174
RICE UNIVERSITY STUDIES
Theorem 1.3. IfO is a hyperbolic point for a smooth function f of the
form (1.2) and for which rank f ≤ 1 in a neighborhood of 0, then
[z,∕]λ = C(Δ)
for all sufficiently small discs Δ centered at 0. In particular, {(z,f(z)): z∈∆}
is rationally convex.
A large class of examples verifying this result is obtained from the func-
tions f = g о Q, where Q is a real quadratic form with two non zero eigen-
values of opposite sign, and g: R→C is a smooth function with g(0) = 0
and g'(0) real and non zero.
It should be noted that the rank condition of Theorem 1.3 is not preserved
by Bishop’s coordinate change. For example, if /(z) = z2 then f is an
open map, but Bishop’s normalization converts it to the function /(z)
= z2 + z2, which has only real values. Moreover, an example is con-
structed in Section 3 for which it is unlikely that there is a coordinate change
converting it to one represented by a function satisfying Theorem 1.3.
2. Proof of Theorem 1.3
Two proofs will be offered ; the first because of its economy and the second
because it avoids the use of the deep Theorem 1.1 and also because it shows
somewhat more promise for generalization.
Proof 1. It will be shown first that if Δ is a disc centered at 0 on which
rank f ≤ 1 then [z,∕]r contains Re/, and second that Δ may be shrunk to
attain [z,Re∕] = C(Δ).
Since f has rank ≤ 1 on Δ, Sard’s Theorem [6] shows that /(Δ) has
measure zero in C. The classical approximation theorem of Hartogs and
Rosenthal [2, p. 47] implies that Ref can be uniformly approximated
on Δ by rational functions of f with poles off /(Δ). That is, [z,∕]r contains
Re/.
Now Δ can be chosen small enough so that for each complex number β,
the level set L11 defined by (1.3) does not disconnect C and has no interior
points. Once this is demonstrated, it follows immediately from Theorem l.l
that [z, Re/] = C(Δ), which will complete the proof.
Since Q is the Hessian form at 0 of Re/, 0 is a non degenerate and therefore
isolated critical point of Re/ [5, Cor. 2.3]. Thus Δ may be shrunk so that 0
is the only critical point of Re/ in Δ. For such Δ it is clear that L∣l has no
interior points. Neither does Lp disconnect the plane, for if B is a non
empty bounded connected component of C — Lp, then B ⊂ Δ since the
unbounded component of C — Lp clearly contains C — Δ. Thus / is defined