Research Problems in Function Theory — Problem 4.5

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $P\in\mathbb C[z]$ have degree $n\ge2$ and zeros $z_1,\ldots,z_n\in\overline{\mathbb D}$, listed with multiplicity. A critical point is a zero of the derivative $P'$.

Statement recorded in the supplied source.

For every zero $z_j$ of $P$, must there be a critical point $w$ with $|w-z_j|\le1$? Equivalently, \[\forall j\in\{1,\ldots,n\}\ \exists w\in\mathbb C:\ P'(w)=0,\quad|w-z_j|\le1. \] The quantifier includes every chosen zero, not only a zero of maximum modulus. [S2]

Short English statement

Does every zero of a polynomial whose zeros lie in the closed unit disc have a critical point within distance one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304005 (AMR-022-4005). 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. Problems 4.5 and 4.30 (the latter explicitly defines the degree-at-least-two class), including complete Update 4.5. https://arxiv.org/pdf/1809.07200.