Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.18270

On the area of polynomial lemniscates

Manjunath Krishnapur·Erik Lundberg·Koushik Ramachandran
View original source ↗Read source document ↗Export metadata ↓

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 $\{|p(z)| 1$ 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 $n \rightarrow \infty$.

Explore related subjects

math.CVmath.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Manjunath Krishnapur, Erik Lundberg, Koushik Ramachandran. 2025-03-24. On the area of polynomial lemniscates. https://arxiv.org/abs/2503.18270

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Explicit Estimates for the Bergman Kernel Form

Let $(L,e^{-ϕ})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $ω=i\partial\overline{\partial}ϕ$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\,ω\leqω$ and the shortest nonconstant closed geodesic has length at least $2π$, then \[ K_{mϕ}\geq \frac{2m-1}{4π}\,ω, \] with sharpness holding for $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound $-ω\leq\mathrm{Ric}\,ω\leqω$ and the same closed-geodesic hypothesis, we also prove \[ K_{mϕ}\leq \frac{mω}{2π} \left(1+\frac{3}{2m}\right). \] The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki--Zwonek and isoperimetric inequalities.

math.CV↗
arXiv2026

The Complete Crouzeix Conjecture in Dimension Three and the Clouâtre-Ostermann-Ransford conjecture

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre--Ostermann--Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

math.CV↗
arXiv2026

Solving non-oscillatory solutions of the Hill equation via the Tumura--Clunie method

We consider the Hill equation $f''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}e^{iz})f=0$ ($†$), where $\mathbf{k}\geq 1$ and $\mathbf{l}\geq 0$ are integers and $b_{-\mathbf{l}}$, $\cdots$, $b_{\mathbf{k}}$ are constants such that $b_{\mathbf{k}}\not=0$. We point out that there is a full correspondence between the class of non-oscillatory solutions such that $λ(f)<\infty$ of equation ($†$) and the class of Liouvillian solutions of equation $x^2u''-(\sum_{i=-\mathbf{l}}^{\mathbf{k}}b_{i}x^{i})u=0$ ($‡$). Then this paper has twofold purposes. First, parallel to Kovacic's algorithms to find the Liouvillian solutions of equation ($‡$), we develop the Tumura--Clunie method to find the non-oscillatory solutions of a higher order version of the Hill equation. Second, for the particular Hill equation $f''-(b_{\mathbf{k}}e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$, where $\mathbf{k}>\mathbf{s}\geq 1$ are integers and $b_{\mathbf{k}}b_{\mathbf{s}}\not=0$, we use the Tumura--Clunie method to determine the non-oscillatory solution $f$ with an additional zero property.

math.CV↗