Abstract
In this paper we discuss a couple of observations related to polynomial convexity. More precisely,
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(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 \(\mathbb {C}^n\).
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(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 \(\mathcal {P}(K)=\mathcal {C}(K)\).
This work is partially supported by an INSPIRE Faculty Fellowship (IFA-11MA-02) funded by DST and is also supported by a research grant under MATRICS scheme (MTR/2017/000974).
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References
de Paepe, P.J.: Eva Kallin’s lemma on polynomial convexity. Bull. Lond. Math. Soc. 33(1), 1–10 (2001)
Duval, J., Sibony, N.: polynomially convexity, rational convexity, and currents. Duke Math. J. 79(2), 487–513 (1995)
Gorai, S.: On polynomially convexity of compact subsets of totally-real submanifolds in \(\mathbb{C}^n\). J. Math. Anal. Appl. 448, 1305–1317 (2017)
Hörmander, L., Wermer, J.: Uniform approximation on compact sets in \(\mathbb{C}^n\). Math. Scand. 23, 5–21 (1968)
Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, vol. 7, 3rd edn. North-Holland Publishing Co., Amsterdam (1990)
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)
Kallin, E.: Polynomial convexity: the three spheres problem. In: Proceedings of Conference Complex Analysis (Minneapolis, 1964), pp. 301-304. Springer, Berlin (1965)
Khudaiberganov, G.: Polynomial and rational convexity of the union of compacta in \(\mathbb{C}^n\) Izv. Vyssh. Uchebn. Zaved., Mat. (2), 70–74 (1987)
Minsker, S.: Some applications of the Stone-Weierstrass theorem to planar rational approximation. Proc. Am. Math. Soc. 58, 94–96 (1976)
NemirovskiÄ, 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))
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)
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)
Stout, E.L.: Polynomial Convexity. Birkhäuser, Boston (2007)
Wermer, J.: Approximations on a disk. Math. Ann. 155, 331–333 (1964)
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.
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Gorai, S. (2020). Some Observations Concerning Polynomial Convexity. In: Roy, P., Cao, X., Li, XZ., Das, P., Deo, S. (eds) Mathematical Analysis and Applications in Modeling. ICMAAM 2018. Springer Proceedings in Mathematics & Statistics, vol 302. Springer, Singapore. https://doi.org/10.1007/978-981-15-0422-8_14
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