The name is absent

LOCAL HOLOMORPHIC CONVEXITY OF A TWO-MANIFOLD IN C² 177

very many, and in fact the set of manifolds which do not have this property
is dense, in a special sense.

It will be shown how to construct a function f₀ of the form (1.2) for
which there exists no non constant function φ, holomorphic in a neigh-
borhood of 0 in C² and such that

(3.1)                          z → ≠(z,∕₀(z))

is real for all z near 0. A function ∕₀ with this property can be found with
any prescribed order of contact at 0 to a given infinitely differentiable
representation of the form (1.2). In particular, there is no prospect for
finding a biholomorphic coordinate change to convert an arbitrary repre-
sentation (1.2) into one of the same form where f takes real values, for one
of the coordinate functions of such a transformation would have to take
real values on M.

These examples probably permit no non constant holomorphic functions
for which (3.1) has rank ≤ 1 near 0, but here it is only shown that (3.1)
cannot be real.

An example can be found easily if it is not required to satisfy (1.2):
/(z) = z²z is a function whose graph permits no non Constantholomorphic
function φ which takes real values on it. This is seen by equating the co-
efficients of the general term z^pz^^, in the power series relation

φ(z, z²z) = fl5(z,z²z).

To find this behavior in a function f of the form (1.2) seems to require
more effort. The construction below produces one whose graph M is in fact
a determining set for real-analytic functions, meaning that any real-analytic
function in a connected neighborhood U of zero which is zero on M vanishes
identically on U. Once this is established and φ is holomorphic on U and
real on M, then Im φ vanishes on U, which implies that φ is constant.

The idea of the construction is roughly that M should be a determining
set if it has infinite order of contact with the graph of Q at O, but is not
contained in any of the obvious real-analytic varieties through the graph
of Q. It is convenient to make the construction in several steps.

Lemma 3.1. Suppose {(x_n, JF,,)} is a sequence in R² such that x_ll ≠ O
and y„ ≠ O for any n, x_n→ O and for any integer p ≥ O,

y_n∣χ^pn → θ∙

Ifh is real-analytic in a connected neighborhood of O in R² and h(x_n,y_n) = O
for all n, then h = O.

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