Research Problems in Function Theory — Problem 7.51

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 $D\subset\mathbb C^n$, $n>1$, be a bounded domain with $C^\infty$ strictly pseudoconvex boundary. Strict pseudoconvexity means a smooth defining function has positive Levi form on the complex tangent directions of the boundary. Let $A^\infty(D)$ be the holomorphic functions on $D$ whose complex derivatives of every order extend continuously to $\overline D$. For a multi-index $\alpha\in\mathbb Z_{\ge0}^n$ write $\partial^\alpha f$ for the corresponding derivative. In the question's convention, a closed $E\subset\partial D$ is a flat non-uniqueness set if some $f\in A^\infty(D)$, $f\not\equiv0$, has zero set exactly $E$ on $\overline D$ and $\partial^\alpha f=0$ on $E$ for every $\alpha$.

Statement recorded in the supplied source.

If $E$ is such a flat non-uniqueness set, is every closed subset $F\subset E$ also such a set? In particular, does each $F$ admit a nonzero function in $A^\infty(D)$ with zero set exactly $F$ and all derivatives vanishing on $F$? [S2] The source uses its explicit exact-zero-set and all-derivatives convention and notes the affirmative result for the one-dimensional unit disc. [S2]

Short English statement

Is the class of exact flat zero sets of smooth holomorphic functions on a strictly pseudoconvex domain closed under taking closed subsets?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307051 (AMR-022-7051). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.51 and its complete update. https://arxiv.org/abs/1809.07200.