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Trong bài báo này, chúng ta nghiên cứu sự tồn tại của các giải pháp liên tục không meromorphic f và g của P phương trình chức năng (f) = Q (g), P (z) và Q (z) là các đa thức phi tuyến với hệ số phức tạplĩnh vực C. | Vietnam Journal of Mathematics 34 3 2006 317-329 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 On the Functional Equation P f Q g in Complex Numbers Field Nguyen Trong Hoa Daklak Pedagogical College Buon Ma Thuot Province Vietnam Received November 9 2005 Revised March 23 2006 Abstract. In this paper we study the existence of non-constant meromorphic solutions f and g of the functional equation P f Q g where P z and Q z are given nonlinear polynomials with coefficients in the complex field C. 2000 Mathematics Subject Classification 32H20 30D35. Keywords Functional equation unique range set meromorphic function algebraic curves. 1. Introduction Let C be the complex number field. In 3 Li and Yang introduced the following definition. Definition. A non-constant polynomial P z defined over C is called a uniqueness polynomial for entire or meromorphic functions if the condition P f P g for entire or meromorphic functions f and g implies that f g. P z is called a strong uniqueness polynomial if the condition P f CP g for entire or meromorphic functions f and g and some non-zero constant C implies that C 1 and f g. Recently there has been considerable progress in the study of uniqueness polynomials Boutabaa Escassut and Hadadd 10 showed that a complex poly- This work was partially supported by the National Basic Research Program of Vietnam 318 Nguyen Trong Hoa nomial P is a strong uniqueness polynomial for the family of complex polynomials if and only if no non-trivial affine transformation preserves its set of zeros. As for the case of complex meromorphic functions some sufficient conditions were found by Fujimoto in 8 . When P is injective on the roots of its derivative P necessary and sufficient conditions were given in 5 . Recently Khoai and Yang generalized the above studies by considering a pair of two nonlinear polynomials P z and Q z such that the only meromorphic solutions f g satisfying P f Q g are constants. By using the singularity theory and the calculation
