Polynomial convexity

Let $\mathcal{P}$ 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 $p$th 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 $k$th homology group of $X$ with coefficients in an Abelian group $G$ and $\pi _ { k } ( X )$ is the $k$th 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].

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