TÍNH CHẤT LỒI ĐA THỨC ĐỊA PHƯƠNG CỦA HỢP CÁC ĐỒ THỊ HOÀN TOÀN THỰC CỦA CÁC HÀM HAI BIẾN SỐ TRONG ℂ3
Nội dung chính của bài viết
Tóm tắt
Trong bài báo này, chúng tôi thiết lập các điều kiện đủ để hợp của một đồ thị hoàn toàn thực của một hàm hai biến và các không gian tiếp xúc của nó là lồi đa thức địa phương tại gốc trong ℂ3. Hơn nữa, chúng tôi đưa ra một số ví dụ nhằm minh họa rằng các điều kiện được đặt ra là cần thiết và hợp lý
Từ khóa
tính lồi đa thức, hoàn toàn thực, bổ đề Kallin
Chi tiết bài viết
Tài liệu tham khảo
H. Alexander and J. Wermer, Several Complex Variables and Banach Algebras, Grad. Texts in Math., 35, Springer- Verlag, New York, 1998.
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.
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.