Research Problems in Function Theory — Problem 7.52

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.

For a nonempty compact $K\subset\mathbb C^n$, let \[\widehat K=\{z\in\mathbb C^n:|p(z)|\le\sup_{w\in K}|p(w)|\text{ for every complex polynomial }p\}\] be its polynomial hull. Let $P(K)$ be the uniform closure in $C(K)$ of the restrictions of complex polynomials. Its Šilov boundary $\operatorname{Sh}(P(K))$ is the smallest closed subset of $K$ on which every member of $P(K)$ attains its maximum modulus. Write $\partial\widehat K$ for the Euclidean topological boundary of the hull in $\mathbb C^n$.

Statement recorded in the supplied source.

For each $n>1$, characterise all compact sets $K\subset\mathbb C^n$ for which \[\operatorname{Sh}(P(K))=\partial\widehat K.\] The empty set can be included with both boundaries empty. [S2]

Short English statement

Characterise compact sets in several complex variables whose polynomial algebra has Šilov boundary equal to the entire topological boundary of the polynomial hull.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307052 (AMR-022-7052). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.52 and its complete update. https://arxiv.org/abs/1809.07200.