Polynomially convex embeddings and CR singularities of real manifolds

It is proved that any smooth manifold $\mathcal{M}$ of dimension $m$ admits a smooth polynomially convex embedding into ${\mathbb{C}}^n$ when $n\ge \lfloor 5m\mathrm{/}4\rfloor$. Further, such embeddings are dense in the space of smooth maps from $\mathcal{M}$ into ${\mathbb{C}}^n$ in the ${\mathcal{C}}^3$-topology. The components of any such embedding give smooth generators of the algebra of complex-valued continuous functions on $\mathcal{M}$. A key ingredient of the proof is a coordinate-free description of certain notions of (non)degeneracy, as defined by Webster and Coffman, for CR-singularities of order one of an embedded real manifold in ${\mathbb{C}}^n$. The main result is obtained by inductively perturbing each stratum of degeneracy to produce a global polynomially convex embedding.