Polynomial approximation, local polynomial convexity, and degenerate CR singularities

Abstract

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 ∈ C 1 ( D ) , this question is complicated by the presence of points in the surface S : = graph D ¯ ( 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.