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Vietnam Journal of Mathematics 34:3(2006) 317-329  On the Functional Equation $P(f)=Q(g)$ in Complex Numbers Field Nguyen Trong Hoa 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 $\mathbb C$.   2000 Mathematics Subject Classification: 32H20, 30D35. Keywords: Functional equation, unique rang set, meromorphic function, algebraic curves.

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