  We provide some conditions for the graph of a Hoelder-continuous function on
\bar{D}, where \bar{D} is a closed disc in the complex plane, to be
polynomially convex. Almost all sufficient conditions known to date ---
provided the function (say F) is smooth --- arise from versions of the
Weierstrass Approximation Theorem on \bar{D}. These conditions often fail to
yield any conclusion if rank_R(DF) is not maximal on a sufficiently large
subset of \bar{D}. We bypass this difficulty by introducing a technique that
relies on the interplay of certain plurisubharmonic functions. This technique
also allows us to make some observations on the polynomial hull of a graph in
C^2 at an isolated complex tangency.
