The polynomially convex embedding dimension of real manifolds of dimension leq 11
Pith reviewed 2026-05-22 22:25 UTC · model grok-4.3
The pith
Any compact smooth real manifold of dimension at most 11 embeds smoothly into C^{n+1} as a polynomially convex set.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Any compact smooth real n-dimensional manifold M with n≤11 can be smoothly embedded into C^{n+1} as a polynomially convex set. In general, there is no such embedding into C^n. Additionally, the image is stratified totally real. As a consequence, each continuous complex-valued function on the image is the uniform limit of holomorphic polynomials in C^{n+1}.
What carries the argument
The jet transversality theorem combined with an improved perturbation result to achieve polynomial convexity.
If this is right
- The polynomially convex embedding dimension is n+1 for manifolds of dimension at most 11.
- Continuous functions on the embedded manifold are approximable by holomorphic polynomials.
- The result applies universally to all compact smooth real manifolds in the given dimension range.
Where Pith is reading between the lines
- The perturbation method might extend to higher dimensions with further refinements.
- Stratified totally real embeddings could be useful for approximation problems on other real submanifolds.
- Explicit constructions for standard manifolds like the sphere could verify the general result.
Load-bearing premise
The jet transversality theorem together with the authors' slight improvement of their earlier perturbation result suffice to produce the required polynomially convex embeddings for every compact smooth real n-manifold when n≤11.
What would settle it
A counterexample consisting of a compact smooth 11-dimensional manifold without a polynomially convex smooth embedding into C^{12} would disprove the main result.
read the original abstract
We show that any compact smooth real $n$-dimensional manifold $M$ with $n\leq 11$ can be smoothly embedded into $\mathbb{C}^{n+1}$ as a polynomially convex set. In general, there is no such embedding into $\mathbb{C}^n$. This solves a problem by Izzo and Stout for $n\leq 11$. Additionally, we show that the image $\widetilde{M}$ of $M$ in $\mathbb{C}^{n+1}$ is stratified totally real. As a consequence, by a result in [13], each continuous complex-valued functions on $\widetilde{M}$ is the uniform limit on $\widetilde{M}$ of holomorphic polynomials in $\mathbb{C}^{n+1}$. Our proof is based on the jet transversality theorem and a slight improvement of a perturbation result by the first and the third author.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that every compact smooth real n-dimensional manifold M with n≤11 admits a smooth embedding into ℂ^{n+1} whose image is polynomially convex. It further shows that this image is stratified totally real, so that by a result in [13] every continuous complex-valued function on the image is a uniform limit of holomorphic polynomials in ℂ^{n+1}. The proof relies on the jet transversality theorem together with a modest strengthening of a perturbation result previously obtained by two of the authors. The result solves the Izzo–Stout problem in dimensions ≤11 and notes that no such embedding into ℂ^n exists in general.
Significance. If the central claim holds, the work resolves an open question on the polynomially convex embedding dimension of real manifolds up to dimension 11 and supplies a useful stratified totally real property that yields a polynomial approximation theorem. The authors are credited for the explicit improvement of their earlier perturbation lemma, which is the ingredient that reaches the cutoff at dimension 11, and for the direct application of jet transversality to produce the required embeddings without additional topological restrictions on M.
minor comments (3)
- [Introduction or the section containing the perturbation lemma] The abstract states that the perturbation result is only 'slightly' improved; the manuscript should contain an explicit comparison (in the section presenting the perturbation lemma) of the new statement with the version in the authors' earlier work, including the precise gain in dimension or regularity that permits n=11.
- [The section applying jet transversality] The jet-transversality argument is invoked to produce the embedding; the manuscript should verify in a dedicated paragraph that the relevant jet space for the totally-real and polynomial-convexity conditions has the expected dimension so that transversality applies without further restrictions when n≤11.
- Reference [13] is used for the approximation consequence; confirm that the citation is complete in the bibliography and that the stratified totally real property established here matches the hypotheses of [13] exactly.
Simulated Author's Rebuttal
We thank the referee for their positive summary of the manuscript, recognition of its significance in resolving the Izzo–Stout problem up to dimension 11, and recommendation of minor revision. No specific major comments were provided in the report.
Circularity Check
Minor self-citation to prior perturbation result, not load-bearing for central existence claim
full rationale
The derivation is an existence proof for polynomially convex embeddings of compact smooth real n-manifolds (n≤11) into C^{n+1}, resting on the external jet transversality theorem together with a new slight improvement (contained in this paper) of an earlier perturbation lemma by two of the authors. No equation or claim reduces by construction to a fitted input, self-definition, or renamed known result. The self-citation is minor and non-load-bearing because the paper supplies an independent strengthening and combines it with an external theorem; the central statement therefore retains independent mathematical content against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Jet transversality theorem
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.