  It is shown that there exists a compact set $X$ in ${\bf C}^N$ ($N\geq 2$)
such that $\widehat X\setminus X$ is nonempty and the uniform algebra $P(X)$
has a dense set of invertible elements, a large Gleason part, and an abundance
of nonzero bounded point derivations. The existence of a Swiss cheese $X$ such
that $R(X)$ has a Gleason part of full planar measure and a nonzero bounded
point derivation at almost every point is established. An analogous result in
${\bf C}^N$ is presented. The analogue for rational hulls of a result of Duval
and Levenberg on polynomial hulls containing no analytic discs is established.
The results presented address questions raised by Dales and Feinstein.

Complex Variables