tailieunhanh - Second main theorem for holomorphic mappings from the discs into the projective spaces

In this paper, we establish a second main theorem for holomorphic mappings from a disc (R) into Pn(C) and families of hyperplanes in subgeneral position. Our result is an extension the classical second main theorem of Cartan-Nochka and the second main theorem of Fujimoto. | JOURNAL OF SCIENCE OF HNUE DOI Mathematical and Physical Sci. 2015 Vol. 60 No. 7 pp. 21-29 This paper is available online at http SECOND MAIN THEOREM FOR HOLOMORPHIC MAPPINGS FROM THE DISCS INTO THE PROJECTIVE SPACES Nguyen Van An1 and Nguyen Thi Nhung2 1 Division of Mathematics Banking Academy 2 Department of Mathematics and Informatics Thang Long University Abstract. In this paper we establish a second main theorem for holomorphic mappings from a disc R into Pn C and families of hyperplanes in subgeneral position. Our result is an extension the classical second main theorem of Cartan-Nochka and the second main theorem of Fujimoto. Keywords Second main theorem holomorphic mapping subgeneral position. 1. Introduction Let f be a holomorphic mapping from C into Pn C with a reduced presentation f f0 fn in a fixed homogeneous coordinate system ω0 ωn of Pn C . Let H be a hyperplane in Pn C defined by H ω0 ωn a0 ω0 an ωn 0 where ai C 0 i n. If there is no confusion we will also denote by H the linear form H ω0 . . . ωn a0 ω0 an ωn . We set ω ω0 2 ωn 2 1 2 f f0 2 fn 2 1 2 H a0 2 an 2 1 2 and H f a0 f0 an fn . Then the function H f depends on the choice of the reduce representation of f and representation of H. However its divisor of the zeros νH f does not depend on these choices. Here we may consider νH f as a function whose value at a point z0 is the multiple intersection of the image of f and H at the point f z0 . The classical second main theorem for holomorphic mappings into projective spaces of H. Cartan which is the most important second main theorem in Nevanlinna theory is stated as follows Theorem A see 4 . Let f be linearly nondegenerate holomorphic mapping of C into Pn C and Hj qj 1 be hyperplanes of Pn C in general position where q n 2. Then q X q n 1 Tf r Nn r νHj f o Tf r . j 1 Received November 16 2015. Accepted December 10 2015. Contact Nguyen Thi Nhung e-mail address hoangnhung227@ 21 Nguyen Van An .