bibkey: minadiaz2025chebyshev
authors: Erwin Miña-Díaz; Olof Rubin
year: 2025
title: "Chebyshev polynomials on equipotential curves"
doi: null
url: https://arxiv.org/abs/2505.03967v1
claim: "Problem 2.2 asks whether equality of a monic Chebyshev polynomial and a monic Faber polynomial on K persists on every exterior equipotential curve."
strata_touched: []
license: citation-only
triage: anchor

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DOI: null

Source: https://arxiv.org/abs/2505.03967v1

Section 1, equation (1.2), specifies the exterior-map normalization. Section 2, Theorem 2.1, gives convergence to the monic Faber polynomial as the equipotential radius tends to infinity. Problem 2.2 asks:

Suppose the compact set $K$ is such that for some $n\in \mathbb{N}$, $T_{n,K}={\hat{F}}_n$, where ${\hat{F}}_n$ is the $n$th monic Faber polynomial associated to the map $\varphi$ that satisfies (1.2). Is it true then that $T_{n,L_r}={\hat{F}}_n$ for all $r>1$?

The arXiv version is the source locator; no DOI is recorded.

Chebyshev polynomials and enclosing circles

Let $\Omega$ be the unbounded component of $\mathbb{C}\setminus K$. The normalization is $g_{\Omega }(z,\infty )=\log |\varphi (z)|$ and $\varphi (z)=cz+c_0+c_{-1}/z+\cdots$, where $c=\mathrm{cap}(K{)}^{-1}$. When $\Omega$ is simply connected, $\psi ={\varphi }^{-1}$ maps $|w|>1$ onto $\Omega$. Write $L_r=\{z\in \Omega :|\varphi (z)|=r\}$, let $F_n$ be the polynomial part of ${\varphi }^n$ at infinity, and set ${\hat{F}}_n=c^{-n}{F}_n$. The polynomial $T_{n,E}$ is the unique monic degree-$n$ minimizer of the sup norm on $E$.

For degree one, $T_{1,E}(z)=z-h$, where $h$ is the centre of the minimal enclosing circle of $E$.

Repository-derived counterexample. The declaration D5.S3.Analytic.Chebyshev.FaberEquipotentialRefutation.result refutes the propagation assertion using the map

$$\psi (w)=w+\frac{1}{4w}+\frac{1}{100}(w^{-2}-w^{-4})$$

has compact complement contained in the disc of radius $5/4$ and containing $\pm 5/4$. Its monic degree-one Faber polynomial is $z$, which is also Chebyshev on that complement. On the radius-two image circle, $z-3/1600$ has sup norm at most $17/8$, whereas $|\psi (2)|=3403/1600>17/8$. Thus exact equality need not propagate. The geometric subject is enclosing circles and their centres under exterior conformal maps.

Related literature and scope

O. Rubin, Chebyshev polynomials in the complex plane and on the real line, arXiv:2411.14175v2, revised 2026-02-18, lists the same question as Problem 5.6. Hübner–Rubin, arXiv:2405.05067, gives rotation-symmetric examples with $T=\hat{F}=z^{\ell }$ preserved on equipotentials. Miña-Díaz–Rubin–Wennman, arXiv:2509.22588, concerns $n\to \infty$ asymptotics, rather than this fixed-degree implication. These readings do not establish an exhaustive priority claim.

Xiao–Qiu, Acta Mathematica Sinica, Chinese Series 53(2) (2010), 323–328, gives an example where Chebyshev polynomials on a Julia set and its equipotentials differ. ASSUMED-UNVERIFIED: its full text was not available, so whether its Julia set satisfies $T=\hat{F}$ was not determined. In the monic degree-$m$ normalization, the relevant centre condition is $h=-a_{m-1}/m$, where $a_{m-1}$ is the coefficient of $z^{m-1}$.

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