Research Problems in Function Theory — Problem 4.22

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.Assume $z_1,\ldots,z_n\in\overline{\mathbb D}$, so $0\in E_f$. Let \[L_f=\inf\{\ell(\gamma):\gamma:[0,1]\to E_f\text{ is a rectifiable continuous path},\ \gamma(0)=0,\ |\gamma(1)|=1\},\] with infimum $+\infty$ if no rectifiable path exists. Define the worst-case quantity \[L(n)=\sup_{z_1,\ldots,z_n\in\overline{\mathbb D}}L_f. \] The source records existence of a path from $0$ to $\mathbb T$ in $E_f$.

Statement recorded in the supplied source.

Estimate the shortest-path length $L_f$ and determine optimal degree-dependent upper bounds, equivalently the size and growth of $L(n)$. In particular, is the worst-case length unbounded as $n\to\infty$, and how slowly can the optimal upper bound grow? [S2]

Short English statement

How long may the shortest path from the origin to the unit circle inside a polynomial’s lemniscate have to be, when all zeros lie in the unit disc?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304022 (AMR-022-4022). 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 4.22, its complete Update 4.22, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.