Let M be a smooth real two
dimensional submanifold of C2. Near a point of M where its tangent space is complex
linear it may, after a certain local biholomorphic change of coordinates, be
represented as the graph near 0 in C2 of a smooth function
in which z is the first coordinate of C2 and Q is a real valued quadratic form. This
paper is concerned with the polynomially convex hull of a small compact set K in M
near 0 and associated descriptions of the Banach algebra P(M) of continuous
functions uniformly approximable on K by polynomials in two complex variables. It
treats the very special case where f has rank ≦ 1 near 0 (as an R2-valued map). It is
shown that if Q has nonzero eigenvalues of opposite sign, then all sufficiently small
compact sets K are polynomially convex, and P(K) is the full algebra of
continuous functions. If Q has nonzero eigenvalues of the same sign, the
polynomial hull of K is described in terms of a foliation by certain simple
analytic sets in C2, and P(K) is isomorphic to the algebra of continuous
functions on the hull whose restriction to the interior of each analytic set is
holomorphic.
