Polynomial convexity
Let denote the set of holomorphic polynomials on \mathbf{C} ^ { n } (cf. also Analytic function). Let K be a compact set in \mathbf{C} ^ { n } and let \| P \| _ { K } = \operatorname { max } _ { z \in K } | P ( z ) | be the sup-norm of P \in \mathcal{P} on K. The set
\begin{equation*} \hat { K } = \{ z \in {\bf C} ^ { n } : | P ( z ) | \leq \| P \| _ { K } , \forall P \in {\cal P} \}, \end{equation*}
is called the polynomially convex hull of K. If \hat { K } = K one says that K is polynomially convex.
An up-to-date (as of 1998) text dealing with polynomial convexity is [a3], while [a13] and [a27] contain some sections on polynomial convexity, background and older results. The paper [a24] is an early study on polynomial convexity.
Polynomial convexity arises naturally in the context of function algebras (cf. also Algebra of functions): Let P ( K ) denote the uniform algebra generated by the holomorphic polynomials on K with the sup-norm. The maximal ideal space M of P ( K ) is the set of homomorphisms mapping P ( K ) onto \mathbf{C}, endowed with the topology inherited from the dual space P ( K ) ^ { * }. It can be identified with via
\begin{equation*} z \in \widehat { K } \leftrightarrow m _ { z }, \end{equation*}
\begin{equation*} P \mapsto P ( z ) , P \in \mathcal{P}. \end{equation*}
Moreover, if A is any finitely generated function algebra on a compact Hausdorff space, then A is isomorphic to P ( K ), where for K one can take the joint spectrum of the generators of A (cf. also Spectrum of an operator).
By the Riesz representation theorem (cf. Riesz theorem) there exists for every z \in \hat { K } at least one representing measure \mu _ { z }, that is, a probability measure \mu _ { z } on K such that
\begin{equation*} P ( z ) = m _ { z } ( P ) = \int _ { K } P ( \zeta ) d \mu _ { z } ( \zeta ) , P \in \mathcal{P}. \end{equation*}
One calls \mu _ { z } a Jensen measure if it has the stronger property
\begin{equation*} \operatorname { log } | P ( z ) | \leq \int _ { K } \operatorname { log } | P ( \zeta ) | d \mu _ { z } ( \zeta ) , P \in \mathcal{P}. \end{equation*}
It can be shown that for each z \in \hat { K } there exists a Jensen measure \mu _ { z }. See e.g. [a27].
For compact sets K in \mathbf{C} one obtains by "filling in the holes" of K, that is, \hat { K } = \mathbf{C} \backslash \Omega _ { \infty }, where \Omega _ { \infty } is the unbounded component of \mathbf{C} \backslash K. In \mathbf{C} ^ { n }, n > 1, there is no such a simple topological description.
Early results on polynomial convexity, cf. [a13], are
Oka's theorem: If K is a polynomially convex set in \mathbf{C} ^ { n } and f is holomorphic on a neighbourhood of K, then f can be written on K as a uniform limit of polynomials. Cf. also Oka theorems.
Browder's theorem: If K is polynomially convex in \mathbf{C} ^ { n }, then H ^ { p } ( K , {\bf C} ) = 0 for p \geq n.
Here, H ^ { p } ( K , \mathbf{C} ) is the pth Čech cohomology group. More recently (1994), the following topological result was obtained, cf. [a9], [a3]:
Forstnerič' theorem: Let K be a polynomially convex set in \mathbf{C} ^ { n }, n \geq 2. Then
\begin{equation*} H _ { k } ( \mathbf{C} ^ { n } \backslash K ; G ) = 0,1 \leq k \leq n - 1, \end{equation*}
and
\begin{equation*} \pi _ { k } ( \mathbf{C} ^ { n } \backslash K ) = 0,1 \leq k \leq n - 1. \end{equation*}
Here, H _ { k } ( X , G ) denotes the kth homology group of X with coefficients in an Abelian group G and \pi _ { k } ( X ) is the kth homotopy group of X.
One method to find is by means of analytic discs. Let \Delta be the unit disc in \mathbf{C} and let T be its boundary. An analytic disc is (the image of) a holomorphic mapping f : \Delta \rightarrow {\bf C} ^ { n } such that f is continuous up to T. Similarly one defines an H ^ { \infty }-disc as a bounded holomorphic mapping f : \Delta \rightarrow {\bf C} ^ { n }. Its components are elements of the usual Hardy space H ^ { \infty } ( \Delta ) (cf. Hardy spaces).
Now, let K be compact in \mathbf{C} ^ { n } and suppose that f ( T ) \subset K for some analytic disc f. Then f ( \Delta ) \subset \hat { K } by the maximum principle applied to P \circ f for polynomials P \in \mathcal{P}. The same goes for H ^ { \infty }-discs whose boundary values are almost everywhere in K. One says that the disc f is glued to K. Next, one says that has analytic structure at p \in \hat{K} if there exists a non-constant analytic disc f such that f ( 0 ) = p and the image of f is contained in
.
It was a major question whether \hat{K} \backslash K always has analytic structure. Moreover, when is obtained by glueing discs to K? One positive result in this direction is due to H. Alexander [a1]; a corollary of his work is as follows: If K is a rectifiable curve in \mathbf{C} ^ { n }, then either \hat { K } = K and P ( K ) = C ( K ), or \hat{K} \backslash K is a pure 1-dimensional analytic subset of \mathbf{C} ^ { n } \backslash K (cf. also Analytic set). If K is a rectifiable arc, K is polynomially convex and P ( K ) = C ( K ).
See [a1] for the complete formulation. Alexander's result is an extension of pioneering work of J. Wermer, cf. [a30], E. Bishop and, later, G. Stolzenberg [a26], who dealt with real-analytic, respectively C ^ { 1 }, curves. Wermer [a29] gave the first example of an arc in \mathbf{C} ^ { 3 } that is not polynomially convex, cf. [a3]. However, Gel'fand's problem (i.e., let \gamma be an arc in \mathbf{C} ^ { n } such that \hat{\gamma} = \gamma; is it true that P ( \gamma ) = C ( \gamma )?) is still open (2000). Under the additional assumption that its projections into the complex coordinate planes have 2-dimensional Hausdorff measure 0, the answer is positive, see [a3].
F.R. Harvey and H.B. Lawson gave a generalization to higher-dimensional K, cf. [a12], which includes the following.
Let p \geq 1. If K is a C ^ { 2 } ( 2 p + 1 )-dimensional submanifold of \mathbf{C} ^ { n } and at each point of K the tangent space to K contains a p-dimensional complex subspace, then K is the boundary of an analytic variety (in the sense of Stokes' theorem).
Another positive result is contained in the work of E. Bedford and W. Klingenberg, cf. [a4]: Suppose \Gamma \subset {\bf C} ^ { 2 } is the graph of a C ^ { 2 }-function \phi over the boundary of a strictly convex domain \Omega \subset \mathbf{C} \times \mathbf{R}. Then \widehat{\Gamma} is the graph of a Lipschitz-continuous extension \Phi of \phi on \Omega. Moreover, \widehat{\Gamma} is foliated with analytic discs (cf. also Foliation).
The work of Bedford and Klingenberg has been generalized in various directions in [a16], [a21] and [a7]. One ingredient of this theorem is work of Bishop [a5], which gives conditions that guarantee locally the existence of analytic discs with boundary in real submanifolds of sufficiently high dimension. See [a11], [a32] and [a15] for results along this line.
A third situation that is fairly well understood is when K \subset \mathbf{C} ^ { n + 1 } is a compact set fibred over T, that is, K is of the form K = \{ ( z , w ) : z \in T , w \in K _ { z } \}, where K _ { z } is a compact set in \mathbf{C} ^ { n } depending on z.
In this case the following is true: Let K \subset \mathbf{C} ^ { 2 } be a compact fibration over the circle T and suppose that for each z the fibre K _ { z } is connected and simply connected. Then \hat{K} \backslash K is the union of graphs \Gamma _ { f }, where f \in H ^ { \infty } ( \Delta ) and the boundary values f ^ { * } ( z ) are in K _ { z } for almost all z \in T.
Of course, it is possible that \hat{K} \backslash K is empty. The present theorem is due to Z. Slodkowski, [a22], earlier results are in [a2] and [a10]. Slodkowski proved a similar theorem in \mathbf{C} ^ { n + 1} under the assumption that the fibres are convex, see [a23].
Despite these positive results, in general \hat{K} \backslash K need not have analytic structure. This has become clear from examples by Stolzenberg [a25] and Wermer [a31]. Presently (2000) it is not known whether has analytic structure everywhere if K is a (real) submanifold of \mathbf{C} ^ { n }, nor is it known under what conditions
is obtained by glueing discs to K.
However, it has been shown that in a weaker sense there is always a kind of analytic structure in polynomial hulls. Let d \theta denote Lebesgue measure on the circle T and let f ^ { * } d \theta denote the push-forward of d \theta under a continuous mapping f : T \rightarrow \mathbf{C} ^ { n }. Let also K be a compact set in \mathbf{C} ^ { n }. The following are equivalent:
1) z \in \hat { K } and \mu _ { z } is a Jensen measure for z supported on K;
2) There exists a sequence of analytic discs f _ { j } : \Delta \rightarrow \mathbf{C} ^ { n } such that f _ { j } ( 0 ) \rightarrow z and f _ { j } ^ { * } d \theta / 2 \pi \rightarrow \mu _ { z } in the weak-* sense (cf. also Weak topology).
This was proved in [a6]; [a8] and [a20] contain more information about additional nice properties that can be required from the sequence of analytic discs. Under suitable regularity conditions on K, it is shown in [a19] that consists of analytic discs f such that f ^ { - 1 } ( K ) \cap T has Lebesgue measure arbitrary close to 2 \pi.
Another problem is to describe P ( K ) assuming that K = \hat { K } and given reasonable additional conditions on K. In particular, when can one conclude that P ( K ) = C ( K )? Recall that a real submanifold M of \mathbf{C} ^ { n } is totally real at p \in M if the tangent space in p does not contain a complex line (cf. also CR-submanifold). The Hörmander–Wermer theorem is as follows, cf. [a14]: Let M be a sufficiently smooth real submanifold of \mathbf{C} ^ { n } and let K _ { 0 } be the subset of M consisting of points that are not totally real. If K \subset M is a compact polynomially convex set that contains an M-neighbourhood of K _ { 0 }, then P ( K ) contains all continuous functions on K that are on K _ { 0 } the uniform limit of functions holomorphic in a neighbourhood of K _ { 0 }.
See [a17] for a variation on this theme. One can deal with some situations where the manifold is replaced by a union of manifolds; e.g., B.M. Weinstock [a28] gives necessary and sufficient conditions for any compact subset of the union of two totally real n-dimensional subspaces of \mathbf{C} ^ { n } to be polynomially convex; then also P ( K ) = C ( K ). See also [a18].
References
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Polynomial convexity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polynomial_convexity&oldid=50772