition 1.2 (The monic Faber normalization).

Formalization. D5/S3/Analytic/Chebyshev/FaberEquipotentialRefutation.IsMonicFaber (✓ std3).

Citation. Erwin Miña-Díaz; Olof Rubin (2025). Chebyshev polynomials on equipotential curves. URL: https://arxiv.org/abs/2505.03967v1.

Commentary.

Write κ=cap(K)=1/c for the leading coefficient of ψ=φ⁻¹. A monic degree-n polynomial F is the monic Faber polynomial when F(ψ(w))−(κw)ⁿ tends to zero at infinity. Here infinity is the cobounded filter on ℂ. In the paper F_n is the polynomial part of φⁿ and F̂_n=c⁻ⁿF_n. The normalization retains the constant term as well as the leading coefficient.

Definition 1.3 (Equality on K and on its equipotentials).

Formalization. D5/S3/Analytic/Chebyshev/FaberEquipotentialRefutation.claim (✓ std3).

Citation. Erwin Miña-Díaz; Olof Rubin (2025). Chebyshev polynomials on equipotential curves. URL: https://arxiv.org/abs/2505.03967v1.

Commentary.

Section 2, Problem 2.2, states verbatim: “Suppose the compact set K is such that for some n∈ℕ, T_{n,K}=F̂_n, where F̂_n is the nth monic Faber polynomial associated to the map φ that satisfies (1.2). Is it true then that T_{n,L_r}=F̂_n for all r>1?” The unbounded component Ω of ℂ∖K has g_Ω(z,∞)=log|φ(z)|, with φ(z)=cz+c₀+c₋₁/z+… and c=cap(K)⁻¹. When Ω is simply connected, ψ=φ⁻¹ and L_r=ψ({w:|w|=r}). In the displayed statement U is {w:1<‖w‖}, K is the complement of ψ(U), and L_r is the image of the radius-r sphere. The bounded-correction hypothesis is written for ‖w‖>2; n≥1 expresses positive degree. Differentiability, injectivity, positive κ and compact complement specify the exterior-map setting.

Theorem 1.4 (Equality need not persist).

Proof. Machine-checked in Lean as D5/S3/Analytic/Chebyshev/FaberEquipotentialRefutation.result (✓ std3). ∎

Resolves. Problems/minadiaz-rubin-2025-chebyshev-faber-equipotential (refuted) by D5/S3/Analytic/Chebyshev/FaberEquipotentialRefutation.result.

Source. Repository-derived.

Acknowledgement. Erwin Miña-Díaz; Olof Rubin (2025). Chebyshev polynomials on equipotential curves. URL: https://arxiv.org/abs/2505.03967v1.

Commentary.

Take ψ(w)=w+1/(4w)+(1/100)(w⁻²−w⁻⁴), κ=1, n=1 and F=X. For |u|,|v|≥1 the perturbation difference is at most (31/100)|u−v|. The power-difference bound D5.S3.AnalyticClosure.ComplexPowerDifference.norm_pow_sub_pow_le at H=1 supplies this estimate. The perturbation estimate gives injectivity on the closed exterior, an upper Lipschitz bound 131/100 and the lower bound |u−v|≤2|ψ(u)−ψ(v)|. The open mapping theorem and closedness of the closed exterior image show that K is compact and contained in the disc of radius 5/4. The boundary values ψ(±1)=±5/4 lie in K, forcing every monic linear competitor to have norm at least 5/4. Thus X is Chebyshev on K; ψ(w)−w→0 makes it the monic Faber polynomial. For r≥1 and |w|=r, the circle estimate is |ψ(w)−(1/100)(r⁻²−r⁻⁴)|≤r+1/(4r). Its proof expands the norm square on a normalized circle and bounds the cross and residual terms. At r=2, X−C(3/1600) has norm at most 17/8=3400/1600 on L₂, whereas ψ(2)=3403/1600 lies on L₂. Therefore X is not Chebyshev there. The supporting lemmas are private: perturbation_lipschitz and injective_closed supply injectivity, compact_complement supplies the compact disc containment, circle_algebra_bound and circle_bound supply the norm estimate, and chebyshev_of_disc and faber establish the initial equality. The conclusion uses one radius and does not assert a formula for the minimizer on every outer curve. Theorem 2.1’s convergence as r tends to infinity is literature-attested and not formalized here; it is consistent with the centre expression tending to zero.

References