Polynomial approximation, local polynomial convexity, and degenerate CR singularities

We begin with the following question: given a closed disc D C and a complex-valued function F ∈ C(D), is the uniform algebra on D generated by z and F equal to C(D)? When F ∈ C1(D), this question is complicated by the presence of points in the surface S := graphD(F ) that have complex tangents. Such points are called CR singularities. Let p ∈ 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. © 2006 Elsevier Inc. All rights reserved.