Ghosh–Ramachandran's connected-set Fekete area question

S. Ghosh and K. Ramachandran, A note on the Erdős minimal area problem, arXiv:2604.03036v3, immediately after Proposition 3.6:

While the above proposition proves that Fekete polynomials are not area minimizers in general, it would be of interest to know if they are minimzers when the set $K$ is connected. We leave this as an open question.

In Section 1, $P_n(K)$ consists of monic degree-$n$ polynomials whose zeros all lie in the compact set $K\subseteq\mathbb C$. Write $\Lambda_p={z:|p(z)|<1}$ and $A_n(K)=\inf_{p\in P_n(K)}m(\Lambda_p)$, with $m$ planar Lebesgue measure. A Fekete tuple $a=(a_1,\ldots,a_n)$ maximizes $\prod_{i<j}|a_i-a_j|$ on $K^n$, and defines $F_a(z)=\prod_i(z-a_i)$.

The Lean claim says that for every compact connected $K$, every $n\geq1$, every Fekete tuple $a$ in $K$, and every $p\in P_n(K)$, $m(\Lambda_{F_a})\leq m(\Lambda_p)$. Thus result : ¬ claim refutes exact area minimization in the stated connected-set setting.

Theorem 3.1 gives logarithmic-scale asymptotic optimality for $\operatorname{cap}(K)>1$. Exact minimization is a stronger assertion. The declaration D5/S3/Analytic/Lemniscates/FeketeLemniscateAreaRefutation.result exhibits a connected compact interval where a degree-four competitor has strictly smaller unit-lemniscate area than a Fekete polynomial.

The source leaves the connected case as an open question; the paper's Proposition 3.6 uses the disconnected set $2\overline{\mathbb D}\cup{a}$. No later answer was found: arXiv:2606.17097 (sharp order in Erdős's minimum-area problem) does not mention Fekete polynomials or this question; keyword searches of the arXiv listings, the web, conjectures.io and Formal Conjectures return no entry for it. The source note is Library/Geometry/ghosh2026erdos.md.

Take $K=[-20,20]$ embedded in $\mathbb C$, and $n=4$. For every $R>0$, the tuple $(-R,-u,u,R)$ with $u>0$ and $5u^2=R^2$ is Fekete on $[-R,R]$. Its maximal Vandermonde product is

$$ 4Ru(R^2-u^2)^2=\frac{64R^6}{25\sqrt5}. $$

At $R=20$, this gives $u=\sqrt{80}$ and $F(z)=(z^2-400)(z^2-80)$. Use the competitor $G(z)=(z^2-400)(z^2-100)$, whose roots are $\pm20,\pm10$.

| Certificate | Centres | Radius at each centre | | --- | --- | --- | | Disjoint discs inside $\Lambda_F$ | $\pm20$ | $97/1250000$ | | Disjoint discs inside $\Lambda_F$ | $\pm\sqrt{80}$ | $217/1250000$ | | Discs covering $\Lambda_G$ | $\pm20$ | $53/625000$ | | Discs covering $\Lambda_G$ | $\pm10$ | $849/5000000$ |

The inner certificate uses $r_k\prod_{j\ne k}(|a_k-a_j|+r_k)<1$. The outer certificate uses the general three-regime cover lemma with $e=1/10$ and $t=5$. Summing disc areas gives the exact rational bounds

$$ m(\Lambda_F)\geq\pi\frac{7231744}{10^{14}} =\pi,7.231744\cdot10^{-8}

\pi,7.204616\cdot10^{-8} =\pi\frac{7204616}{10^{14}}\geq m(\Lambda_G). $$

The strict area comparison on this single connected compact interval falsifies the universal minimization claim. The competitor is monic, has degree four, and has all roots in $K$. The proof also certifies the Fekete property, the inner-disc disjointness, and the outer cover; these conditions are required for the area comparison to address the source question.

The committed Lean module defines vandermonde, IsFeketeTuple, InP, lemniscateArea and claim. Its public theorems are isFeketeTuple_interval_four, ball_subset_lemniscate, lemniscate_subset_balls and result : ¬ claim. The exact rational disc bounds, compactness, connectedness and admissible competitor establish the contradiction. The accepted axiom closure is ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$; there is no sorry or additional axiom.

The supplied numerical calculation uses mpmath radial integration and gives $m(\Lambda_F)\approx2.30097118\cdot10^{-7}$ and $m(\Lambda_G)\approx2.18166157\cdot10^{-7}$. These are computed values, not the rational certificate used by Lean; no certified numerical error bound is supplied.

What the settlement shows

1. Proved (Lean) — general mechanisms. ball_subset_lemniscate puts a disc inside the unit lemniscate of an arbitrary finite root family under its product bound. lemniscate_subset_balls covers the lemniscate of a nonempty root family under separation and three distance-regime bounds. isFeketeTuple_interval_four proves that $(-R,-R/\sqrt5,R/\sqrt5,R)$ is Fekete for every $R>0$. This theorem exhibits maximizers; it does not classify all maximizers.

2. Computed — numerical areas. mpmath radial integration gives $m(\Lambda_F)\approx2.30097118\cdot10^{-7}$ and $m(\Lambda_G)\approx2.18166157\cdot10^{-7}$. The strict comparison in result relies on the rational disc certificate, independently of these numerical approximations.

3. Open here — small-level asymptotics and scaling. For a polynomial with simple zeros $a_i$, the proposed relation is

   $$ \frac{m{z:|p(z)|<t}}{\pi t^2} \longrightarrow\sum_i|p'(a_i)|^{-2}\qquad(t\downarrow0). $$

   Computed — derivative sums. For $f(z)=(z^2-1)(z^2-1/5)$ and $g(z)=(z^2-1)(z^2-1/4)$, $S(f)=75/16>40/9=S(g)$, where $S(p)=\sum_i|p'(a_i)|^{-2}$. Open here (heuristic) — interval family. With $F_R(z)=R^4f(z/R)$ and $G_R(z)=R^4g(z/R)$, the level is $R^{-4}$ and the area scaling factor is $R^2$. The proposed asymptotic relation would imply that every sufficiently large $R$ gives a degree-four counterexample on $[-R,R]$. Neither the limiting relation nor this uniform eventual statement is proved here.

4. Open (heuristic) — every even degree. A proposed extension to every even $n\geq4$ is based on incompatibility of the unweighted Fekete balance equations with the weighted stationarity equations of $\sum_i|p'(a_i)|^{-2}$. Such incompatibility and a corresponding family of finite certificates remain unproved here.

5. Proved — exact optimality fails for connected sets. The counterexample prevents upgrading the paper's Theorem 3.1 logarithmic asymptotic conclusion to universal exact optimality, even when $K$ is connected. Proved (paper, Theorem 3.1) — the logarithmic conclusion survives. Theorem 3.1 for $\operatorname{cap}(K)>1$ is not contradicted by this finite example. Open (heuristic) — local explanation. The derivative sums suggest a local obstruction at the zeros rather than a topological one; the Lean certificate proves that disconnectedness of $K$ is not necessary, without proving the derivative-weight asymptotic mechanism.

6. Open — neighbouring exact-minimization questions. Are Fekete polynomials exact minimizers for connected $K$ with $\operatorname{cap}(K)\leq1$, or for the closed unit disc? Does the answer change when the level $t$ is required to make ${z:|p(z)|<t}$ connected? The present unit-level interval counterexample settles none of these restrictions.

The forward-citation list of arXiv:2604.03036 was not obtained in full, so an answer in a citing paper outside the searched listings is not excluded. The numerical areas carry no certified error bound. Small-level asymptotics, the eventual large-interval family, the derivative-weight explanation and the extension to every even degree are not proved.