Research Problems in Function Theory — Problem 4.9

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\ge1$ and let $f(z)=\prod_{j=1}^n(z-z_j)$ be a monic polynomial of degree $n$, with zeros repeated according to multiplicity. Its filled unit lemniscate is the compact set \[E_f=\{z\in\mathbb C:|f(z)|\le1\}. \] The monic normalization is the one used in the source's Problem 4.7 and its following lemniscate questions.For a compact set $C\subset\mathbb C$, write $\operatorname{diam}C=\sup\{|z-w|:z,w\in C\}$. Connected components are counted as subsets of $E_f$.

Statement recorded in the supplied source.

For each $c>0$, is there a constant $A(c)$ independent of $n$ and $f$ such that the number of components of $E_f$ having diameter greater than $1+c^2$ is at most $A(c)$? The source's Update 4.9 records counterexamples to this uniform-constant conjecture and asks instead whether, for each fixed $c>0$, the number of components of diameter greater than $1+c$ is uniformly $o(n)$, or even $o(n^\varepsilon)$ for every fixed $\varepsilon>0$, as $n\to\infty$. Here uniformly means the supremum over all monic degree-$n$ polynomials. [S2]

Short English statement

Can the number of large components of a monic polynomial’s lemniscate be bounded independently of its degree? The source gives counterexamples and asks for smaller-than-linear, or smaller-than-every-power, growth instead.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304009 (AMR-022-4009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.9, its complete Update 4.9, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.