On the functional equation \(P(f)=Q(g)\) in complex numbers field (Q879682)

The authors 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 \(\mathbb C\). To this end, the author studies the hyperbolicity of the algebraic curve \( \{ P(x)-Q(y)=0\}\) and estimates its genus following a method of \textit{T. T. H. An, J. T.-Y. Wang}, and \textit{P.-M. Wong} [Acta Arith. 109, No.~3, 259--280 (2003; Zbl 1020.30049)] and extending some other related results on this functional equations.