Research Problems in Function Theory — Problem 8.3

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 $D\subset\mathbb C$ be any bounded domain and put $H^\infty(D)=\{h\in\mathcal O(D):\sup_D|h|<\infty\}$. Define \[N(D)=\{a/b:a,b\in H^\infty(D),\ b(z)\ne0\text{ for every }z\in D\}. \] This is the source's bounded-characteristic ring. To generate its full ring means to have coefficients $u_j\in N(D)$ with $\sum_j u_jf_j=1$. In either ring, the generated ideal is formed from finite sums of generator multiples. Nonvanishing means no zeros anywhere in $D$, without a uniform positive lower bound.

Statement recorded in the supplied source.

For $f_1,\ldots,f_N\in N(D)$ with no common zero, find necessary and sufficient conditions for them to generate all of $N(D)$. Equivalently, for $g_1,\ldots,g_N\in H^\infty(D)$, characterize when their ideal in $H^\infty(D)$, or in $N(D)$, contains a nonvanishing function. Seek in particular a corona-type pointwise lower estimate on $\sum_j|g_j|$ for $D=\mathbb D$. The source update records a solution in the disc, leaving the stated programme on general bounded domains. [S2]

Short English statement

Characterize when holomorphic generators produce the whole bounded-characteristic ring, or a zero-free element in their ideals, for general bounded plane domains.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308003 (AMR-022-8003). 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 8.3, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.