On the area of polynomial lemniscates

TL;DR

This work resolves Erdős's minimal area problem for lemniscates by establishing near-optimal two-sided bounds for the area of $\Lambda_p$ across levels and constraints. The authors develop a robust framework using potential theory, the Nazarov–Polterovich–Sodin (NPS) results, and a novel zero-pushing lemma to compare disk- and circle-constrained minimizers, achieving upper bounds of order $O(1/\log\log n)$ and matching lower bounds up to logarithmic factors. They extend the analysis to sublevel sets $\{|p(z)|<t\}$, with sharp decay rates for $t>1$ and power-law behavior for $0<t<1$, and they prove that minimal area under unit-capacity constraints $K$ with smooth boundary collapses to zero as $n\to\infty$, with zeros asymptotically equidistributing toward the equilibrium measure. The paper also establishes a quantitative inradius bound $\rho_n \ge c/(n\sqrt{\log n})$, and investigates normality of area minimizers, culminating in a substantial numerical study that suggests extremal zeros concentrate near the unit circle and often align with carefully structured $2n$-th root-of-unity configurations. Overall, the work advances the understanding of lemniscate geometry, connecting area minimization, zero distribution, and potential theory with practical implications for extremal polynomial problems.

Abstract

Erdös posed in 1940 the extremal problem of studying the minimal area of the lemniscate $\{|p(z)|<1\}$ of a monic polynomial $p$ of degree $n$ all of whose zeros are in the closed unit disc. In this article, we prove that there exist positive constants $c,C$ independent of the degree $n$ such that \[ \dfrac{c}{\log n} \leq \min \text{Area}( \{ |p(z)|<1 \} ) \leq \frac{C}{\log \log n},\] improving substantially the previously best known lower bound (due to Pommerenke in 1961) as well as improving the best known upper bound (due to Wagner in 1988). We also study the inradius (radius of the largest inscribed disc); we provide an estimate for the inradius in terms of the area that confirms a 2009 conjecture of Solynin and Williams, and we use this to give a lower bound of order $(n \sqrt{\log n})^{-1}$ on the inradius, addressing a 1958 problem posed by Erdös, Herzog, and Piranian (confirming their conjecture up to the logarithmic factor). In addition to studying the area of $\{|p(z)|<1\}$, we consider other sublevel sets $\{|p(z)|<t\}$, proving both upper and lower bounds of the same order $1/\log \log n$ when $t>1$ and proving power law upper and lower bounds when $0<t<1$. We also consider the minimal area problem under a more general constraint, namely, replacing the unit disc with a compact set $K$ of unit capacity, where we show that the minimal area converges to zero as $n \rightarrow \infty$ (giving an affirmative answer to another question of Erdös, Herzog, Piranian); we also investigate the structure of the area minimizing polynomials, showing that the normalized zero-counting measure converges to the equilibrium measure of $K$ as the degree $n \rightarrow \infty$.

Figures (2)

• Figure 1: Suppose $\arg z$ varies from $0$ to $1$. The lemniscate area is shown for these configurations with $n=3$: Left: $\{1,z,\bar{z}\}$, Middle: $\{1,1,z\}$, Right: $\{1,-1,z\}$. Conclusion: $\{1,1,-1\}$ is a local minimum in that changing any one root increases the lemniscate area. But it is not a global minimum.
• Figure 2: The empirical minimizers for $n=3,4,5,6,7,8$. For reference, we also plot the (possibly rotated) Erdös lemniscate of corresponding degree as well as the unit circle. For $n=3$ the empirical minimizer is the Erdös lemniscate. In the other cases the minimizer consists of a large oval, formed by "merging" some petals in the corresponding Erdös lemniscate, along with small ovals within the remaining petals).

Theorems & Definitions (49)

• Theorem
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 4
• Remark 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 9