The polynomially convex embedding dimension of real manifolds of dimension $\leq 11$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913043354484736 |
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| author | Arosio, Leandro Kalm, Håkan Samuelsson Wold, Erlend F. |
| author_facet | Arosio, Leandro Kalm, Håkan Samuelsson Wold, Erlend F. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19765 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The polynomially convex embedding dimension of real manifolds of dimension $\leq 11$ Arosio, Leandro Kalm, Håkan Samuelsson Wold, Erlend F. Complex Variables 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. |
| title | The polynomially convex embedding dimension of real manifolds of dimension $\leq 11$ |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2503.19765 |