The Local Polynomial Hull near a Degenerate Cr Singularity – Bishop Discs Revisited

Let S be a smooth real surface in C and let p ∈ S be a point at which the tangent plane is a complex line. Many problems in function theory depend on knowing whether S is locally polynomially convex at such a p — i.e. at a CR singularity. Even when the order of contact of Tp(S) with S at p equals 2, no clean characterisation exists; difficulties are posed by parabolic points. Hence, we study non-parabolic CR singularities. We show that the presence or absence of Bishop discs around certain non-parabolic CR singularities is completely determined by a Maslovtype index. This result subsumes all known results of this flavour for order-two, non-parabolic CR singularities. Sufficient conditions for Bishop discs have been investigated by Wiegerinck at a CR singularity p ∈ S where the order of contact with Tp(S) degenerates. His investigations depended upon a subharmonicity condition. Yet, Bishop discs exist in many cases where Wiegerinck’s condition fails. We give examples showing that this condition fails and focus on how Bishop discs still arise.