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Polynomial approximation, local polynomial convexity,and degenerate CR singularities

Bharali, Gautam (2006) Polynomial approximation, local polynomial convexity,and degenerate CR singularities. In: Journal Of Functional Analysis, 236 (1). 351 -368.

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Abstract

We begin with the following question: given a closed disc (D) over bar subset of C and a complex-valued function F is an element of C((D) over bar), is the uniform algebra on (D) over bar generated by z and F equal to C((D) over bar)? When F F is an element of C-1 (D), this question is complicated by the presence of points in the surface S := graph((D) over bar)(F) that have complex tangents. Such points are called CR singularities. Let p is an element of 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.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Elsavier.
Keywords: CR singularity;Polynomial approximation;Polynomially convex.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 06 Apr 2009 09:53
Last Modified: 19 Sep 2010 04:58
URI: http://eprints.iisc.ernet.in/id/eprint/17775

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