Research Problems in Function Theory — Problem 4.30

Definitions and mathematical statement

Let $n\geq2$ and let $p(z)=\prod_{\nu=1}^{n}(z-\zeta_\nu)$ be a monic polynomial whose zeros satisfy $|\zeta_\nu|\leq1$. Repeated zeros are allowed. Write $K(p)$ for the convex hull of these zeros in the real plane $\mathbb C$, and write $C(p)=\{w:p'(w)=0\}$ for the set of critical points.

Consider the assertion \[ \text{for every }\zeta\in K(p),\qquad \min_{w\in C(p)}|\zeta-w|\leq1. \] The point $\zeta$ may lie anywhere in the convex hull, including its interior and boundary; it need not itself be a zero of $p$. The minimum is over the nonempty critical set and multiplicities do not affect this distance.

Does the assertion hold for each of the following two classes, considered separately? First, all such polynomials with $p(0)=0$. Second, all such polynomials of the form $p(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_0$ with every $a_j$ real and $a_j\leq0$ for $0\leq j<n$. The closed-unit-disk condition on every zero remains in force in both classes. A proof for the zeros of $p$ alone would leave the convex-hull assertion unanswered. The task is to prove the displayed covering property in both classes or provide a counterexample in one of them, with the class identified.

Short English statement

Is every point of the convex hull within distance one of a critical point in each of two special polynomial classes?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2304030 (AMR-022-4030), 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 4.30. Original source indexed by AMR-022-4030. https://arxiv.org/abs/1809.07200.
• S3 T. Tao, A digestion of the proof of Sendov's conjecture, 2026; distinguishes the remaining Schmeisser strengthening. https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/.