  We begin with the following question: given a closed disc $\bar{D}$ in the
complex plane and a complex-valued function F in $C(\bar{D})$, is the uniform
algebra on $\bar{D}$ generated by z and F equal to $C(\bar{D})$ ? When F is in
$C^1(\bar{D})$, this question is complicated by the presence of points in the
surface S:=graph(F) that have complex tangents. Such points are called CR
singularities. Let $p\in S$ be a CR singularity at which the order of contact
of the tangent plane with S is greater than 2; i.e. a degenerate CR
singularity. We provide sufficient conditions for S to be locally polynomially
convex at the degenerate singularity p. This is useful because it is essential
to know whether S is locally polynomially convex at a CR singularity in order
to answer the initial question. To this end, we also present a general theorem
on the uniform algebra generated by z and F, which we use in our
investigations. This result may be of independent interest because it is
applicable even to non-smooth, complex-valued F.
