Research Problems in Function Theory — Problem 9.8

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 $n\ge2$ and let $\gamma\subset\mathbb C^n$ be a Jordan arc, the image of a continuous injective map from $[0,1]$. Let $\pi_j:\mathbb C^n\to\mathbb C$ be the $j$th coordinate projection, and assume that each $\pi_j(\gamma)$ has two-dimensional Lebesgue measure zero. Let $C(\gamma)$ have its uniform norm. Write $P(\gamma)$ for the uniform closure of restrictions of complex polynomials in $n$ variables, and $R(\gamma)$ for the uniform closure of restrictions of rational functions with no poles on $\gamma$.

Statement recorded in the supplied source.

Must $P(\gamma)=C(\gamma)$? The source records $R(\gamma)=C(\gamma)$ under the same projection hypotheses. [S2]

Short English statement

Does polynomial approximation of all continuous functions hold on every higher-dimensional Jordan arc whose coordinate projections have area zero?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309008 (AMR-022-9008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 9.8, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.