The polynomial hull of unions of convex sets in ℂ n

Ulf Backlund; Anders Fällström

Colloquium Mathematicae (1996)

  • Volume: 70, Issue: 1, page 7-11
  • ISSN: 0010-1354

Abstract

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We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form ( z 1 , z 2 , z 3 ) ∈ ℂ 3 : | z 1 | 2 + | z 2 | 2 + | z 3 | 2 m ≤ 1 , such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.

How to cite

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Backlund, Ulf, and Fällström, Anders. "The polynomial hull of unions of convex sets in $ℂ^n$." Colloquium Mathematicae 70.1 (1996): 7-11. <http://eudml.org/doc/210399>.

@article{Backlund1996,
abstract = {We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form $\{(z_1,z_2,z_3) ∈ ℂ^3: |z_1|^2 + |z_2|^2 + |z_3|^\{2m\} ≤ 1\}$, such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.},
author = {Backlund, Ulf, Fällström, Anders},
journal = {Colloquium Mathematicae},
keywords = {polynomial convexity},
language = {eng},
number = {1},
pages = {7-11},
title = {The polynomial hull of unions of convex sets in $ℂ^n$},
url = {http://eudml.org/doc/210399},
volume = {70},
year = {1996},
}

TY - JOUR
AU - Backlund, Ulf
AU - Fällström, Anders
TI - The polynomial hull of unions of convex sets in $ℂ^n$
JO - Colloquium Mathematicae
PY - 1996
VL - 70
IS - 1
SP - 7
EP - 11
AB - We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form ${(z_1,z_2,z_3) ∈ ℂ^3: |z_1|^2 + |z_2|^2 + |z_3|^{2m} ≤ 1}$, such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.
LA - eng
KW - polynomial convexity
UR - http://eudml.org/doc/210399
ER -

References

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  1. [BePi] E. Bedford and S. Pinchuk, Domains in ℂ n + 1 with noncompact automorphism group, J. Geom. Anal. 1 (1991), 165-192. 
  2. [Kal] E. Kallin, Polynomial convexity: The three spheres problem, in: Proceedings of the Conference on Complex Analysis, Minneapolis 1964, H. Röhrl, A. Aeppli and E. Calabi (eds.), Springer, 1965, 301-304. 
  3. [Khud] G. Khudaĭberganov, On polynomial and rational convexity of unions of compacts in ℂ n , Izv. Vuz. Mat. 2 (1987), 70-74 (in Russian). 
  4. [KyKh] A. M. Kytmanov and G. Khudaĭberganov, An example of a nonpolynomially convex compact set consisting of three non-intersecting ellipsoids, Sibirsk. Mat. Zh. 25 (5) (1984), 196-198 (in Russian). 
  5. [Ros] J.-P. Rosay, The polynomial hull of non-connected tube domains, and an example of E. Kallin, Bull. London Math. Soc. 21 (1989), 73-78. Zbl0634.32011
  6. [Wer1] J. Wermer, Polynomial approximation on an arc in ℂ 3 , Ann. of Math. 62 (1955), 269-270. 
  7. [Wer2] J. Wermer, An example concerning polynomial convexity, Math. Ann. 139 (1959), 147-150. Zbl0094.28302

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