# The polynomial hull of unions of convex sets in ${\u2102}^{n}$

Ulf Backlund; Anders Fällström

Colloquium Mathematicae (1996)

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

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topBacklund, 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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- [Khud] G. Khudaĭberganov, On polynomial and rational convexity of unions of compacts in ${\u2102}^{n}$, Izv. Vuz. Mat. 2 (1987), 70-74 (in Russian).
- [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).
- [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
- [Wer1] J. Wermer, Polynomial approximation on an arc in ${\u2102}^{3}$, Ann. of Math. 62 (1955), 269-270.
- [Wer2] J. Wermer, An example concerning polynomial convexity, Math. Ann. 139 (1959), 147-150. Zbl0094.28302

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