Research Problems in Function Theory — Problem 5.24

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. All Taylor coefficients below are taken at $0$.Let $H^\infty(\mathbb D)$ be the commutative ring of bounded holomorphic functions on $\mathbb D$, with pointwise addition and multiplication. An ideal generated by $f_1,\ldots,f_m$ is \[(f_1,\ldots,f_m)=\left\{\sum_{j=1}^m h_j f_j:h_j\in H^\infty(\mathbb D)\right\}. \] Finite generation is algebraic; no norm closure is applied.

Statement recorded in the supplied source.

For every two finitely generated ideals $I,J\subset H^\infty(\mathbb D)$, is $I\cap J$ finitely generated as an ideal of $H^\infty(\mathbb D)$? [S2]

Short English statement

Is the intersection of two finitely generated ideals of bounded holomorphic disc functions again finitely generated?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305024 (AMR-022-5024). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, Chapter 5, Problem 5.24, its update and the preceding shared notation. https://arxiv.org/abs/1809.07200.