Polynomial convexity

Polynomial convexity is a mathematical property of sets im , which is considered in the function theory of several variables. It plays a role in the approximation of holomorphic functions by polynomials . ${\displaystyle \mathbb {C} ^{n}}$

definition

For a compact subset is called ${\displaystyle K\subset \mathbb {C} ^{n}}$

${\displaystyle {\hat {K}}\,:=\,\{(z_{1},\ldots ,z_{n})\in \mathbb {C} ^{n};\,|p(z_{1},\ldots ,z_{n})|\leq \|p\|_{K}\,{\mbox{ für alle Polynome }}p\}}$

the polynomial convex hull of . It is the supremum on . ${\displaystyle K}$ ${\displaystyle \|\cdot \|_{K}}$ ${\displaystyle K}$

A subset is called a polynomial convex if also holds for every compact subset . ${\displaystyle S\subset \mathbb {C} ^{n}}$ ${\displaystyle K\subset S}$ ${\displaystyle {\hat {K}}\subset S}$

Remarks

Are and compact subsets of , then obviously ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle \mathbb {C} ^{n}}$

• ${\displaystyle K\subset {\hat {K}}}$
• From follows ${\displaystyle K\subset L}$ ${\displaystyle {\hat {K}}\subset {\hat {L}}}$
• ${\displaystyle {\hat {\hat {K}}}={\hat {K}}}$.

This justifies the term envelope in analogy to the convex envelope. This analogy can be carried further: Note that the closed convex hull of a compact subset is equal to the set of all vectors , so that for all linear functionals . In the above definition, the linear functionals are replaced by polynomials. This analogy motivates the terms polynomial convex hull and polynomial convex . ${\displaystyle C\subset \mathbb {R} ^{n}}$ ${\displaystyle x}$ ${\displaystyle |f(x)|\leq \|f\|_{C}}$ ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$

Furthermore, this consideration shows that a compact set is polynomial convex if and only if . In particular, a compact set is polynomial convex if and only if there is a polynomial for each with ${\displaystyle K\subset \mathbb {C} ^{n}}$ ${\displaystyle K={\hat {K}}}$ ${\displaystyle K\subset \mathbb {C} ^{n}}$ ${\displaystyle (z_{1},\ldots ,z_{n})\in \mathbb {C} ^{n}\setminus K}$ ${\displaystyle p}$

• ${\displaystyle p(z_{1},\ldots ,z_{n})=1}$
• ${\displaystyle |p(w_{1},\ldots ,w_{n})|<1}$for everyone . ${\displaystyle (w_{1},\ldots ,w_{n})\in K}$

Examples

• If polynomial convex, then is connected . In this case , the reverse applies, in this case the reverse is false. ${\displaystyle K\subset \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}\setminus K}$ ${\displaystyle n=1}$ ${\displaystyle n>1}$
• Poly cylinders are polynomial convex.
• Compact convex sets im are polynomial convex. ${\displaystyle \mathbb {C} ^{n}}$
• The union of two disjoint convex sets im is polynomial convex; this does not generally apply to three sets. ${\displaystyle \mathbb {C} ^{n}}$
• ${\displaystyle \{(z_{1},z_{2})\in \mathbb {C} ^{2};\,|z_{1}|=1=|z_{2}|\}}$ is not polynomial convex.

A sentence from Oka

Polynomial convex sets play an important role in the approximation of holomorphic functions by polynomials. The one-dimensional case is exactly Runge's approximation theorem .

The Oka theorem can be reproduced in the following versions:

• Let be a compact, polynomial convex set. Then every holomorphic function defined in a neighborhood of can be approximated uniformly by polynomials. ${\displaystyle K\subset \mathbb {C} ^{n}}$ ${\displaystyle K}$ ${\displaystyle K}$
• Let be a polynomial convex domain. Then every holomorphic function defined for it can be approximated compactly and uniformly by polynomials. ${\displaystyle D\subset \mathbb {C} ^{n}}$ ${\displaystyle D}$

literature

• Gunning - Rossi : Analytic functions of several complex variables . Prentice Hall 1965
• Lars Hörmander : An Introduction to Complex Analysis in Several Variables , North-Holland Mathematical Library 1973