Polynomial approximation, local polynomial convexity, and degenerate CR singularities

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 ∈ 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 co...

Polynomial Approximation, Local Polynomial Convexity, and Degenerate Cr Singularities

We begin with the following question: given a closed disc D b C and a complexvalued 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 contac...

Local Polynomial Convexity

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) ? 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 tange...

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 no...

Linearity Testing Using Local Polynomial Approximation