The name is absent

LOCAL HOLOMORPHIC CONVEXITY OF A TWO-MANIFOLD IN Ci 173

has no interior points and does not separate the plane, then every con-
tinuous complex valued function on Δ is a uniform limit of polynomial
combinations of f and the identity function z→z.

This and standard facts about the maximal ideal space of finitely generated
Banach algebras implies that the graph

M — {(z,∕(z)): z∈∆}

is polynomially convex in C². Of course, no mention has been made of
smoothness, but it is not difficult to see that if 0 is a hyperbolic point of a
smooth real function f then f satisfies the hypothesis of Theorem 1.1 for
small discs Δ. This follows from the proof of Theorem 1.3 or from Morse’s
Lemma [5, Lemma 2.2].

Very little is known if f is permitted to take complex values. It is shown
here as a consequence of Theorem 1.3 that if 0 is a hyperbolic point of a
smooth function f of the form (1.2) taking complex values, but in such a
way that rank f ≤ 1 near 0, then the graph of f is locally rationally convex.
The rank condition means that the ordinary Jacobian determinant of f,
as a map of the underlying real space R², vanishes on a neighborhood of 0.
The force of it is that f still has a thin image in C. This fact is important in
the proof ofTheorem 1.3, but there seems to be no reason to believe that
it is essential to the conclusion.

Among the several equivalent definitions of rational convexity [2, p. 69],
the one which is most convenient here is

Definition 1.2. A compact set M in Cⁿ is rationally convex if for each
point p not in M, there exists a rational function r holomorphic in a
neighborhood of M but with a singularity at p.

Any compact set in C is rationally convex, as is any polynomially
convex set in C". The graph of a function f : Δ → C is rationally convex
in C² if every continuous complex-valued function on Δ is a uniform limit
of rational combinations of z and f [2]. That is, when

[ɪ,ʃ]a = C(Δ),

where [z,∕]_r is the uniform closure on Δ of the functions z→r(z,∕(z)),
with r rational and holomorphic in a neighborhood of the graph of f, and
where C(Δ) is the full algebra of continuous functions. This assertion
follows from the same arguments [2, pp. 68-69] that establish polynomial
convexity of the graph when the closure [z,ʃ] of polynomials in z and f is
C(Δ).

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