LOCAL POLYNOMIAL CONVEXITY OF UNION OF TOTALLY REAL GRAPHS OF FUNCTIONS IN TWO VARIABLES IN ℂ3
Main Article Content
Abstract
In this note, we establish sufficient conditions under which the union of a totally real graph of a function in two variables and its tangent spaces is locally polynomially convex at the origin in ℂ3. Furthermore, several examples are provided to demonstrate that the conditions we impose are both necessary and reasonable.
Keywords
polynomially convex, totally real, Kallin's lemma
Article Details
References
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Nguyen Quang Dieu, Local hull of perturbations of a union of totally real planes meeting along a real line, Publ. Mat., 45 (2001), no. 2, 335-341.
Nguyen, Quang Dieu and P. De Paepe, Function algebras on disks, Complex Var. Theory Appl., 47 (2002), no. 5, 447-451.
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P.J. de Paepe, Approximation on a disk I, Math. Zeit., 212, 145-152 (1993).
P.J. de Paepe, Eva Kallin’s lemma on polynomial convexity, Bull. of London Math. Soc., 33 (2001), 1-10.
K.J. Preskenis Approximation on disks, Tran. Amer. Math. Soc., 171 (1972), 445-467.
E. L. Stout, Polynomial convexity, Progress in Mathematics, 261. Birkháuser Boston, 2007.
G. Stolzenberg, Polynomially and rationally convex sets, Acta Math., 109 (1963), 259-269.
M. T.Tan and K. P. Chi. Local polynomial convexity of the union of two tangential totally real graphs in ℂ3, Acta Math. Vietnam., 50 (2025), no. 3, 411-418.
B.M. Weinstock, On the polynomial convexity of the union of two maximal totally real subspaces of ℂ , Math. Ann., 282 (1988), no. 1, 131-138.
J. Wermer, Approximation on a disk, Math Ann.,155 (1964) 331-333.
J. Wiegerinck, Locally polynomially convex hulls at degenerated CR singularities of surfaces in ℂ2, Indiana Univ. Math. J., 44 (1995), 897-915.
E. M. Chirka, Approximation by holomorphic functions on smooth manifolds in ℂ , Mat. Sb. (N.S.), 78 (1969), no. 10, 101-123.
Kieu Phuong Chi, Local polynomial convexity of graphs of functions in several variables, J. Math. Anal. Appl., 433 (2016), no. 1, 473-486.
Nguyen Quang Dieu, Local polynomial convexity of tangential unions of totally real graphs in ℂ2, Indagationes Math., 10 (1999), no. 3, 349-355.
Nguyen Quang Dieu, Local hull of perturbations of a union of totally real planes meeting along a real line, Publ. Mat., 45 (2001), no. 2, 335-341.
Nguyen, Quang Dieu and P. De Paepe, Function algebras on disks, Complex Var. Theory Appl., 47 (2002), no. 5, 447-451.
S. Gorai, A note on polynomial convexity of the union of finitely many totally-real planes in ℂ2, J. Math. Anal. Appl., 418 (2014), no. 2, 842-851.
L. Hörmander, An Introduction to Complex Analysis in Several Variables, Third edition. North-Holland Mathematical Library, 7. North-Holland Publishing Co., Amsterdam, 1990. xii+254 pp.
E. Kallin, Fat polynomially convex sets, Function Algebras, (Proc. Inter. Symp. on Function Algebras, Tulane Univ, 1965, Scott Foresman, Chicago, 149-152 (1966).
A. G. O’Farrell, K.J. Preskenis and D. Walsh, Holomorphic approximation in Lipschitz norms, Contemp. Math., 32(1984), 187-194.
P.J. de Paepe, Approximation on a disk I, Math. Zeit., 212, 145-152 (1993).
P.J. de Paepe, Eva Kallin’s lemma on polynomial convexity, Bull. of London Math. Soc., 33 (2001), 1-10.
K.J. Preskenis Approximation on disks, Tran. Amer. Math. Soc., 171 (1972), 445-467.
E. L. Stout, Polynomial convexity, Progress in Mathematics, 261. Birkháuser Boston, 2007.
G. Stolzenberg, Polynomially and rationally convex sets, Acta Math., 109 (1963), 259-269.
M. T.Tan and K. P. Chi. Local polynomial convexity of the union of two tangential totally real graphs in ℂ3, Acta Math. Vietnam., 50 (2025), no. 3, 411-418.
B.M. Weinstock, On the polynomial convexity of the union of two maximal totally real subspaces of ℂ , Math. Ann., 282 (1988), no. 1, 131-138.
J. Wermer, Approximation on a disk, Math Ann.,155 (1964) 331-333.
J. Wiegerinck, Locally polynomially convex hulls at degenerated CR singularities of surfaces in ℂ2, Indiana Univ. Math. J., 44 (1995), 897-915.