Báo cáo toán học: On the Functional Equation P(f)=Q(g) in Complex Numbers Field

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.
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Báo cáo toán học: " On the Functional Equation P(f)=Q(g) in Complex Numbers Field" Vietnam Journal of Mathematics 34:3 (2006) 317–329 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 ‹ 9$ 67 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, 2006Abstract. In this paper, we study the existence of non-constant meromorphic so-lutions f and g of the functional equation P (f ) = Q(g), where P (z ) and Q(z ) aregiven nonlinear polynomials with coefficients in the complex field C.2000 Mathematics Subject Classification: 32H20, 30D35.Keywords: Functional equation, unique range set, meromorphic function, algebraiccurves.1. IntroductionLet C be the complex number field. In [3], Li and Yang introduced the followingdefinition.Definition. A non-constant polynomial P (z ) defined over C is called a unique-ness 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 ) iscalled 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, impliesthat C = 1 and f ≡ g. Recently, there has been considerable progress in the study of uniquenesspolynomials, Boutabaa, Escassut and Hadadd [10] showed that a complex poly-∗ This work was partially supported by the National Basic Research Program of Vietnam318 Nguyen Trong Hoanomial P is a strong uniqueness polynomial for the family of complex polyno-mials if and only if no non-trivial affine transformation preserves its set of zeros.As for the case of complex meromorphic functions, some sufficient conditionswere found by Fujimoto in [8]. When P is injective on the roots of its derivativeP , necessary and sufficient conditions were given in [5]. Recently, Khoai andYang generalized the above studies by considering a pair of two nonlinear poly-nomials P (z ) and Q(z ) such that the only meromorphic solutions f, g satisfyingP (f ) = Q(g) are constants. By using the singularity theory and the calculationof the genus of algebraic curves based on Newton polygons as the main tools,they gave some sufficient conditions on the degrees of P and Q for the problem(see [1]). After that, by using value distribution theory, in [2], Yang-Li gavemore sufficient conditions related to this problem in general, and also gave somemore explicit conditions for the cases when the degrees of P and Q are 2, 3, 4. In this paper, we solve this functional equation by studying the hyperbolic-ity of the algebraic curve {P (x) − Q(y) = 0}. Using different from Khoai andYang’s method, we estimate the genus by giving sufficiently many linear inde-pendent regular 1-forms of Wronskian type on that curve. This method was firstintroduced in [4] by An-Wang-Wong.2. Main TheoremsDefinition. Let P (z ) be a nonlinear polynomial of degree n whose derivative isgiven by P (z ) = c(z − α1)n1 . . . (z − αk )nk ,where n1 + · · · + nk = n − 1 and α1 , . . . , αk are distinct zeros of P . The numberk is called the derivative index of P. The polynomial P (z ) is said to satisfy the condition separating the roots ofP (separation condition) if P (αi) = P (αj ) for all i = j, i, j = 1, 2. . . . , k. Here we only consider two nonlinear polynomials of degrees n and m, respec-tively P (x) = an xn + . . . + a1 x + a0 , Q(y) = bm ym + . . . + b1y + b0, (1)in C so that P (x) − Q(y) has no linear factors of the form ax + by + c. Assume that P (x) = nan(x − α1)n1 . . . (x − αk )nk , ...