A compact subset of $\mathbb{C}^n$ is polynomially convex if it is defined by a family of polynomial inequalities. The minimum complex dimension into which all compact real manifolds of a fixed dimension admit smooth polynomially convex embeddings is not known (although there are some obvious bounds).

In this talk, we will discuss some recent improvements on the previously known bounds, especially focusing on the odd-dimensional case, where the embeddings cannot be produced by classical (local) perturbation techniques. This is joint work with R. Shafikov.