The polynomial hull of unions of convex sets in $ℂ^n$

• Autorzy: Ulf Backlund , Anders Fällström
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1996
• Tom: 70
• Numer: 1
• Strony: 7-11

Daty

wydano

1996

otrzymano

1995-01-23

Twórcy

autor

Ulf Backlund

• Department of Mathematics, Umeå University, S-901 87 Umeå, Sweden

autor

Anders Fällström

• Department of Mathematics, Umeå University, S-901 87 Umeå, Sweden

Bibliografia

• [BePi] E. Bedford and S. Pinchuk, Domains in $ℂ^n+1$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), 165-192.
• [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.
• [Khud] G. Khudaĭberganov, On polynomial and rational convexity of unions of compacts in $ℂ^n$, 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.
• [Wer1] J. Wermer, Polynomial approximation on an arc in $ℂ^3$, Ann. of Math. 62 (1955), 269-270.
• [Wer2] J. Wermer, An example concerning polynomial convexity, Math. Ann. 139 (1959), 147-150.