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RICE UNIVERSITY STUDIES
The theorems of Sard and Hartogs and Rosenthal show as before that
w→ w is in R(M), and hence so is w→(w- β)(w — β) = ∣∙w — β∣2. From
(2.2) and the fact that λ is multiplicative on R(M),
O = Λ(w - β)λ(w - β) = Λ(∣ w - ∕i∣2) = ʃ I w -β∖2dμ.
Since both μ and the integrand are non negative, this implies that support
μ c M ∩{(z,w)tw = β}. Note that the projection (z,w)→z carries the
latter set onto Lf.
Therefore if p is any polynomial in z,
(2.3) Ip(α)∣ = ∣λ(p)I = I ʃpdμ∖ ≤ sup{∣p(z)∣:zeLf}.
Now as in Proof 1 it may be assumed that O is the only critical point of
Re/ in Δ, and hence that C — Lfl is connected. So if α φ Lft, Runge’s Theorem
yields a polynomial p such that ∣ p(α)∣ > sup {∣ p(z)∣ : z ∈ Lfi}, in contra-
diction to (2.3). Thus α∈L^; that is, /(α) = β, or (a,β)eM.
A. theorem of Wermer [9] is applicable, now that Mr = M, to show
that [z,∕]r = C(Δ). As stated, Wermer’s theorem applies to polynomially
convex sets M and concludes that [z,f] = C(A) lin that case. However,
his techniques will establish the rational version of this result needed here,
and so complete the proof.
It seems likely that under the hypotheses of Theorem 1.3 M is in fact
polynomially convex, and that a method similar to the second proof could
be used to show it. If so, Wermer’s theorem would give a deeper result on
polynomial approximation more closely comparable with Theorem 1.1:
It would yield that [z,f] = C(A) if f = Q + o(∣z∣2) has rank ≤ 1 near O
and Q is a real quadratic form with non zero eigenvalues of opposite sign.*
3. A Counterexample
The techniques employed to prove Theorem 1.3 might be generalized
to obtain the same result in the situation where M admits a non constant
function which is holomorphic in a neighborhood of O in C2 and which
takes real values on M. The idea is that this function would play the same
general role in the proofs as that of the coordinate function w. This is an
attractive possibility, especially if there are very many manifolds M which
admit such a holomorphic function. Unfortunately, there do not seem to be
* (Added in proof.) This result is true as stated (for small Δ). The proof, along
the indicated lines, will appear in [10]. An exposition of it is in [11].