Research Problems in Function Theory — Problem 4.10

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.Let $L_f=\{z\in\mathbb C:|f(z)|=1\}$ and let $\ell(L_f)=\mathcal H^1(L_f)$ be its total Euclidean arc length, with $\mathcal H^1$ denoting one-dimensional Hausdorff measure. All components are counted once, including when the curve has singular meeting points.

Statement recorded in the supplied source.

For every $n\ge1$ and every monic degree-$n$ polynomial $f$, is \[\ell(L_f)\le\ell\left(\{z:|z^n-1|=1\}\right)?\] That is, does $z^n-1$ maximize total unit-lemniscate length in each degree? [S2]

Short English statement

Among all monic polynomials of a fixed degree, does z to that degree minus one have the longest unit lemniscate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304010 (AMR-022-4010). 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 4.10, its complete Update 4.10, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.