Research Problems in Function Theory — Problem 8.5

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.

For a connected open $G\subset\mathbb C$, let $H(G)=\mathcal O(G)$ with pointwise addition and multiplication, regarded as a structure in the ring language $\{0,1,+,-,\cdot\}$. Two such structures are elementarily equivalent if they satisfy the same first-order sentences in that language. This differs from $H_{\mathbb C}(G)$ as a complex algebra, whose language additionally names the complex scalar constants. Conformal or anticonformal equivalence means the existence of a holomorphic or antiholomorphic bijection between domains.

Statement recorded in the supplied source.

What geometric or conformal restrictions on $G_1,G_2$ follow from $H(G_1)\equiv H(G_2)$? Characterize elementary equivalence of these rings, in contrast with ring isomorphism, which the source identifies with conformal or anticonformal equivalence. Retain the annulus subcase $G_j=\{z:r_j<|z|<R_j\}$, $0<r_j<R_j<\infty$, singled out in the update. Results for the scalar-named complex algebra do not replace the ring-language question. [S2]

Short English statement

Determine what elementary equivalence of holomorphic-function rings implies about their domains, including the annulus case and the distinction from complex-algebra equivalence.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308005 (AMR-022-8005). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 8.5, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.