95 research outputs found
Polynomial approximation, local polynomial convexity, and degenerate CR singularities
We begin with the following question: given a closed disc in the
complex plane and a complex-valued function F in , is the uniform
algebra on generated by z and F equal to ? When F is in
, 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 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.Comment: 17 pages; final version; restated Thm.1.2 using slightly clearer
notation, corrected minor typo
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