Research Problems in Function Theory — Problem 4.7

Definitions and mathematical statement

Let $n\geq1$ and let $P(z)=z^n+a_1z^{n-1}+\cdots+a_n$ be a monic complex polynomial. Its unit lemniscate is the compact set \[ E_P=\{z\in\mathbb C:|P(z)|\leq1\}. \] For a finite family of closed Euclidean disks $D_j=\{z:|z-c_j|\leq r_j\}$, with arbitrary centers $c_j\in\mathbb C$ and radii $r_j\geq0$, define the cost of the covering to be $\sum_j r_j$. Radii, rather than diameters or disk areas, are being summed; overlaps between disks are allowed.

Does every such $P$, in every degree, admit a disk covering of $E_P$ of cost at most $2$? Explicitly, must there exist an integer $s\geq1$ and choices $c_1,\ldots,c_s,r_1,\ldots,r_s$ such that \[ E_P\subseteq\bigcup_{j=1}^{s}D_j, \qquad \sum_{j=1}^{s}r_j\leq2? \] The number of disks is not prescribed, and the claim must hold without any connectedness assumption on the lemniscate. Monicity fixes the natural scale: for a general leading coefficient $a\ne0$ and a positive level $t$, the corresponding radius scale is $(t/|a|)^{1/n}$. Thus the displayed normalized statement asks for a constant independent of degree, coefficients, and the number or geometry of components. Covering only the polynomial's zeros does not suffice; the entire closed sublevel set must be covered.

Short English statement

Can every monic polynomial's unit lemniscate be covered by disks whose radii sum to at most two?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2304007 (AMR-022-4007), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 4.7. Original source indexed by AMR-022-4007. https://arxiv.org/abs/1809.07200.