Research Problems in Function Theory — Problem 8.17

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 $n\ge1$, let $\Omega=\mathbb B^n=\{z\in\mathbb C^n:\sum_{j=1}^n|z_j|^2<1\}$ or $\Omega=\mathbb D^n=\{z:\max_j|z_j|<1\}$. Write $H^{\infty}(\Omega)$ for the commutative unital ring of bounded holomorphic functions. A finitely generated ideal has the form \[I(f_1,\ldots,f_m)=\left\{\sum_{j=1}^mf_jh_j:h_j\in H^{\infty}(\Omega)\right\}.\]

Statement recorded in the supplied source.

For every such $\Omega$, is the intersection of any two finitely generated ideals of $H^{\infty}(\Omega)$ itself finitely generated? [S2]

Short English statement

Are intersections of finitely generated ideals finitely generated in the bounded-holomorphic-function rings of balls and polydisks?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308017 (AMR-022-8017). 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 8.17, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.