Optimal Dimensions for Polynomially Convex Embeddings

Definitions and mathematical statement

For a compact nonempty set $K\subseteq{\mathbb{C}}^n$, its polynomial hull is \[ \widehat K=\left\{z\in{\mathbb{C}}^n: |p(z)|\leq\sup_{w\in K}|p(w)|\ \text{for every }p\in{\mathbb{C}}[z_1,\ldots,z_n]\right\}. \] The set $K$ is polynomially convex if $\widehat K=K$; the empty set is also taken to be polynomially convex. A smooth embedding $f:M\to{\mathbb{C}}^n$, with ${\mathbb{C}}^n$ viewed as a real smooth manifold, is a smooth injective immersion that is a homeomorphism onto its image. It is polynomially convex when $f(M)$ is polynomially convex.

The supplied problem asks for the least ambient complex dimension that works for every compact smooth real manifold of a given dimension. Its boundary convention is unspecified. To display this qualification explicitly, for an integer $m\geq0$, let $\mathcal M_m^{(0)}$ be the class of compact smooth real $m$-manifolds without boundary, and let $\mathcal M_m^{(1)}$ be the class of compact smooth real $m$-manifolds with possibly nonempty boundary. No connectedness or orientability restriction is stated.

For a fixed boundary convention $b\in\{0,1\}$ and an ambient-dimension convention $D\in\{{\mathbb{Z}}_{\geq0},{\mathbb{Z}}_{\geq1}\}$, the corresponding precise optimization problem is to determine \[ n_{b,D}(m)=\min\left\{n\in D:\ \forall M\in\mathcal M_m^{(b)}\ \exists\text{ a smooth embedding }f:M\to{\mathbb{C}}^n\text{ with }\widehat{f(M)}=f(M)\right\}, \] whenever this minimum exists. The embedding may depend on $M$. If $0\in D$, use ${\mathbb{C}}^0=\{0\}$. The source does not select between the boundary conventions, specify whether zero ambient dimension is permitted, or explicitly address the zero-dimensional case; these alternatives are displayed rather than silently resolved.

The supplied registry labels the problem open. Its dated literature assessment reports improved universal upper bounds, including the expression $n\geq\lfloor5m/4\rfloor$, while stating that the least possible dimension is not known for every $m$. The record does not give the full hypotheses or dimension range of that bound. This assessment is supplied curation, not an independently verified theorem or status determination.

Short English statement

Determine, as a function of the real dimension $m$, the smallest complex dimension $n$ such that every compact smooth real $m$-manifold has a smooth embedding into ${\mathbb{C}}^n$ whose image equals its polynomial hull. The source leaves boundary and zero-dimension conventions unresolved. Its open status and literature assessment are supplied dated curation requiring independent review.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30003929$, title Optimal Dimensions for Polynomially Convex Embeddings, Oberwolfach Reports Open Problems collection, exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, pp. 2253--2302, DOI 10.4171/owr/2018/37. https://doi.org/10.4171/owr/2018/37. The supplied background attributes the collection to mathematical workshop reports and their open-problem discussions.
• Supplied literature source: Polynomially convex embeddings and CR singularities of real manifolds. https://arxiv.org/abs/2504.01895.
• Supplied literature source: Polynomially convex embeddings of even-dimensional compact manifolds. https://doi.org/10.2422/2036-2145.201811_008.
• Literature checked at: 2026-08-21. This is the exact date of the supplied curation, not an independent verification. That assessment labels the problem open, reports improved universal upper bounds including $n\geq\lfloor5m/4\rfloor$ for smooth manifolds, and says the optimal dimension remains undetermined for every $m$. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.