The polynomially convex embedding dimension of real manifolds of dimension ≤ 11

2025/03/25 by Arosio, Leandro, Kalm, Håkan Samuelsson, Wold, Erlend F.
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.19765

Abstract

We show that any compact smooth real n-dimensional manifold M with n≤ 11 can be smoothly embedded into ℂⁿ⁺¹ as a polynomially convex set. In general, there is no such embedding into ℂⁿ. This solves a problem by Izzo and Stout for n≤ 11. Additionally, we show that the image \widetildeM of M in ℂⁿ⁺¹ is stratified totally real. As a consequence, by a result in [13], each continuous complex-valued functions on \widetildeM is the uniform limit on \widetildeM of holomorphic polynomials in ℂⁿ⁺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.

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