  The totally-real embeddability of any $2k$-dimensional compact manifold $M$
into $\mathbb C^n$, $n\geq 3k$, has several consequences: the genericity of
polynomially convex embeddings of $M$ into $\mathbb C^n$, the existence of $n$
smooth generators for the Banach algebra $\mathcal C(M)$, the existence of
nonpolynomially convex embeddings with no analytic disks in their hulls, and
the existence of special plurisubharmonic defining functions. We show that
these results can be recovered even when $n=3k-1$, $k>1$, despite the presence
of complex tangencies, thus lowering the known bound for the optimal $n$ in
these (related but inequivalent) questions.
