Advertisement

Some Observations Concerning Polynomial Convexity

  • Sushil GoraiEmail author
Conference paper
  • 38 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 302)

Abstract

In this paper we discuss a couple of observations related to polynomial convexity. More precisely,
  1. (i)

    We observe that the union of finitely many disjoint closed balls with centres in \(\bigcup _{\theta \in [0,\pi /2]}e^{i\theta }V\) is polynomially convex, where V is a Lagrangian subspace of \(\mathbb {C}^n\).

     
  2. (ii)

    We show that any compact subset K of \(\{(z,w)\in \mathbb {C}^2:q(w)=\overline{p(z)}\}\), where p and q are two non-constant holomorphic polynomials in one variable, is polynomially convex and \(\mathcal {P}(K)=\mathcal {C}(K)\).

     

Keywords

Polynomial convexity Closed ball Totally real Lagrangians 

2010 Mathematics Subject Classification

Primary: 32E20 

Notes

Acknowledgements

I would like to thank the referee for valuable comments. I would also like to thank Professor Peter de Paepe for pointing out a mistake in the earlier version of Corollary  4.1.

References

  1. 1.
    de Paepe, P.J.: Eva Kallin’s lemma on polynomial convexity. Bull. Lond. Math. Soc. 33(1), 1–10 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Duval, J., Sibony, N.: polynomially convexity, rational convexity, and currents. Duke Math. J. 79(2), 487–513 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gorai, S.: On polynomially convexity of compact subsets of totally-real submanifolds in \(\mathbb{C}^n\). J. Math. Anal. Appl. 448, 1305–1317 (2017)Google Scholar
  4. 4.
    Hörmander, L., Wermer, J.: Uniform approximation on compact sets in \(\mathbb{C}^n\). Math. Scand. 23, 5–21 (1968)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, vol. 7, 3rd edn. North-Holland Publishing Co., Amsterdam (1990)Google Scholar
  6. 6.
    Kallin, E.: Fat polynomially convex sets, function algebras. In: Proceedings of International Symposium on Function Algebras, pp. 149–152. Tulane University, 1965, Scott Foresman, Chicago, IL (1966)Google Scholar
  7. 7.
    Kallin, E.: Polynomial convexity: the three spheres problem. In: Proceedings of Conference Complex Analysis (Minneapolis, 1964), pp. 301-304. Springer, Berlin (1965)CrossRefGoogle Scholar
  8. 8.
    Khudaiberganov, G.: Polynomial and rational convexity of the union of compacta in \(\mathbb{C}^n\) Izv. Vyssh. Uchebn. Zaved., Mat. (2), 70–74 (1987)Google Scholar
  9. 9.
    Minsker, S.: Some applications of the Stone-Weierstrass theorem to planar rational approximation. Proc. Am. Math. Soc. 58, 94–96 (1976)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nemirovskiĭ, S.Y.: Finite unions of balls in \(\mathbb{C}^n\) are rationally convex, Uspekhi Mat. Nauk 63(2)(380), 157–158 (2008) (translation in Russian Math. Surv. 63(2), 381–382 (2008))Google Scholar
  11. 11.
    O’Farrell, A.G., Preskenis, K.J., Walsh, D.: Holomorphic approximation in Lipschitz norms. In: Proceedings of the Conference on Banach Algebras and Several Complex Variables (New Haven, Conn., 1983), pp. 187–194. Contemp. Math., 32, Amer. Math. Soc., Providence, RI (1984)Google Scholar
  12. 12.
    Smirnov, M.M., Chirka, E.M.: Polynomial convexity of some sets in \(\mathbb{C}^n\). Mat. Zametki 50(5), 81–89 (1991); translation in Math. Notes 50(5–6), 1151–1157 (1991)Google Scholar
  13. 13.
    Stout, E.L.: Polynomial Convexity. Birkhäuser, Boston (2007)zbMATHGoogle Scholar
  14. 14.
    Wermer, J.: Approximations on a disk. Math. Ann. 155, 331–333 (1964)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIndian Institute of Science Education and Research KolkataNadiaIndia

Personalised recommendations