The polynomially convex embedding dimension of real manifolds of dimension $\le 11$

We show that any compact smooth real $n$-dimensional manifold $M$ with $n\le 11$ 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 $n\le 11$. Additionally, we show that the image $\tilde{M}$ 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 $\tilde{M}$ is the uniform limit on $\tilde{M}$ 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.