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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301001.png" /> denote the set of holomorphic polynomials on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301002.png" /> (cf. also [[Analytic function|Analytic function]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301003.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301004.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301005.png" /> be the sup-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301006.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301007.png" />. The set
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301008.png" /></td> </tr></table>
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is called the polynomially convex hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p1301009.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010010.png" /> one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010011.png" /> is polynomially convex.
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Let $\mathcal{P}$ denote the set of holomorphic polynomials on $\mathbf{C} ^ { n }$ (cf. also [[Analytic function|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
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\begin{equation*} \hat { K } = \{ z \in {\bf C} ^ { n } : | P ( z ) | \leq \| P \| _ { K } , \forall P \in {\cal P} \}, \end{equation*}
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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 [[#References|[a3]]], while [[#References|[a13]]] and [[#References|[a27]]] contain some sections on polynomial convexity, background and older results. The paper [[#References|[a24]]] is an early study on polynomial convexity.
 
An up-to-date (as of 1998) text dealing with polynomial convexity is [[#References|[a3]]], while [[#References|[a13]]] and [[#References|[a27]]] contain some sections on polynomial convexity, background and older results. The paper [[#References|[a24]]] is an early study on polynomial convexity.
  
Polynomial convexity arises naturally in the context of function algebras (cf. also [[Algebra of functions|Algebra of functions]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010012.png" /> denote the [[Uniform algebra|uniform algebra]] generated by the holomorphic polynomials on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010013.png" /> with the sup-norm. The maximal ideal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010015.png" /> is the set of homomorphisms mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010016.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010017.png" />, endowed with the topology inherited from the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010018.png" />. It can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010019.png" /> via
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Polynomial convexity arises naturally in the context of function algebras (cf. also [[Algebra of functions|Algebra of functions]]): Let $P ( K )$ denote the [[Uniform algebra|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010019.png"/> via
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010020.png" /></td> </tr></table>
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\begin{equation*} z \in \widehat { K } \leftrightarrow m _ { z }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010021.png" /></td> </tr></table>
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\begin{equation*} P \mapsto P ( z ) , P \in \mathcal{P}. \end{equation*}
  
Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010022.png" /> is any finitely generated function algebra on a compact Hausdorff space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010023.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010024.png" />, where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010025.png" /> one can take the joint spectrum of the generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010026.png" /> (cf. also [[Spectrum of an operator|Spectrum of an operator]]).
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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|Spectrum of an operator]]).
  
By the Riesz representation theorem (cf. [[Riesz theorem(2)|Riesz theorem]]) there exists for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010027.png" /> at least one representing measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010028.png" />, that is, a [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010030.png" /> such that
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By the Riesz representation theorem (cf. [[Riesz theorem(2)|Riesz theorem]]) there exists for every $z \in \hat { K }$ at least one representing measure $\mu _ { z }$, that is, a [[Probability measure|probability measure]] $\mu _ { z }$ on $K$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010031.png" /></td> </tr></table>
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\begin{equation*} P ( z ) = m _ { z } ( P ) = \int _ { K } P ( \zeta ) d \mu _ { z } ( \zeta ) , P \in \mathcal{P}. \end{equation*}
  
One calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010032.png" /> a Jensen measure if it has the stronger property
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One calls $\mu _ { z }$ a Jensen measure if it has the stronger property
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010033.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010034.png" /> there exists a Jensen measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010035.png" />. See e.g. [[#References|[a27]]].
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It can be shown that for each $z \in \hat { K }$ there exists a Jensen measure $\mu _ { z }$. See e.g. [[#References|[a27]]].
  
For compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010037.png" /> one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010038.png" /> by "filling in the holes" of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010039.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010041.png" /> is the unbounded component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010042.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010044.png" />, there is no such a simple topological description.
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For compact sets $K$ in $\mathbf{C}$ one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010038.png"/> 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 $\mathbf{C} ^ { n }$, $n &gt; 1$, there is no such a simple topological description.
  
 
Early results on polynomial convexity, cf. [[#References|[a13]]], are
 
Early results on polynomial convexity, cf. [[#References|[a13]]], are
  
Oka's theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010045.png" /> is a polynomially convex set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010047.png" /> is holomorphic on a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010049.png" /> can be written on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010050.png" /> as a uniform limit of polynomials. Cf. also [[Oka theorems|Oka theorems]].
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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|Oka theorems]].
  
Browder's theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010051.png" /> is polynomially convex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010053.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010054.png" />.
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Browder's theorem: If $K$ is polynomially convex in $\mathbf{C} ^ { n }$, then $H ^ { p } ( K , {\bf C} ) = 0$ for $p \geq n$.
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010055.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010056.png" />th [[Čech cohomology|Čech cohomology]] group. More recently (1994), the following topological result was obtained, cf. [[#References|[a9]]], [[#References|[a3]]]:
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Here, $H ^ { p } ( K , \mathbf{C} )$ is the $p$th [[Čech cohomology|Čech cohomology]] group. More recently (1994), the following topological result was obtained, cf. [[#References|[a9]]], [[#References|[a3]]]:
  
Forstnerič' theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010057.png" /> be a polynomially convex set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010059.png" />. Then
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Forstnerič' theorem: Let $K$ be a polynomially convex set in $\mathbf{C} ^ { n }$, $n \geq 2$. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010060.png" /></td> </tr></table>
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\begin{equation*} H _ { k } ( \mathbf{C} ^ { n } \backslash K ; G ) = 0,1 \leq k \leq n - 1, \end{equation*}
  
 
and
 
and
  
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\begin{equation*} \pi _ { k } ( \mathbf{C} ^ { n } \backslash K ) = 0,1 \leq k \leq n - 1. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010062.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010063.png" />th [[Homology group|homology group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010064.png" /> with coefficients in an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010066.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010067.png" />th homotopy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010068.png" />.
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Here, $H _ { k } ( X , G )$ denotes the $k$th [[Homology group|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010069.png" /> is by means of analytic discs. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010070.png" /> be the unit disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010071.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010072.png" /> be its boundary. An analytic disc is (the image of) a holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010073.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010074.png" /> is continuous up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010075.png" />. Similarly one defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010077.png" />-disc as a bounded holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010078.png" />. Its components are elements of the usual Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010079.png" /> (cf. [[Hardy spaces|Hardy spaces]]).
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One method to find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010069.png"/> 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|Hardy spaces]]).
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010080.png" /> be compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010081.png" /> and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010082.png" /> for some analytic disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010083.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010084.png" /> by the [[Maximum principle|maximum principle]] applied to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010085.png" /> for polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010086.png" />. The same goes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010087.png" />-discs whose boundary values are almost everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010088.png" />. One says that the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010089.png" /> is glued to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010090.png" />. Next, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010091.png" /> has analytic structure at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010092.png" /> if there exists a non-constant analytic disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010093.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010094.png" /> and the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010095.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010096.png" />.
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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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010091.png"/> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010096.png"/>.
  
It was a major question whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010097.png" /> always has analytic structure. Moreover, when is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010098.png" /> obtained by glueing discs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010099.png" />? One positive result in this direction is due to H. Alexander [[#References|[a1]]]; a corollary of his work is as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100100.png" /> is a rectifiable curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100101.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100103.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100104.png" /> is a pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100105.png" />-dimensional analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100106.png" /> (cf. also [[Analytic set|Analytic set]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100107.png" /> is a rectifiable arc, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100108.png" /> is polynomially convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100109.png" />.
+
It was a major question whether $\hat{K} \backslash K$ always has analytic structure. Moreover, when is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p13010098.png"/> obtained by glueing discs to $K$? One positive result in this direction is due to H. Alexander [[#References|[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|Analytic set]]). If $K$ is a rectifiable arc, $K$ is polynomially convex and $P ( K ) = C ( K )$.
  
See [[#References|[a1]]] for the complete formulation. Alexander's result is an extension of pioneering work of J. Wermer, cf. [[#References|[a30]]], E. Bishop and, later, G. Stolzenberg [[#References|[a26]]], who dealt with real-analytic, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100110.png" />, curves. Wermer [[#References|[a29]]] gave the first example of an arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100111.png" /> that is not polynomially convex, cf. [[#References|[a3]]]. However, Gel'fand's problem (i.e., let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100112.png" /> be an arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100113.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100114.png" />; is it true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100115.png" />?) is still open (2000). Under the additional assumption that its projections into the complex coordinate planes have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100116.png" />-dimensional [[Hausdorff measure|Hausdorff measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100117.png" />, the answer is positive, see [[#References|[a3]]].
+
See [[#References|[a1]]] for the complete formulation. Alexander's result is an extension of pioneering work of J. Wermer, cf. [[#References|[a30]]], E. Bishop and, later, G. Stolzenberg [[#References|[a26]]], who dealt with real-analytic, respectively $C ^ { 1 }$, curves. Wermer [[#References|[a29]]] gave the first example of an arc in $\mathbf{C} ^ { 3 }$ that is not polynomially convex, cf. [[#References|[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|Hausdorff measure]] $0$, the answer is positive, see [[#References|[a3]]].
  
F.R. Harvey and H.B. Lawson gave a generalization to higher-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100118.png" />, cf. [[#References|[a12]]], which includes the following.
+
F.R. Harvey and H.B. Lawson gave a generalization to higher-dimensional $K$, cf. [[#References|[a12]]], which includes the following.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100119.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100120.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100121.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100122.png" />-dimensional submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100123.png" /> and at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100124.png" /> the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100125.png" /> contains a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100126.png" />-dimensional complex subspace, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100127.png" /> is the boundary of an analytic variety (in the sense of Stokes' theorem).
+
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. [[#References|[a4]]]: Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100128.png" /> is the graph of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100129.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100130.png" /> over the boundary of a strictly convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100131.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100132.png" /> is the graph of a Lipschitz-continuous extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100133.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100134.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100135.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100136.png" /> is foliated with analytic discs (cf. also [[Foliation|Foliation]]).
+
Another positive result is contained in the work of E. Bedford and W. Klingenberg, cf. [[#References|[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|Foliation]]).
  
 
The work of Bedford and Klingenberg has been generalized in various directions in [[#References|[a16]]], [[#References|[a21]]] and [[#References|[a7]]]. One ingredient of this theorem is work of Bishop [[#References|[a5]]], which gives conditions that guarantee locally the existence of analytic discs with boundary in real submanifolds of sufficiently high dimension. See [[#References|[a11]]], [[#References|[a32]]] and [[#References|[a15]]] for results along this line.
 
The work of Bedford and Klingenberg has been generalized in various directions in [[#References|[a16]]], [[#References|[a21]]] and [[#References|[a7]]]. One ingredient of this theorem is work of Bishop [[#References|[a5]]], which gives conditions that guarantee locally the existence of analytic discs with boundary in real submanifolds of sufficiently high dimension. See [[#References|[a11]]], [[#References|[a32]]] and [[#References|[a15]]] for results along this line.
  
A third situation that is fairly well understood is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100137.png" /> is a compact set fibred over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100138.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100139.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100140.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100141.png" /> is a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100142.png" /> depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100143.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100144.png" /> be a compact fibration over the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100145.png" /> and suppose that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100146.png" /> the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100147.png" /> is connected and simply connected. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100148.png" /> is the union of graphs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100149.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100150.png" /> and the boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100151.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100152.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100153.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100154.png" /> is empty. The present theorem is due to Z. Slodkowski, [[#References|[a22]]], earlier results are in [[#References|[a2]]] and [[#References|[a10]]]. Slodkowski proved a similar theorem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100155.png" /> under the assumption that the fibres are convex, see [[#References|[a23]]].
+
Of course, it is possible that $\hat{K} \backslash K$ is empty. The present theorem is due to Z. Slodkowski, [[#References|[a22]]], earlier results are in [[#References|[a2]]] and [[#References|[a10]]]. Slodkowski proved a similar theorem in $\mathbf{C} ^ { n  + 1}$ under the assumption that the fibres are convex, see [[#References|[a23]]].
  
Despite these positive results, in general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100156.png" /> need not have analytic structure. This has become clear from examples by Stolzenberg [[#References|[a25]]] and Wermer [[#References|[a31]]]. Presently (2000) it is not known whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100157.png" /> has analytic structure everywhere if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100158.png" /> is a (real) submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100159.png" />, nor is it known under what conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100160.png" /> is obtained by glueing discs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100161.png" />.
+
Despite these positive results, in general $\hat{K} \backslash K$ need not have analytic structure. This has become clear from examples by Stolzenberg [[#References|[a25]]] and Wermer [[#References|[a31]]]. Presently (2000) it is not known whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100157.png"/> has analytic structure everywhere if $K$ is a (real) submanifold of $\mathbf{C} ^ { n }$, nor is it known under what conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100160.png"/> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100162.png" /> denote [[Lebesgue measure|Lebesgue measure]] on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100163.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100164.png" /> denote the push-forward of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100165.png" /> under a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100166.png" />. Let also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100167.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100168.png" />. The following are equivalent:
+
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|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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100170.png" /> is a Jensen measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100171.png" /> supported on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100172.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100173.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100174.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100175.png" /> in the weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100176.png" /> sense (cf. also [[Weak topology|Weak topology]]).
+
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|Weak topology]]).
  
This was proved in [[#References|[a6]]]; [[#References|[a8]]] and [[#References|[a20]]] contain more information about additional nice properties that can be required from the sequence of analytic discs. Under suitable regularity conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100177.png" />, it is shown in [[#References|[a19]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100178.png" /> consists of analytic discs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100179.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100180.png" /> has Lebesgue measure arbitrary close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100181.png" />.
+
This was proved in [[#References|[a6]]]; [[#References|[a8]]] and [[#References|[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 [[#References|[a19]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100178.png"/> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100182.png" /> assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100183.png" /> and given reasonable additional conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100184.png" />. In particular, when can one conclude that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100185.png" />? Recall that a real submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100186.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100187.png" /> is totally real at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100188.png" /> if the tangent space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100189.png" /> does not contain a complex line (cf. also [[CR-submanifold|CR-submanifold]]). The Hörmander–Wermer theorem is as follows, cf. [[#References|[a14]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100190.png" /> be a sufficiently smooth real submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100191.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100192.png" /> be the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100193.png" /> consisting of points that are not totally real. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100194.png" /> is a compact polynomially convex set that contains an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100195.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100196.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100197.png" /> contains all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100198.png" /> that are on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100199.png" /> the uniform limit of functions holomorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100200.png" />.
+
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|CR-submanifold]]). The Hörmander–Wermer theorem is as follows, cf. [[#References|[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 [[#References|[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 [[#References|[a28]]] gives necessary and sufficient conditions for any compact subset of the union of two totally real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100201.png" />-dimensional subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100202.png" /> to be polynomially convex; then also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100203.png" />. See also [[#References|[a18]]].
+
See [[#References|[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 [[#References|[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 [[#References|[a18]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Alexander, "Polynomial approximation and hulls in sets of finite linear measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100204.png" />" ''Amer J. Math.'' , '''62''' (1971) pp. 65–74 {{MR|0284617}} {{ZBL|0221.32011}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Alexander, J. Wermer, "Polynomial hulls with convex fibres" ''Math. Ann.'' , '''281''' (1988) pp. 13–22</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Alexander, J. Wermer, "Several complex variables and Banach algebras" , Springer (1998) {{MR|1482798}} {{ZBL|0894.46037}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Bedford, W. Klingenberg Jr., "On the envelope of holomorphy of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100205.png" />-sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100206.png" />" ''J. Amer. Math. Soc.'' , '''4''' (1991) pp. 623–646 {{MR|1094437}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E. Bishop, "Differentiable manifolds in Euclidean space" ''Duke Math. J.'' , '''32''' (1965) pp. 1–21 {{MR|200476}} {{ZBL|0154.08501}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S. Bu, W. Schachermayer, "Approximation of Jensen measures by image measures under holomorphic functions and applications" ''Trans. Amer. Math. Soc.'' , '''331''' (1992) pp. 585–608 {{MR|1035999}} {{ZBL|0758.46014}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> E.M. Chirka, N.V. Shcherbina, "Pseudoconvexity of rigid domains and foliations of hulls of graphs" ''Ann. Scuola Norm. Sup. Pisa'' , '''22''' (1995) pp. 707–735 {{MR|1375316}} {{ZBL|0868.32020}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Duval, N. Sibony, "Polynomial convexity, rational convexity and currents" ''Duke Math. J.'' , '''79''' (1995) pp. 487–513 {{MR|1344768}} {{ZBL|0838.32006}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> F. Forstnerič, "Complements of Runge domains and holomorphic hulls" ''Michigan Math. J.'' , '''41''' (1994) pp. 297–308 {{MR|1278436}} {{ZBL|0811.32007}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> F. Forstnerič, "Polynomial hulls of sets fibered over the circle" ''Indiana Univ. Math. J.'' , '''37''' (1988) pp. 869–889 {{MR|0982834}} {{ZBL|0647.32017}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> F. Forstnerič, E.L. Stout, "A new class of polynomially convex sets" ''Ark. Mat.'' , '''29''' (1991) pp. 51–62 {{MR|1115074}} {{ZBL|0734.32006}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> F.R. Harvey, H.B. Lawson Jr., "On boundaries of complex analytic varieties I" ''Ann. of Math.'' , '''102''' (1975) pp. 223–290 {{MR|0425173}} {{ZBL|0317.32017}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> L. Hörmander, J. Wermer, "Uniform approximation on compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100207.png" />" ''Math. Scand.'' , '''23''' (1968) pp. 5–21 {{MR|0254275}} {{ZBL|0181.36201}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> B. Jöricke, "Local polynomial hulls of discs near isolated parabolic points" ''Indiana Univ. Math. J.'' , '''46''' : 3 (1997) pp. 789–826 {{MR|1488338}} {{ZBL|0901.32010}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> N.G. Kruzhilin, "Two-dimensional spheres in the boundaries of strictly pseudoconvex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100208.png" />" ''Math. USSR Izv.'' , '''39''' (1992) pp. 1151–1187 (In Russian) {{MR|1152210}} {{ZBL|0778.32003}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> A.G. O'Farrell, K.J. Preskenis, D. Walsh, "Holomorphic approximation in Lipschitz norms" , ''Proc. Conf. Banach Algebras and Several Complex Variables (New Haven, Conn., 1983)'' , ''Contemp. Math.'' , '''32''' (1983) pp. 187–194 {{MR|0769507}} {{ZBL|0553.32015}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> P.J. de Paepe, "Approximation on a disk I" ''Math. Z.'' , '''212''' (1993) pp. 145–152 {{MR|}} {{ZBL|0789.30027}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> E.A. Poletsky, "Holomorphic currents" ''Indiana Univ. Math. J.'' , '''42''' (1993) pp. 85–144 {{MR|1218708}} {{ZBL|0811.32010}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> E.A. Poletsky, "Analytic geometry on compacta in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100209.png" />" ''Math. Z.'' , '''222''' (1996) pp. 407–424 {{MR|1400200}} {{ZBL|0849.32009}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> N. Shcherbina, "On the polynomial hull of a graph" ''Indiana Univ. Math. J.'' , '''42''' (1993) pp. 477–503 {{MR|1237056}} {{ZBL|0798.32026}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> Z. Slodkowski, "Polynomial hulls with convex convex sections and interpolating spaces" ''Proc. Amer. Math. Soc.'' , '''96''' (1986) pp. 255–260 {{MR|818455}} {{ZBL|0588.32017}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> Z. Slodkowski, "Polynomial hulls in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100210.png" /> and quasi circles" ''Ann. Scuola Norm. Sup. Pisa'' , '''16''' (1989) pp. 367–391 {{MR|1050332}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> G. Stolzenberg, "Polynomially and rationally convex sets" ''Acta Math.'' , '''109''' (1963) pp. 259–289 {{MR|0146407}} {{ZBL|0122.08404}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> G. Stolzenberg, "A hull with no analytic structure" ''J. Math. Mech.'' , '''12''' (1963) pp. 103–112 {{MR|0143061}} {{ZBL|0113.29101}} </TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> G. Stolzenberg, "Uniform approximation on smooth curves" ''Acta Math.'' , '''115''' (1966) pp. 185–198 {{MR|0192080}} {{ZBL|0143.30005}} </TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> E.L. Stout, "The theory of uniform algebras" , Bogden and Quigley (1971) {{MR|0423083}} {{ZBL|0286.46049}} </TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> B.M. Weinstock, "On the polynomial convexity of the union of two maximal totally real subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100211.png" />" ''Math. Ann.'' , '''282''' (1988) pp. 131–138 {{MR|0960837}} {{ZBL|0628.32015}} </TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> J. Wermer, "Polynomial approximation on an arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100212.png" />" ''Ann. of Math.'' , '''62''' (1955) pp. 269–270 {{MR|0072260}} {{ZBL|0067.05001}} </TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> J. Wermer, "The hull of a curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100213.png" />" ''Ann. of Math.'' , '''68''' (1958) pp. 550–561 {{MR|0100102}} {{ZBL|0084.33402}} </TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> J. Wermer, "On an example of Stolzenberg" , ''Symp. Several Complex Variables, Park City, Utah'' , ''Lecture Notes in Mathematics'' , '''184''' , Springer (1970) {{MR|0298428}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> J. Wiegerinck, "Local polynomially convex hulls at degenerated CR singularities of surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130100/p130100214.png" />" ''Indiana Univ. Math. J.'' , '''44''' (1995) pp. 897–915 {{MR|1375355}} {{ZBL|0847.32013}} </TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> H. Alexander, "Polynomial approximation and hulls in sets of finite linear measure in $\mathbf{C} ^ { n }$" ''Amer J. Math.'' , '''62''' (1971) pp. 65–74 {{MR|0284617}} {{ZBL|0221.32011}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> H. Alexander, J. Wermer, "Polynomial hulls with convex fibres" ''Math. Ann.'' , '''281''' (1988) pp. 13–22</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> H. Alexander, J. Wermer, "Several complex variables and Banach algebras" , Springer (1998) {{MR|1482798}} {{ZBL|0894.46037}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> E. Bedford, W. Klingenberg Jr., "On the envelope of holomorphy of a $2$-sphere in $\mathbf{C} ^ { 2 }$" ''J. Amer. Math. Soc.'' , '''4''' (1991) pp. 623–646 {{MR|1094437}} {{ZBL|}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> E. Bishop, "Differentiable manifolds in Euclidean space" ''Duke Math. J.'' , '''32''' (1965) pp. 1–21 {{MR|200476}} {{ZBL|0154.08501}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> S. Bu, W. Schachermayer, "Approximation of Jensen measures by image measures under holomorphic functions and applications" ''Trans. Amer. Math. Soc.'' , '''331''' (1992) pp. 585–608 {{MR|1035999}} {{ZBL|0758.46014}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> E.M. Chirka, N.V. Shcherbina, "Pseudoconvexity of rigid domains and foliations of hulls of graphs" ''Ann. Scuola Norm. Sup. Pisa'' , '''22''' (1995) pp. 707–735 {{MR|1375316}} {{ZBL|0868.32020}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> J. Duval, N. Sibony, "Polynomial convexity, rational convexity and currents" ''Duke Math. J.'' , '''79''' (1995) pp. 487–513 {{MR|1344768}} {{ZBL|0838.32006}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> F. Forstnerič, "Complements of Runge domains and holomorphic hulls" ''Michigan Math. J.'' , '''41''' (1994) pp. 297–308 {{MR|1278436}} {{ZBL|0811.32007}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> F. Forstnerič, "Polynomial hulls of sets fibered over the circle" ''Indiana Univ. Math. J.'' , '''37''' (1988) pp. 869–889 {{MR|0982834}} {{ZBL|0647.32017}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> F. Forstnerič, E.L. Stout, "A new class of polynomially convex sets" ''Ark. Mat.'' , '''29''' (1991) pp. 51–62 {{MR|1115074}} {{ZBL|0734.32006}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> F.R. Harvey, H.B. Lawson Jr., "On boundaries of complex analytic varieties I" ''Ann. of Math.'' , '''102''' (1975) pp. 223–290 {{MR|0425173}} {{ZBL|0317.32017}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.32001}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> L. Hörmander, J. Wermer, "Uniform approximation on compact sets in $\mathbf{C} ^ { n }$" ''Math. Scand.'' , '''23''' (1968) pp. 5–21 {{MR|0254275}} {{ZBL|0181.36201}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> B. Jöricke, "Local polynomial hulls of discs near isolated parabolic points" ''Indiana Univ. Math. J.'' , '''46''' : 3 (1997) pp. 789–826 {{MR|1488338}} {{ZBL|0901.32010}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> N.G. Kruzhilin, "Two-dimensional spheres in the boundaries of strictly pseudoconvex domains in $\mathbf{C} ^ { 2 }$" ''Math. USSR Izv.'' , '''39''' (1992) pp. 1151–1187 (In Russian) {{MR|1152210}} {{ZBL|0778.32003}} </td></tr><tr><td valign="top">[a17]</td> <td valign="top"> A.G. O'Farrell, K.J. Preskenis, D. Walsh, "Holomorphic approximation in Lipschitz norms" , ''Proc. Conf. Banach Algebras and Several Complex Variables (New Haven, Conn., 1983)'' , ''Contemp. Math.'' , '''32''' (1983) pp. 187–194 {{MR|0769507}} {{ZBL|0553.32015}} </td></tr><tr><td valign="top">[a18]</td> <td valign="top"> P.J. de Paepe, "Approximation on a disk I" ''Math. Z.'' , '''212''' (1993) pp. 145–152 {{MR|}} {{ZBL|0789.30027}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> E.A. Poletsky, "Holomorphic currents" ''Indiana Univ. Math. J.'' , '''42''' (1993) pp. 85–144 {{MR|1218708}} {{ZBL|0811.32010}} </td></tr><tr><td valign="top">[a20]</td> <td valign="top"> E.A. Poletsky, "Analytic geometry on compacta in $\mathbf{C} ^ { n }$" ''Math. Z.'' , '''222''' (1996) pp. 407–424 {{MR|1400200}} {{ZBL|0849.32009}} </td></tr><tr><td valign="top">[a21]</td> <td valign="top"> N. Shcherbina, "On the polynomial hull of a graph" ''Indiana Univ. Math. J.'' , '''42''' (1993) pp. 477–503 {{MR|1237056}} {{ZBL|0798.32026}} </td></tr><tr><td valign="top">[a22]</td> <td valign="top"> Z. Slodkowski, "Polynomial hulls with convex convex sections and interpolating spaces" ''Proc. Amer. Math. Soc.'' , '''96''' (1986) pp. 255–260 {{MR|818455}} {{ZBL|0588.32017}} </td></tr><tr><td valign="top">[a23]</td> <td valign="top"> Z. Slodkowski, "Polynomial hulls in $\mathbf{C} ^ { 2 }$ and quasi circles" ''Ann. Scuola Norm. Sup. Pisa'' , '''16''' (1989) pp. 367–391 {{MR|1050332}} {{ZBL|}} </td></tr><tr><td valign="top">[a24]</td> <td valign="top"> G. Stolzenberg, "Polynomially and rationally convex sets" ''Acta Math.'' , '''109''' (1963) pp. 259–289 {{MR|0146407}} {{ZBL|0122.08404}} </td></tr><tr><td valign="top">[a25]</td> <td valign="top"> G. Stolzenberg, "A hull with no analytic structure" ''J. Math. Mech.'' , '''12''' (1963) pp. 103–112 {{MR|0143061}} {{ZBL|0113.29101}} </td></tr><tr><td valign="top">[a26]</td> <td valign="top"> G. Stolzenberg, "Uniform approximation on smooth curves" ''Acta Math.'' , '''115''' (1966) pp. 185–198 {{MR|0192080}} {{ZBL|0143.30005}} </td></tr><tr><td valign="top">[a27]</td> <td valign="top"> E.L. Stout, "The theory of uniform algebras" , Bogden and Quigley (1971) {{MR|0423083}} {{ZBL|0286.46049}} </td></tr><tr><td valign="top">[a28]</td> <td valign="top"> B.M. Weinstock, "On the polynomial convexity of the union of two maximal totally real subspaces of $\mathbf{C} ^ { n }$" ''Math. Ann.'' , '''282''' (1988) pp. 131–138 {{MR|0960837}} {{ZBL|0628.32015}} </td></tr><tr><td valign="top">[a29]</td> <td valign="top"> J. Wermer, "Polynomial approximation on an arc in $\mathbf{C} ^ { 3 }$" ''Ann. of Math.'' , '''62''' (1955) pp. 269–270 {{MR|0072260}} {{ZBL|0067.05001}} </td></tr><tr><td valign="top">[a30]</td> <td valign="top"> J. Wermer, "The hull of a curve in $\mathbf{C} ^ { n }$" ''Ann. of Math.'' , '''68''' (1958) pp. 550–561 {{MR|0100102}} {{ZBL|0084.33402}} </td></tr><tr><td valign="top">[a31]</td> <td valign="top"> J. Wermer, "On an example of Stolzenberg" , ''Symp. Several Complex Variables, Park City, Utah'' , ''Lecture Notes in Mathematics'' , '''184''' , Springer (1970) {{MR|0298428}} {{ZBL|}} </td></tr><tr><td valign="top">[a32]</td> <td valign="top"> J. Wiegerinck, "Local polynomially convex hulls at degenerated CR singularities of surfaces in $\mathbf{C} ^ { 2 }$" ''Indiana Univ. Math. J.'' , '''44''' (1995) pp. 897–915 {{MR|1375355}} {{ZBL|0847.32013}} </td></tr></table>

Revision as of 15:31, 1 July 2020

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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How to Cite This Entry:
Polynomial convexity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polynomial_convexity&oldid=49938
This article was adapted from an original article by Jan Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article